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3.5.2 DG for the acoustic wave equation

We consider the acoustic problem \begin{align} \partial_{t} \mathbf{u} - c \nabla p &= 0 \quad \text{ in } \Omega \times I, \\ \partial_{t} p - c \operatorname{div}(\mathbf{u}) &= 0 \quad \text{ in } \Omega \times I, \\ % p (0,\cdot) &= p(1,\cdot), \quad % p (\cdot,0) = p(\cdot,1), \label{eq:per2b}\\ % \mathbf{q} (0,\cdot) &= \mathbf{q}(1,\cdot), \quad % \mathbf{q} (\cdot,0) = \mathbf{q}(\cdot,1), \label{eq:per2bb}\\ p &= p_0 \!\!\! \quad \text{ on } \Omega \times \{0\},\\ u &= 0 \quad \text{ on } \Omega \times \{0\}. \end{align}

Here \(p\) is the acoustic pressure (the local deviation from the ambient pressure) and \(\mathbf{u}\) is the local velocity.

+ suitable boundary conditions

A simple grid:

#imports, geometry and mesh:
from netgen.occ import *
from netgen.webgui import Draw as DrawGeo
geo = OCCGeometry(unit_square_shape.Scale((0,0,0),2).Move((-1,-1,0)), dim=2)
from ngsolve import *
from ngsolve.webgui import Draw
mesh = Mesh(geo.GenerateMesh(maxh=0.1))
Draw(mesh)

BaseWebGuiScene

Find \(p: [0,T] \to \bigoplus_{T\in\mathcal{T}_h} \mathcal{P}^{k+1}(T)\) and \(\mathbf{u}: [0,T] \to \bigoplus_{T\in\mathcal{T}_h} [\mathcal{P}^k(T)]^d\) so that

\begin{align*} (\partial_t \mathbf{u}, v) &= b_h(p,v) & &&& \forall v,\\ (\partial_t p, q) &= & - b_h(q,\mathbf{u}) &&& \forall q\\ \end{align*}

with the centered flux approximation:

\[b_h(p,v) = \sum_{T} \int_T \nabla p \cdot v + \int_{\partial T} ( \hat{p} - p) v_n\]

Here \(\hat{p}\) is the centered approximation i.e. \(\hat{p} = \{\!\!\{p\}\!\!\}\).

order = 6
fes_p = L2(mesh, order=order+1, all_dofs_together=True)
fes_u = VectorL2(mesh, order=order, piola=True, all_dofs_together=True)
fes = fes_p*fes_u
gfu = GridFunction(fes)

What is the flag piola doing?

The VectorL2 space uses the following definition of basis functions on the mesh.

Let \(\hat{\varphi}(\hat{x})\) be a (vectorial) basis function on the reference element \(\hat{T}\), \(\Phi: \hat{T} \to T\) with \(x = \Phi(\hat{x})\) and \(F= D\Phi\), then

\[\begin{split}\varphi(x) := \left\{ \begin{array}{rlll} \hat{\varphi}(\hat{x}), & \texttt{piola=False}, & \texttt{covariant=False}, & \text{(default)} \\ \frac{1}{|\operatorname{det} F|} \cdot F \cdot \hat{\varphi}(\hat{x}), & \texttt{piola=True}, & \texttt{covariant=False}, \\ F^{-T} \cdot \hat{\varphi}(\hat{x}), & \texttt{piola=False}, & \texttt{covariant=True}. \\ \end{array} \right.\end{split}\]

Inverse mass matrix operations:

Combining Embedding and the inverse mass matrix operation for fes_p allows to realize the following block inverse operations acting on fes:

\[\begin{split}\underbrace{\left( \begin{array}{cc} M_{p}^{-1} & 0 \\ 0 & 0 \end{array} \right)}_{\texttt{invp}} = \underbrace{\left( \begin{array}{c} I \\ 0 \end{array} \right)}_{\texttt{emb_p}} \cdot \underbrace{M_{p}^{-1}}_{\texttt{invmassp}} \cdot \underbrace{\left( \begin{array}{cc} I & 0 \end{array} \right)}_{\texttt{emb_p.T}}\end{split}\]

pdofs = fes.Range(0);
emb_p = Embedding(fes.ndof, pdofs)
invmassp = fes_p.Mass(1).Inverse()
invp = emb_p @ invmassp @ emb_p.T

Analogously, combining Embedding and the inverse mass matrix operation for fes_u allows to realize the following block inverse operations on fes:

\[\begin{split}\underbrace{\left( \begin{array}{cc} 0 & 0 \\ 0 & M_{\mathbf{u}}^{-1} \end{array} \right)}_{\texttt{invu}} = \underbrace{\left( \begin{array}{c} I \\ 0 \end{array} \right)}_{\texttt{emb_u}} \cdot \underbrace{M_{\mathbf{u}}^{-1}}_{\texttt{invmassu}} \cdot \underbrace{\left( \begin{array}{cc} I & 0 \end{array} \right)}_{\texttt{emb_u.T}}\end{split}\]

udofs = fes.Range(1)
emb_u = Embedding(fes.ndof, udofs)
invmassu = fes_u.Mass(Id(mesh.dim)).Inverse()
invu = emb_u @ invmassu @ emb_u.T

Time loop

Assuming the operators B,BT: fes_p\(\times\) fes_u \(\to\)(fes_p\(\times\) fes_u )\('\) corresponding to \(B_h((u,p),(v,q)=b_h(p,v)\) are given, the symplectic Euler time stepping method takes the form:

def Run(B, BT, t0 = 0, tend = 0.25, tau = 1e-3,
        backward = False, scenes = []):
    t = 0
    with TaskManager():
        while t < (tend-t0) - tau/2:
            t += tau
            if not backward:
                gfu.vec.data += -tau * invp @ BT * gfu.vec
                gfu.vec.data += tau * invu @ B * gfu.vec
                print("\r t = {:}".format(t0 + t),end="")
            else:
                gfu.vec.data += -tau * invu @ B * gfu.vec
                gfu.vec.data += tau * invp @ BT * gfu.vec
                print("\r t = {:}".format(tend - t),end="")
            for sc in scenes: sc.Redraw() # blocking=False)
        print("")

Initial values (density ring):

gfu.components[0].Set (exp(-50*(x**2+y**2))-exp(-100*(x**2+y**2)))
gfu.components[1].vec[:] = 0
Draw(gfu.components[0], mesh, "p", min=-0.02, max=0.08, autoscale=False, order=3)

BaseWebGuiScene

Version 1:

The bilinear form for application on the full space fes

n = specialcf.normal(mesh.dim)
(p,u),(q,v) = fes.TnT()
B = BilinearForm(fes, nonassemble=True)
B += grad(p)*v * dx + 0.5*(p.Other()-p)*(v*n) * dx(element_boundary=True)
BT = BilinearForm(fes, nonassemble=True)
BT += grad(q)*u * dx + 0.5*(q.Other()-q)*(u*n) * dx(element_boundary=True)

scenep=Draw(gfu.components[0], mesh, "p", min=-0.02, max=0.08,
            autoscale=False, order=3, deformation=True)
%time Run(B.mat, BT.mat, backward=False, scenes=[scenep])
%time Run(B.mat, BT.mat, backward=True, scenes=[scenep])

more-to-come:

t = 0.001 t = 0.002 t = 0.003 t = 0.004 t = 0.005 t = 0.006 t = 0.007 t = 0.008 t = 0.009000000000000001 t = 0.010000000000000002 t = 0.011000000000000003 t = 0.012000000000000004 t = 0.013000000000000005 t = 0.014000000000000005 t = 0.015000000000000006 t = 0.016000000000000007

</pre>

t = 0.001 t = 0.002 t = 0.003 t = 0.004 t = 0.005 t = 0.006 t = 0.007 t = 0.008 t = 0.009000000000000001 t = 0.010000000000000002 t = 0.011000000000000003 t = 0.012000000000000004 t = 0.013000000000000005 t = 0.014000000000000005 t = 0.015000000000000006 t = 0.016000000000000007

end{sphinxVerbatim}

t = 0.001 t = 0.002 t = 0.003 t = 0.004 t = 0.005 t = 0.006 t = 0.007 t = 0.008 t = 0.009000000000000001 t = 0.010000000000000002 t = 0.011000000000000003 t = 0.012000000000000004 t = 0.013000000000000005 t = 0.014000000000000005 t = 0.015000000000000006 t = 0.016000000000000007

more-to-come:

t = 0.017000000000000008 t = 0.01800000000000001 t = 0.01900000000000001 t = 0.02000000000000001 t = 0.02100000000000001 t = 0.022000000000000013 t = 0.023000000000000013 t = 0.024000000000000014 t = 0.025000000000000015 t = 0.026000000000000016 t = 0.027000000000000017 t = 0.028000000000000018 t = 0.02900000000000002 t = 0.03000000000000002 t = 0.03100000000000002 t = 0.03200000000000002

</pre>

t = 0.017000000000000008 t = 0.01800000000000001 t = 0.01900000000000001 t = 0.02000000000000001 t = 0.02100000000000001 t = 0.022000000000000013 t = 0.023000000000000013 t = 0.024000000000000014 t = 0.025000000000000015 t = 0.026000000000000016 t = 0.027000000000000017 t = 0.028000000000000018 t = 0.02900000000000002 t = 0.03000000000000002 t = 0.03100000000000002 t = 0.03200000000000002

end{sphinxVerbatim}

t = 0.017000000000000008 t = 0.01800000000000001 t = 0.01900000000000001 t = 0.02000000000000001 t = 0.02100000000000001 t = 0.022000000000000013 t = 0.023000000000000013 t = 0.024000000000000014 t = 0.025000000000000015 t = 0.026000000000000016 t = 0.027000000000000017 t = 0.028000000000000018 t = 0.02900000000000002 t = 0.03000000000000002 t = 0.03100000000000002 t = 0.03200000000000002

more-to-come:

t = 0.03300000000000002 t = 0.03400000000000002 t = 0.035000000000000024 t = 0.036000000000000025 t = 0.037000000000000026 t = 0.03800000000000003 t = 0.03900000000000003 t = 0.04000000000000003 t = 0.04100000000000003 t = 0.04200000000000003 t = 0.04300000000000003 t = 0.04400000000000003 t = 0.04500000000000003 t = 0.046000000000000034 t = 0.047000000000000035 t = 0.048000000000000036 t = 0.04900000000000004 t = 0.05000000000000004 t = 0.05100000000000004

</pre>

t = 0.03300000000000002 t = 0.03400000000000002 t = 0.035000000000000024 t = 0.036000000000000025 t = 0.037000000000000026 t = 0.03800000000000003 t = 0.03900000000000003 t = 0.04000000000000003 t = 0.04100000000000003 t = 0.04200000000000003 t = 0.04300000000000003 t = 0.04400000000000003 t = 0.04500000000000003 t = 0.046000000000000034 t = 0.047000000000000035 t = 0.048000000000000036 t = 0.04900000000000004 t = 0.05000000000000004 t = 0.05100000000000004

end{sphinxVerbatim}

t = 0.03300000000000002 t = 0.03400000000000002 t = 0.035000000000000024 t = 0.036000000000000025 t = 0.037000000000000026 t = 0.03800000000000003 t = 0.03900000000000003 t = 0.04000000000000003 t = 0.04100000000000003 t = 0.04200000000000003 t = 0.04300000000000003 t = 0.04400000000000003 t = 0.04500000000000003 t = 0.046000000000000034 t = 0.047000000000000035 t = 0.048000000000000036 t = 0.04900000000000004 t = 0.05000000000000004 t = 0.05100000000000004

more-to-come:

t = 0.05200000000000004 t = 0.05300000000000004 t = 0.05400000000000004 t = 0.05500000000000004 t = 0.05600000000000004 t = 0.057000000000000044 t = 0.058000000000000045 t = 0.059000000000000045 t = 0.060000000000000046 t = 0.06100000000000005 t = 0.06200000000000005 t = 0.06300000000000004 t = 0.06400000000000004 t = 0.06500000000000004 t = 0.06600000000000004 t = 0.06700000000000005

</pre>

t = 0.05200000000000004 t = 0.05300000000000004 t = 0.05400000000000004 t = 0.05500000000000004 t = 0.05600000000000004 t = 0.057000000000000044 t = 0.058000000000000045 t = 0.059000000000000045 t = 0.060000000000000046 t = 0.06100000000000005 t = 0.06200000000000005 t = 0.06300000000000004 t = 0.06400000000000004 t = 0.06500000000000004 t = 0.06600000000000004 t = 0.06700000000000005

end{sphinxVerbatim}

t = 0.05200000000000004 t = 0.05300000000000004 t = 0.05400000000000004 t = 0.05500000000000004 t = 0.05600000000000004 t = 0.057000000000000044 t = 0.058000000000000045 t = 0.059000000000000045 t = 0.060000000000000046 t = 0.06100000000000005 t = 0.06200000000000005 t = 0.06300000000000004 t = 0.06400000000000004 t = 0.06500000000000004 t = 0.06600000000000004 t = 0.06700000000000005

more-to-come:

t = 0.06800000000000005 t = 0.06900000000000005 t = 0.07000000000000005 t = 0.07100000000000005 t = 0.07200000000000005 t = 0.07300000000000005 t = 0.07400000000000005 t = 0.07500000000000005 t = 0.07600000000000005 t = 0.07700000000000005 t = 0.07800000000000006 t = 0.07900000000000006 t = 0.08000000000000006 t = 0.08100000000000006 t = 0.08200000000000006 t = 0.08300000000000006

</pre>

t = 0.06800000000000005 t = 0.06900000000000005 t = 0.07000000000000005 t = 0.07100000000000005 t = 0.07200000000000005 t = 0.07300000000000005 t = 0.07400000000000005 t = 0.07500000000000005 t = 0.07600000000000005 t = 0.07700000000000005 t = 0.07800000000000006 t = 0.07900000000000006 t = 0.08000000000000006 t = 0.08100000000000006 t = 0.08200000000000006 t = 0.08300000000000006

end{sphinxVerbatim}

t = 0.06800000000000005 t = 0.06900000000000005 t = 0.07000000000000005 t = 0.07100000000000005 t = 0.07200000000000005 t = 0.07300000000000005 t = 0.07400000000000005 t = 0.07500000000000005 t = 0.07600000000000005 t = 0.07700000000000005 t = 0.07800000000000006 t = 0.07900000000000006 t = 0.08000000000000006 t = 0.08100000000000006 t = 0.08200000000000006 t = 0.08300000000000006

more-to-come:

t = 0.08400000000000006 t = 0.08500000000000006 t = 0.08600000000000006 t = 0.08700000000000006 t = 0.08800000000000006 t = 0.08900000000000007 t = 0.09000000000000007 t = 0.09100000000000007 t = 0.09200000000000007 t = 0.09300000000000007 t = 0.09400000000000007 t = 0.09500000000000007 t = 0.09600000000000007 t = 0.09700000000000007 t = 0.09800000000000007 t = 0.09900000000000007

</pre>

t = 0.08400000000000006 t = 0.08500000000000006 t = 0.08600000000000006 t = 0.08700000000000006 t = 0.08800000000000006 t = 0.08900000000000007 t = 0.09000000000000007 t = 0.09100000000000007 t = 0.09200000000000007 t = 0.09300000000000007 t = 0.09400000000000007 t = 0.09500000000000007 t = 0.09600000000000007 t = 0.09700000000000007 t = 0.09800000000000007 t = 0.09900000000000007

end{sphinxVerbatim}

t = 0.08400000000000006 t = 0.08500000000000006 t = 0.08600000000000006 t = 0.08700000000000006 t = 0.08800000000000006 t = 0.08900000000000007 t = 0.09000000000000007 t = 0.09100000000000007 t = 0.09200000000000007 t = 0.09300000000000007 t = 0.09400000000000007 t = 0.09500000000000007 t = 0.09600000000000007 t = 0.09700000000000007 t = 0.09800000000000007 t = 0.09900000000000007

more-to-come:

t = 0.10000000000000007 t = 0.10100000000000008 t = 0.10200000000000008 t = 0.10300000000000008 t = 0.10400000000000008 t = 0.10500000000000008 t = 0.10600000000000008 t = 0.10700000000000008 t = 0.10800000000000008 t = 0.10900000000000008 t = 0.11000000000000008 t = 0.11100000000000008 t = 0.11200000000000009 t = 0.11300000000000009 t = 0.11400000000000009 t = 0.11500000000000009 t = 0.11600000000000009

</pre>

t = 0.10000000000000007 t = 0.10100000000000008 t = 0.10200000000000008 t = 0.10300000000000008 t = 0.10400000000000008 t = 0.10500000000000008 t = 0.10600000000000008 t = 0.10700000000000008 t = 0.10800000000000008 t = 0.10900000000000008 t = 0.11000000000000008 t = 0.11100000000000008 t = 0.11200000000000009 t = 0.11300000000000009 t = 0.11400000000000009 t = 0.11500000000000009 t = 0.11600000000000009

end{sphinxVerbatim}

t = 0.10000000000000007 t = 0.10100000000000008 t = 0.10200000000000008 t = 0.10300000000000008 t = 0.10400000000000008 t = 0.10500000000000008 t = 0.10600000000000008 t = 0.10700000000000008 t = 0.10800000000000008 t = 0.10900000000000008 t = 0.11000000000000008 t = 0.11100000000000008 t = 0.11200000000000009 t = 0.11300000000000009 t = 0.11400000000000009 t = 0.11500000000000009 t = 0.11600000000000009

more-to-come:

t = 0.11700000000000009 t = 0.11800000000000009 t = 0.11900000000000009 t = 0.12000000000000009 t = 0.1210000000000001 t = 0.1220000000000001 t = 0.1230000000000001 t = 0.1240000000000001 t = 0.12500000000000008 t = 0.12600000000000008 t = 0.12700000000000009 t = 0.12800000000000009 t = 0.1290000000000001 t = 0.1300000000000001 t = 0.1310000000000001 t = 0.1320000000000001

</pre>

t = 0.11700000000000009 t = 0.11800000000000009 t = 0.11900000000000009 t = 0.12000000000000009 t = 0.1210000000000001 t = 0.1220000000000001 t = 0.1230000000000001 t = 0.1240000000000001 t = 0.12500000000000008 t = 0.12600000000000008 t = 0.12700000000000009 t = 0.12800000000000009 t = 0.1290000000000001 t = 0.1300000000000001 t = 0.1310000000000001 t = 0.1320000000000001

end{sphinxVerbatim}

t = 0.11700000000000009 t = 0.11800000000000009 t = 0.11900000000000009 t = 0.12000000000000009 t = 0.1210000000000001 t = 0.1220000000000001 t = 0.1230000000000001 t = 0.1240000000000001 t = 0.12500000000000008 t = 0.12600000000000008 t = 0.12700000000000009 t = 0.12800000000000009 t = 0.1290000000000001 t = 0.1300000000000001 t = 0.1310000000000001 t = 0.1320000000000001

more-to-come:

t = 0.1330000000000001 t = 0.1340000000000001 t = 0.1350000000000001 t = 0.1360000000000001 t = 0.1370000000000001 t = 0.1380000000000001 t = 0.1390000000000001 t = 0.1400000000000001 t = 0.1410000000000001 t = 0.1420000000000001 t = 0.1430000000000001 t = 0.1440000000000001 t = 0.1450000000000001 t = 0.1460000000000001 t = 0.1470000000000001 t = 0.1480000000000001 t = 0.1490000000000001 t = 0.1500000000000001 t = 0.1510000000000001

</pre>

t = 0.1330000000000001 t = 0.1340000000000001 t = 0.1350000000000001 t = 0.1360000000000001 t = 0.1370000000000001 t = 0.1380000000000001 t = 0.1390000000000001 t = 0.1400000000000001 t = 0.1410000000000001 t = 0.1420000000000001 t = 0.1430000000000001 t = 0.1440000000000001 t = 0.1450000000000001 t = 0.1460000000000001 t = 0.1470000000000001 t = 0.1480000000000001 t = 0.1490000000000001 t = 0.1500000000000001 t = 0.1510000000000001

end{sphinxVerbatim}

t = 0.1330000000000001 t = 0.1340000000000001 t = 0.1350000000000001 t = 0.1360000000000001 t = 0.1370000000000001 t = 0.1380000000000001 t = 0.1390000000000001 t = 0.1400000000000001 t = 0.1410000000000001 t = 0.1420000000000001 t = 0.1430000000000001 t = 0.1440000000000001 t = 0.1450000000000001 t = 0.1460000000000001 t = 0.1470000000000001 t = 0.1480000000000001 t = 0.1490000000000001 t = 0.1500000000000001 t = 0.1510000000000001

more-to-come:

t = 0.1520000000000001 t = 0.1530000000000001 t = 0.1540000000000001 t = 0.1550000000000001 t = 0.1560000000000001 t = 0.1570000000000001 t = 0.1580000000000001 t = 0.1590000000000001 t = 0.16000000000000011 t = 0.16100000000000012 t = 0.16200000000000012 t = 0.16300000000000012 t = 0.16400000000000012 t = 0.16500000000000012 t = 0.16600000000000012 t = 0.16700000000000012 t = 0.16800000000000012 t = 0.16900000000000012 t = 0.17000000000000012

</pre>

t = 0.1520000000000001 t = 0.1530000000000001 t = 0.1540000000000001 t = 0.1550000000000001 t = 0.1560000000000001 t = 0.1570000000000001 t = 0.1580000000000001 t = 0.1590000000000001 t = 0.16000000000000011 t = 0.16100000000000012 t = 0.16200000000000012 t = 0.16300000000000012 t = 0.16400000000000012 t = 0.16500000000000012 t = 0.16600000000000012 t = 0.16700000000000012 t = 0.16800000000000012 t = 0.16900000000000012 t = 0.17000000000000012

end{sphinxVerbatim}

t = 0.1520000000000001 t = 0.1530000000000001 t = 0.1540000000000001 t = 0.1550000000000001 t = 0.1560000000000001 t = 0.1570000000000001 t = 0.1580000000000001 t = 0.1590000000000001 t = 0.16000000000000011 t = 0.16100000000000012 t = 0.16200000000000012 t = 0.16300000000000012 t = 0.16400000000000012 t = 0.16500000000000012 t = 0.16600000000000012 t = 0.16700000000000012 t = 0.16800000000000012 t = 0.16900000000000012 t = 0.17000000000000012

more-to-come:

t = 0.17100000000000012 t = 0.17200000000000013 t = 0.17300000000000013 t = 0.17400000000000013 t = 0.17500000000000013 t = 0.17600000000000013 t = 0.17700000000000013 t = 0.17800000000000013 t = 0.17900000000000013 t = 0.18000000000000013 t = 0.18100000000000013 t = 0.18200000000000013 t = 0.18300000000000013 t = 0.18400000000000014 t = 0.18500000000000014 t = 0.18600000000000014 t = 0.18700000000000014 t = 0.18800000000000014 t = 0.18900000000000014

</pre>

t = 0.17100000000000012 t = 0.17200000000000013 t = 0.17300000000000013 t = 0.17400000000000013 t = 0.17500000000000013 t = 0.17600000000000013 t = 0.17700000000000013 t = 0.17800000000000013 t = 0.17900000000000013 t = 0.18000000000000013 t = 0.18100000000000013 t = 0.18200000000000013 t = 0.18300000000000013 t = 0.18400000000000014 t = 0.18500000000000014 t = 0.18600000000000014 t = 0.18700000000000014 t = 0.18800000000000014 t = 0.18900000000000014

end{sphinxVerbatim}

t = 0.17100000000000012 t = 0.17200000000000013 t = 0.17300000000000013 t = 0.17400000000000013 t = 0.17500000000000013 t = 0.17600000000000013 t = 0.17700000000000013 t = 0.17800000000000013 t = 0.17900000000000013 t = 0.18000000000000013 t = 0.18100000000000013 t = 0.18200000000000013 t = 0.18300000000000013 t = 0.18400000000000014 t = 0.18500000000000014 t = 0.18600000000000014 t = 0.18700000000000014 t = 0.18800000000000014 t = 0.18900000000000014

more-to-come:

t = 0.19000000000000014 t = 0.19100000000000014 t = 0.19200000000000014 t = 0.19300000000000014 t = 0.19400000000000014 t = 0.19500000000000015 t = 0.19600000000000015 t = 0.19700000000000015 t = 0.19800000000000015 t = 0.19900000000000015 t = 0.20000000000000015 t = 0.20100000000000015 t = 0.20200000000000015 t = 0.20300000000000015 t = 0.20400000000000015 t = 0.20500000000000015 t = 0.20600000000000016 t = 0.20700000000000016 t = 0.20800000000000016

</pre>

t = 0.19000000000000014 t = 0.19100000000000014 t = 0.19200000000000014 t = 0.19300000000000014 t = 0.19400000000000014 t = 0.19500000000000015 t = 0.19600000000000015 t = 0.19700000000000015 t = 0.19800000000000015 t = 0.19900000000000015 t = 0.20000000000000015 t = 0.20100000000000015 t = 0.20200000000000015 t = 0.20300000000000015 t = 0.20400000000000015 t = 0.20500000000000015 t = 0.20600000000000016 t = 0.20700000000000016 t = 0.20800000000000016

end{sphinxVerbatim}

t = 0.19000000000000014 t = 0.19100000000000014 t = 0.19200000000000014 t = 0.19300000000000014 t = 0.19400000000000014 t = 0.19500000000000015 t = 0.19600000000000015 t = 0.19700000000000015 t = 0.19800000000000015 t = 0.19900000000000015 t = 0.20000000000000015 t = 0.20100000000000015 t = 0.20200000000000015 t = 0.20300000000000015 t = 0.20400000000000015 t = 0.20500000000000015 t = 0.20600000000000016 t = 0.20700000000000016 t = 0.20800000000000016

more-to-come:

t = 0.20900000000000016 t = 0.21000000000000016 t = 0.21100000000000016 t = 0.21200000000000016 t = 0.21300000000000016 t = 0.21400000000000016 t = 0.21500000000000016 t = 0.21600000000000016 t = 0.21700000000000016 t = 0.21800000000000017 t = 0.21900000000000017 t = 0.22000000000000017 t = 0.22100000000000017 t = 0.22200000000000017 t = 0.22300000000000017 t = 0.22400000000000017 t = 0.22500000000000017 t = 0.22600000000000017 t = 0.22700000000000017

</pre>

t = 0.20900000000000016 t = 0.21000000000000016 t = 0.21100000000000016 t = 0.21200000000000016 t = 0.21300000000000016 t = 0.21400000000000016 t = 0.21500000000000016 t = 0.21600000000000016 t = 0.21700000000000016 t = 0.21800000000000017 t = 0.21900000000000017 t = 0.22000000000000017 t = 0.22100000000000017 t = 0.22200000000000017 t = 0.22300000000000017 t = 0.22400000000000017 t = 0.22500000000000017 t = 0.22600000000000017 t = 0.22700000000000017

end{sphinxVerbatim}

t = 0.20900000000000016 t = 0.21000000000000016 t = 0.21100000000000016 t = 0.21200000000000016 t = 0.21300000000000016 t = 0.21400000000000016 t = 0.21500000000000016 t = 0.21600000000000016 t = 0.21700000000000016 t = 0.21800000000000017 t = 0.21900000000000017 t = 0.22000000000000017 t = 0.22100000000000017 t = 0.22200000000000017 t = 0.22300000000000017 t = 0.22400000000000017 t = 0.22500000000000017 t = 0.22600000000000017 t = 0.22700000000000017

more-to-come:

t = 0.22800000000000017 t = 0.22900000000000018 t = 0.23000000000000018 t = 0.23100000000000018 t = 0.23200000000000018 t = 0.23300000000000018 t = 0.23400000000000018 t = 0.23500000000000018 t = 0.23600000000000018 t = 0.23700000000000018 t = 0.23800000000000018 t = 0.23900000000000018 t = 0.24000000000000019 t = 0.2410000000000002 t = 0.2420000000000002 t = 0.2430000000000002

</pre>

t = 0.22800000000000017 t = 0.22900000000000018 t = 0.23000000000000018 t = 0.23100000000000018 t = 0.23200000000000018 t = 0.23300000000000018 t = 0.23400000000000018 t = 0.23500000000000018 t = 0.23600000000000018 t = 0.23700000000000018 t = 0.23800000000000018 t = 0.23900000000000018 t = 0.24000000000000019 t = 0.2410000000000002 t = 0.2420000000000002 t = 0.2430000000000002

end{sphinxVerbatim}

t = 0.22800000000000017 t = 0.22900000000000018 t = 0.23000000000000018 t = 0.23100000000000018 t = 0.23200000000000018 t = 0.23300000000000018 t = 0.23400000000000018 t = 0.23500000000000018 t = 0.23600000000000018 t = 0.23700000000000018 t = 0.23800000000000018 t = 0.23900000000000018 t = 0.24000000000000019 t = 0.2410000000000002 t = 0.2420000000000002 t = 0.2430000000000002

more-to-come:

t = 0.2440000000000002 t = 0.2450000000000002 t = 0.2460000000000002 t = 0.2470000000000002 t = 0.2480000000000002 t = 0.2490000000000002 t = 0.25000000000000017

CPU times: user 14.1 s, sys: 114 ms, total: 14.3 s Wall time: 3.57 s

t = 0.249 t = 0.248 t = 0.247 t = 0.246 t = 0.245 t = 0.244 t = 0.243 t = 0.242 t = 0.241 t = 0.24 t = 0.239 t = 0.238

</pre>

t = 0.2440000000000002 t = 0.2450000000000002 t = 0.2460000000000002 t = 0.2470000000000002 t = 0.2480000000000002 t = 0.2490000000000002 t = 0.25000000000000017

CPU times: user 14.1 s, sys: 114 ms, total: 14.3 s Wall time: 3.57 s

t = 0.249 t = 0.248 t = 0.247 t = 0.246 t = 0.245 t = 0.244 t = 0.243 t = 0.242 t = 0.241 t = 0.24 t = 0.239 t = 0.238

end{sphinxVerbatim}

t = 0.2440000000000002 t = 0.2450000000000002 t = 0.2460000000000002 t = 0.2470000000000002 t = 0.2480000000000002 t = 0.2490000000000002 t = 0.25000000000000017

CPU times: user 14.1 s, sys: 114 ms, total: 14.3 s Wall time: 3.57 s

t = 0.249 t = 0.248 t = 0.247 t = 0.246 t = 0.245 t = 0.244 t = 0.243 t = 0.242 t = 0.241 t = 0.24 t = 0.239 t = 0.238

more-to-come:

t = 0.237 t = 0.236 t = 0.235 t = 0.23399999999999999 t = 0.23299999999999998 t = 0.23199999999999998 t = 0.23099999999999998 t = 0.22999999999999998 t = 0.22899999999999998 t = 0.22799999999999998 t = 0.22699999999999998 t = 0.22599999999999998 t = 0.22499999999999998 t = 0.22399999999999998 t = 0.22299999999999998 t = 0.22199999999999998 t = 0.22099999999999997

</pre>

t = 0.237 t = 0.236 t = 0.235 t = 0.23399999999999999 t = 0.23299999999999998 t = 0.23199999999999998 t = 0.23099999999999998 t = 0.22999999999999998 t = 0.22899999999999998 t = 0.22799999999999998 t = 0.22699999999999998 t = 0.22599999999999998 t = 0.22499999999999998 t = 0.22399999999999998 t = 0.22299999999999998 t = 0.22199999999999998 t = 0.22099999999999997

end{sphinxVerbatim}

t = 0.237 t = 0.236 t = 0.235 t = 0.23399999999999999 t = 0.23299999999999998 t = 0.23199999999999998 t = 0.23099999999999998 t = 0.22999999999999998 t = 0.22899999999999998 t = 0.22799999999999998 t = 0.22699999999999998 t = 0.22599999999999998 t = 0.22499999999999998 t = 0.22399999999999998 t = 0.22299999999999998 t = 0.22199999999999998 t = 0.22099999999999997

more-to-come:

t = 0.21999999999999997 t = 0.21899999999999997 t = 0.21799999999999997 t = 0.21699999999999997 t = 0.21599999999999997 t = 0.21499999999999997 t = 0.21399999999999997 t = 0.21299999999999997 t = 0.21199999999999997 t = 0.21099999999999997 t = 0.20999999999999996 t = 0.20899999999999996 t = 0.20799999999999996 t = 0.20699999999999996 t = 0.20599999999999996 t = 0.20499999999999996 t = 0.20399999999999996 t = 0.20299999999999996 t = 0.20199999999999996

</pre>

t = 0.21999999999999997 t = 0.21899999999999997 t = 0.21799999999999997 t = 0.21699999999999997 t = 0.21599999999999997 t = 0.21499999999999997 t = 0.21399999999999997 t = 0.21299999999999997 t = 0.21199999999999997 t = 0.21099999999999997 t = 0.20999999999999996 t = 0.20899999999999996 t = 0.20799999999999996 t = 0.20699999999999996 t = 0.20599999999999996 t = 0.20499999999999996 t = 0.20399999999999996 t = 0.20299999999999996 t = 0.20199999999999996

end{sphinxVerbatim}

t = 0.21999999999999997 t = 0.21899999999999997 t = 0.21799999999999997 t = 0.21699999999999997 t = 0.21599999999999997 t = 0.21499999999999997 t = 0.21399999999999997 t = 0.21299999999999997 t = 0.21199999999999997 t = 0.21099999999999997 t = 0.20999999999999996 t = 0.20899999999999996 t = 0.20799999999999996 t = 0.20699999999999996 t = 0.20599999999999996 t = 0.20499999999999996 t = 0.20399999999999996 t = 0.20299999999999996 t = 0.20199999999999996

more-to-come:

t = 0.20099999999999996 t = 0.19999999999999996 t = 0.19899999999999995 t = 0.19799999999999995 t = 0.19699999999999995 t = 0.19599999999999995 t = 0.19499999999999995 t = 0.19399999999999995 t = 0.19299999999999995 t = 0.19199999999999995 t = 0.19099999999999995 t = 0.18999999999999995 t = 0.18899999999999995 t = 0.18799999999999994 t = 0.18699999999999994 t = 0.18599999999999994 t = 0.18499999999999994 t = 0.18399999999999994 t = 0.18299999999999994

</pre>

t = 0.20099999999999996 t = 0.19999999999999996 t = 0.19899999999999995 t = 0.19799999999999995 t = 0.19699999999999995 t = 0.19599999999999995 t = 0.19499999999999995 t = 0.19399999999999995 t = 0.19299999999999995 t = 0.19199999999999995 t = 0.19099999999999995 t = 0.18999999999999995 t = 0.18899999999999995 t = 0.18799999999999994 t = 0.18699999999999994 t = 0.18599999999999994 t = 0.18499999999999994 t = 0.18399999999999994 t = 0.18299999999999994

end{sphinxVerbatim}

t = 0.20099999999999996 t = 0.19999999999999996 t = 0.19899999999999995 t = 0.19799999999999995 t = 0.19699999999999995 t = 0.19599999999999995 t = 0.19499999999999995 t = 0.19399999999999995 t = 0.19299999999999995 t = 0.19199999999999995 t = 0.19099999999999995 t = 0.18999999999999995 t = 0.18899999999999995 t = 0.18799999999999994 t = 0.18699999999999994 t = 0.18599999999999994 t = 0.18499999999999994 t = 0.18399999999999994 t = 0.18299999999999994

more-to-come:

t = 0.18199999999999994 t = 0.18099999999999994 t = 0.17999999999999994 t = 0.17899999999999994 t = 0.17799999999999994 t = 0.17699999999999994 t = 0.17599999999999993 t = 0.17499999999999993 t = 0.17399999999999993 t = 0.17299999999999993 t = 0.17199999999999993 t = 0.17099999999999993 t = 0.16999999999999993 t = 0.16899999999999993 t = 0.16799999999999993 t = 0.16699999999999993

</pre>

t = 0.18199999999999994 t = 0.18099999999999994 t = 0.17999999999999994 t = 0.17899999999999994 t = 0.17799999999999994 t = 0.17699999999999994 t = 0.17599999999999993 t = 0.17499999999999993 t = 0.17399999999999993 t = 0.17299999999999993 t = 0.17199999999999993 t = 0.17099999999999993 t = 0.16999999999999993 t = 0.16899999999999993 t = 0.16799999999999993 t = 0.16699999999999993

end{sphinxVerbatim}

t = 0.18199999999999994 t = 0.18099999999999994 t = 0.17999999999999994 t = 0.17899999999999994 t = 0.17799999999999994 t = 0.17699999999999994 t = 0.17599999999999993 t = 0.17499999999999993 t = 0.17399999999999993 t = 0.17299999999999993 t = 0.17199999999999993 t = 0.17099999999999993 t = 0.16999999999999993 t = 0.16899999999999993 t = 0.16799999999999993 t = 0.16699999999999993

more-to-come:

t = 0.16599999999999993 t = 0.16499999999999992 t = 0.16399999999999992 t = 0.16299999999999992 t = 0.16199999999999992 t = 0.16099999999999992 t = 0.15999999999999992 t = 0.15899999999999992 t = 0.15799999999999992 t = 0.15699999999999992 t = 0.15599999999999992 t = 0.15499999999999992 t = 0.15399999999999991 t = 0.1529999999999999 t = 0.1519999999999999 t = 0.1509999999999999 t = 0.1499999999999999 t = 0.1489999999999999 t = 0.1479999999999999

</pre>

t = 0.16599999999999993 t = 0.16499999999999992 t = 0.16399999999999992 t = 0.16299999999999992 t = 0.16199999999999992 t = 0.16099999999999992 t = 0.15999999999999992 t = 0.15899999999999992 t = 0.15799999999999992 t = 0.15699999999999992 t = 0.15599999999999992 t = 0.15499999999999992 t = 0.15399999999999991 t = 0.1529999999999999 t = 0.1519999999999999 t = 0.1509999999999999 t = 0.1499999999999999 t = 0.1489999999999999 t = 0.1479999999999999

end{sphinxVerbatim}

t = 0.16599999999999993 t = 0.16499999999999992 t = 0.16399999999999992 t = 0.16299999999999992 t = 0.16199999999999992 t = 0.16099999999999992 t = 0.15999999999999992 t = 0.15899999999999992 t = 0.15799999999999992 t = 0.15699999999999992 t = 0.15599999999999992 t = 0.15499999999999992 t = 0.15399999999999991 t = 0.1529999999999999 t = 0.1519999999999999 t = 0.1509999999999999 t = 0.1499999999999999 t = 0.1489999999999999 t = 0.1479999999999999

more-to-come:

t = 0.1469999999999999 t = 0.1459999999999999 t = 0.1449999999999999 t = 0.1439999999999999 t = 0.1429999999999999 t = 0.1419999999999999 t = 0.1409999999999999 t = 0.1399999999999999 t = 0.1389999999999999 t = 0.1379999999999999 t = 0.1369999999999999 t = 0.1359999999999999 t = 0.1349999999999999 t = 0.1339999999999999 t = 0.1329999999999999 t = 0.1319999999999999 t = 0.1309999999999999

</pre>

t = 0.1469999999999999 t = 0.1459999999999999 t = 0.1449999999999999 t = 0.1439999999999999 t = 0.1429999999999999 t = 0.1419999999999999 t = 0.1409999999999999 t = 0.1399999999999999 t = 0.1389999999999999 t = 0.1379999999999999 t = 0.1369999999999999 t = 0.1359999999999999 t = 0.1349999999999999 t = 0.1339999999999999 t = 0.1329999999999999 t = 0.1319999999999999 t = 0.1309999999999999

end{sphinxVerbatim}

t = 0.1469999999999999 t = 0.1459999999999999 t = 0.1449999999999999 t = 0.1439999999999999 t = 0.1429999999999999 t = 0.1419999999999999 t = 0.1409999999999999 t = 0.1399999999999999 t = 0.1389999999999999 t = 0.1379999999999999 t = 0.1369999999999999 t = 0.1359999999999999 t = 0.1349999999999999 t = 0.1339999999999999 t = 0.1329999999999999 t = 0.1319999999999999 t = 0.1309999999999999

more-to-come:

t = 0.1299999999999999 t = 0.1289999999999999 t = 0.1279999999999999 t = 0.1269999999999999 t = 0.1259999999999999 t = 0.12499999999999992 t = 0.12399999999999992 t = 0.12299999999999991 t = 0.12199999999999991 t = 0.12099999999999991 t = 0.11999999999999991 t = 0.11899999999999991 t = 0.11799999999999991 t = 0.11699999999999991 t = 0.11599999999999991 t = 0.11499999999999991 t = 0.11399999999999991 t = 0.1129999999999999 t = 0.1119999999999999

</pre>

t = 0.1299999999999999 t = 0.1289999999999999 t = 0.1279999999999999 t = 0.1269999999999999 t = 0.1259999999999999 t = 0.12499999999999992 t = 0.12399999999999992 t = 0.12299999999999991 t = 0.12199999999999991 t = 0.12099999999999991 t = 0.11999999999999991 t = 0.11899999999999991 t = 0.11799999999999991 t = 0.11699999999999991 t = 0.11599999999999991 t = 0.11499999999999991 t = 0.11399999999999991 t = 0.1129999999999999 t = 0.1119999999999999

end{sphinxVerbatim}

t = 0.1299999999999999 t = 0.1289999999999999 t = 0.1279999999999999 t = 0.1269999999999999 t = 0.1259999999999999 t = 0.12499999999999992 t = 0.12399999999999992 t = 0.12299999999999991 t = 0.12199999999999991 t = 0.12099999999999991 t = 0.11999999999999991 t = 0.11899999999999991 t = 0.11799999999999991 t = 0.11699999999999991 t = 0.11599999999999991 t = 0.11499999999999991 t = 0.11399999999999991 t = 0.1129999999999999 t = 0.1119999999999999

more-to-come:

t = 0.1109999999999999 t = 0.1099999999999999 t = 0.1089999999999999 t = 0.1079999999999999 t = 0.1069999999999999 t = 0.1059999999999999 t = 0.1049999999999999 t = 0.1039999999999999 t = 0.1029999999999999 t = 0.1019999999999999 t = 0.1009999999999999 t = 0.0999999999999999 t = 0.0989999999999999 t = 0.09799999999999989 t = 0.09699999999999989 t = 0.09599999999999989 t = 0.09499999999999989 t = 0.09399999999999989 t = 0.09299999999999989 t = 0.09199999999999989

</pre>

t = 0.1109999999999999 t = 0.1099999999999999 t = 0.1089999999999999 t = 0.1079999999999999 t = 0.1069999999999999 t = 0.1059999999999999 t = 0.1049999999999999 t = 0.1039999999999999 t = 0.1029999999999999 t = 0.1019999999999999 t = 0.1009999999999999 t = 0.0999999999999999 t = 0.0989999999999999 t = 0.09799999999999989 t = 0.09699999999999989 t = 0.09599999999999989 t = 0.09499999999999989 t = 0.09399999999999989 t = 0.09299999999999989 t = 0.09199999999999989

end{sphinxVerbatim}

t = 0.1109999999999999 t = 0.1099999999999999 t = 0.1089999999999999 t = 0.1079999999999999 t = 0.1069999999999999 t = 0.1059999999999999 t = 0.1049999999999999 t = 0.1039999999999999 t = 0.1029999999999999 t = 0.1019999999999999 t = 0.1009999999999999 t = 0.0999999999999999 t = 0.0989999999999999 t = 0.09799999999999989 t = 0.09699999999999989 t = 0.09599999999999989 t = 0.09499999999999989 t = 0.09399999999999989 t = 0.09299999999999989 t = 0.09199999999999989

more-to-come:

t = 0.09099999999999989 t = 0.08999999999999989 t = 0.08899999999999988 t = 0.08799999999999988 t = 0.08699999999999988 t = 0.08599999999999988 t = 0.08499999999999988 t = 0.08399999999999988 t = 0.08299999999999988 t = 0.08199999999999988 t = 0.08099999999999988 t = 0.07999999999999988 t = 0.07899999999999988 t = 0.07799999999999987 t = 0.07699999999999987 t = 0.07599999999999987 t = 0.07499999999999987 t = 0.07399999999999987 t = 0.07299999999999987

</pre>

t = 0.09099999999999989 t = 0.08999999999999989 t = 0.08899999999999988 t = 0.08799999999999988 t = 0.08699999999999988 t = 0.08599999999999988 t = 0.08499999999999988 t = 0.08399999999999988 t = 0.08299999999999988 t = 0.08199999999999988 t = 0.08099999999999988 t = 0.07999999999999988 t = 0.07899999999999988 t = 0.07799999999999987 t = 0.07699999999999987 t = 0.07599999999999987 t = 0.07499999999999987 t = 0.07399999999999987 t = 0.07299999999999987

end{sphinxVerbatim}

t = 0.09099999999999989 t = 0.08999999999999989 t = 0.08899999999999988 t = 0.08799999999999988 t = 0.08699999999999988 t = 0.08599999999999988 t = 0.08499999999999988 t = 0.08399999999999988 t = 0.08299999999999988 t = 0.08199999999999988 t = 0.08099999999999988 t = 0.07999999999999988 t = 0.07899999999999988 t = 0.07799999999999987 t = 0.07699999999999987 t = 0.07599999999999987 t = 0.07499999999999987 t = 0.07399999999999987 t = 0.07299999999999987

more-to-come:

t = 0.07199999999999987 t = 0.07099999999999987 t = 0.06999999999999987 t = 0.06899999999999987 t = 0.06799999999999987 t = 0.06699999999999987 t = 0.06599999999999986 t = 0.06499999999999986 t = 0.06399999999999986 t = 0.06299999999999986 t = 0.06199999999999986 t = 0.06099999999999986 t = 0.05999999999999986 t = 0.05899999999999986 t = 0.05799999999999986 t = 0.056999999999999856 t = 0.055999999999999855 t = 0.054999999999999855 t = 0.053999999999999854

</pre>

t = 0.07199999999999987 t = 0.07099999999999987 t = 0.06999999999999987 t = 0.06899999999999987 t = 0.06799999999999987 t = 0.06699999999999987 t = 0.06599999999999986 t = 0.06499999999999986 t = 0.06399999999999986 t = 0.06299999999999986 t = 0.06199999999999986 t = 0.06099999999999986 t = 0.05999999999999986 t = 0.05899999999999986 t = 0.05799999999999986 t = 0.056999999999999856 t = 0.055999999999999855 t = 0.054999999999999855 t = 0.053999999999999854

end{sphinxVerbatim}

t = 0.07199999999999987 t = 0.07099999999999987 t = 0.06999999999999987 t = 0.06899999999999987 t = 0.06799999999999987 t = 0.06699999999999987 t = 0.06599999999999986 t = 0.06499999999999986 t = 0.06399999999999986 t = 0.06299999999999986 t = 0.06199999999999986 t = 0.06099999999999986 t = 0.05999999999999986 t = 0.05899999999999986 t = 0.05799999999999986 t = 0.056999999999999856 t = 0.055999999999999855 t = 0.054999999999999855 t = 0.053999999999999854

more-to-come:

t = 0.05299999999999985 t = 0.05199999999999985 t = 0.05099999999999985 t = 0.04999999999999985 t = 0.04899999999999985 t = 0.04799999999999985 t = 0.04699999999999985 t = 0.04599999999999985 t = 0.044999999999999846 t = 0.043999999999999845 t = 0.042999999999999844 t = 0.04199999999999984 t = 0.04099999999999984 t = 0.03999999999999984 t = 0.03899999999999984 t = 0.03799999999999984

</pre>

t = 0.05299999999999985 t = 0.05199999999999985 t = 0.05099999999999985 t = 0.04999999999999985 t = 0.04899999999999985 t = 0.04799999999999985 t = 0.04699999999999985 t = 0.04599999999999985 t = 0.044999999999999846 t = 0.043999999999999845 t = 0.042999999999999844 t = 0.04199999999999984 t = 0.04099999999999984 t = 0.03999999999999984 t = 0.03899999999999984 t = 0.03799999999999984

end{sphinxVerbatim}

t = 0.05299999999999985 t = 0.05199999999999985 t = 0.05099999999999985 t = 0.04999999999999985 t = 0.04899999999999985 t = 0.04799999999999985 t = 0.04699999999999985 t = 0.04599999999999985 t = 0.044999999999999846 t = 0.043999999999999845 t = 0.042999999999999844 t = 0.04199999999999984 t = 0.04099999999999984 t = 0.03999999999999984 t = 0.03899999999999984 t = 0.03799999999999984

more-to-come:

t = 0.03699999999999984 t = 0.03599999999999984 t = 0.03499999999999984 t = 0.033999999999999836 t = 0.032999999999999835 t = 0.031999999999999834 t = 0.030999999999999833 t = 0.029999999999999832 t = 0.02899999999999983 t = 0.02799999999999983 t = 0.02699999999999983 t = 0.02599999999999983 t = 0.024999999999999828 t = 0.023999999999999827 t = 0.022999999999999826

</pre>

t = 0.03699999999999984 t = 0.03599999999999984 t = 0.03499999999999984 t = 0.033999999999999836 t = 0.032999999999999835 t = 0.031999999999999834 t = 0.030999999999999833 t = 0.029999999999999832 t = 0.02899999999999983 t = 0.02799999999999983 t = 0.02699999999999983 t = 0.02599999999999983 t = 0.024999999999999828 t = 0.023999999999999827 t = 0.022999999999999826

end{sphinxVerbatim}

t = 0.03699999999999984 t = 0.03599999999999984 t = 0.03499999999999984 t = 0.033999999999999836 t = 0.032999999999999835 t = 0.031999999999999834 t = 0.030999999999999833 t = 0.029999999999999832 t = 0.02899999999999983 t = 0.02799999999999983 t = 0.02699999999999983 t = 0.02599999999999983 t = 0.024999999999999828 t = 0.023999999999999827 t = 0.022999999999999826

more-to-come:

t = 0.021999999999999825 t = 0.020999999999999824 t = 0.019999999999999823 t = 0.018999999999999823 t = 0.01799999999999982 t = 0.01699999999999982 t = 0.01599999999999982 t = 0.014999999999999819 t = 0.013999999999999818 t = 0.012999999999999817 t = 0.011999999999999816 t = 0.010999999999999815 t = 0.009999999999999815 t = 0.008999999999999814 t = 0.007999999999999813 t = 0.006999999999999812

</pre>

t = 0.021999999999999825 t = 0.020999999999999824 t = 0.019999999999999823 t = 0.018999999999999823 t = 0.01799999999999982 t = 0.01699999999999982 t = 0.01599999999999982 t = 0.014999999999999819 t = 0.013999999999999818 t = 0.012999999999999817 t = 0.011999999999999816 t = 0.010999999999999815 t = 0.009999999999999815 t = 0.008999999999999814 t = 0.007999999999999813 t = 0.006999999999999812

end{sphinxVerbatim}

t = 0.021999999999999825 t = 0.020999999999999824 t = 0.019999999999999823 t = 0.018999999999999823 t = 0.01799999999999982 t = 0.01699999999999982 t = 0.01599999999999982 t = 0.014999999999999819 t = 0.013999999999999818 t = 0.012999999999999817 t = 0.011999999999999816 t = 0.010999999999999815 t = 0.009999999999999815 t = 0.008999999999999814 t = 0.007999999999999813 t = 0.006999999999999812

t = 0.005999999999999811 t = 0.00499999999999981 t = 0.003999999999999809 t = 0.0029999999999998084 t = 0.0019999999999998075 t = 0.0009999999999998066 t = -1.6653345369377348e-16

CPU times: user 13.9 s, sys: 147 ms, total: 14.1 s Wall time: 3.52 s </pre>

t = 0.005999999999999811 t = 0.00499999999999981 t = 0.003999999999999809 t = 0.0029999999999998084 t = 0.0019999999999998075 t = 0.0009999999999998066 t = -1.6653345369377348e-16

CPU times: user 13.9 s, sys: 147 ms, total: 14.1 s Wall time: 3.52 s end{sphinxVerbatim}

t = 0.005999999999999811 t = 0.00499999999999981 t = 0.003999999999999809 t = 0.0029999999999998084 t = 0.0019999999999998075 t = 0.0009999999999998066 t = -1.6653345369377348e-16

CPU times: user 13.9 s, sys: 147 ms, total: 14.1 s Wall time: 3.52 s

Version 2:

Using the TraceOperator and assembling of \(b_h\)

We want to use the TraceOperator again:

fes_tr = FacetFESpace(mesh, order=order+1)
traceop = fes_p.TraceOperator(fes_tr, False)

We want to assemble the sub-block matrix and need test/trial functions for single spaces (not the product spaces):

p = fes_p.TrialFunction()
v = fes_u.TestFunction()
phat = fes_tr.TrialFunction()

We split the operator to \(b_h\) into * volume contributions (local) * and couplings between the trace (obtained through the trace op) and the volume:

Bel = BilinearForm(trialspace=fes_p, testspace=fes_u)
Bel += grad(p)*v * dx -p*(v*n) * dx(element_boundary=True)
%time Bel.Assemble()

Btr = BilinearForm(trialspace=fes_tr, testspace=fes_u)
Btr += 0.5 * phat * (v*n) * dx(element_boundary=True)
%time Btr.Assemble()

B = emb_u @ (Bel.mat + Btr.mat @ traceop) @ emb_p.T

CPU times: user 126 ms, sys: 11.6 ms, total: 138 ms
Wall time: 137 ms
CPU times: user 50.9 ms, sys: 3.93 ms, total: 54.8 ms
Wall time: 54.6 ms

Version 2 in action (assembled matrices, TraceOperators and Embeddings can do transposed multiply):

%time Run(B, B.T, backward=False)
%time Run(B, B.T, backward=True)

more-to-come:

t = 0.001 t = 0.002 t = 0.003 t = 0.004 t = 0.005 t = 0.006 t = 0.007

</pre>

t = 0.001 t = 0.002 t = 0.003 t = 0.004 t = 0.005 t = 0.006 t = 0.007

end{sphinxVerbatim}

t = 0.001 t = 0.002 t = 0.003 t = 0.004 t = 0.005 t = 0.006 t = 0.007

more-to-come:

t = 0.008 t = 0.009000000000000001 t = 0.010000000000000002 t = 0.011000000000000003 t = 0.012000000000000004 t = 0.013000000000000005

</pre>

t = 0.008 t = 0.009000000000000001 t = 0.010000000000000002 t = 0.011000000000000003 t = 0.012000000000000004 t = 0.013000000000000005

end{sphinxVerbatim}

t = 0.008 t = 0.009000000000000001 t = 0.010000000000000002 t = 0.011000000000000003 t = 0.012000000000000004 t = 0.013000000000000005

more-to-come:

t = 0.014000000000000005 t = 0.015000000000000006 t = 0.016000000000000007 t = 0.017000000000000008 t = 0.01800000000000001 t = 0.01900000000000001 t = 0.02000000000000001 t = 0.02100000000000001 t = 0.022000000000000013

</pre>

t = 0.014000000000000005 t = 0.015000000000000006 t = 0.016000000000000007 t = 0.017000000000000008 t = 0.01800000000000001 t = 0.01900000000000001 t = 0.02000000000000001 t = 0.02100000000000001 t = 0.022000000000000013

end{sphinxVerbatim}

t = 0.014000000000000005 t = 0.015000000000000006 t = 0.016000000000000007 t = 0.017000000000000008 t = 0.01800000000000001 t = 0.01900000000000001 t = 0.02000000000000001 t = 0.02100000000000001 t = 0.022000000000000013

more-to-come:

t = 0.023000000000000013 t = 0.024000000000000014 t = 0.025000000000000015 t = 0.026000000000000016 t = 0.027000000000000017

</pre>

t = 0.023000000000000013 t = 0.024000000000000014 t = 0.025000000000000015 t = 0.026000000000000016 t = 0.027000000000000017

end{sphinxVerbatim}

t = 0.023000000000000013 t = 0.024000000000000014 t = 0.025000000000000015 t = 0.026000000000000016 t = 0.027000000000000017

more-to-come:

t = 0.028000000000000018 t = 0.02900000000000002 t = 0.03000000000000002 t = 0.03100000000000002 t = 0.03200000000000002 t = 0.03300000000000002 t = 0.03400000000000002 t = 0.035000000000000024 t = 0.036000000000000025

</pre>

t = 0.028000000000000018 t = 0.02900000000000002 t = 0.03000000000000002 t = 0.03100000000000002 t = 0.03200000000000002 t = 0.03300000000000002 t = 0.03400000000000002 t = 0.035000000000000024 t = 0.036000000000000025

end{sphinxVerbatim}

t = 0.028000000000000018 t = 0.02900000000000002 t = 0.03000000000000002 t = 0.03100000000000002 t = 0.03200000000000002 t = 0.03300000000000002 t = 0.03400000000000002 t = 0.035000000000000024 t = 0.036000000000000025

more-to-come:

t = 0.037000000000000026 t = 0.03800000000000003 t = 0.03900000000000003 t = 0.04000000000000003 t = 0.04100000000000003 t = 0.04200000000000003 t = 0.04300000000000003 t = 0.04400000000000003 t = 0.04500000000000003 t = 0.046000000000000034 t = 0.047000000000000035

</pre>

t = 0.037000000000000026 t = 0.03800000000000003 t = 0.03900000000000003 t = 0.04000000000000003 t = 0.04100000000000003 t = 0.04200000000000003 t = 0.04300000000000003 t = 0.04400000000000003 t = 0.04500000000000003 t = 0.046000000000000034 t = 0.047000000000000035

end{sphinxVerbatim}

t = 0.037000000000000026 t = 0.03800000000000003 t = 0.03900000000000003 t = 0.04000000000000003 t = 0.04100000000000003 t = 0.04200000000000003 t = 0.04300000000000003 t = 0.04400000000000003 t = 0.04500000000000003 t = 0.046000000000000034 t = 0.047000000000000035

more-to-come:

t = 0.048000000000000036 t = 0.04900000000000004 t = 0.05000000000000004 t = 0.05100000000000004 t = 0.05200000000000004 t = 0.05300000000000004 t = 0.05400000000000004 t = 0.05500000000000004

</pre>

t = 0.048000000000000036 t = 0.04900000000000004 t = 0.05000000000000004 t = 0.05100000000000004 t = 0.05200000000000004 t = 0.05300000000000004 t = 0.05400000000000004 t = 0.05500000000000004

end{sphinxVerbatim}

t = 0.048000000000000036 t = 0.04900000000000004 t = 0.05000000000000004 t = 0.05100000000000004 t = 0.05200000000000004 t = 0.05300000000000004 t = 0.05400000000000004 t = 0.05500000000000004

more-to-come:

t = 0.05600000000000004 t = 0.057000000000000044 t = 0.058000000000000045 t = 0.059000000000000045 t = 0.060000000000000046 t = 0.06100000000000005 t = 0.06200000000000005 t = 0.06300000000000004 t = 0.06400000000000004 t = 0.06500000000000004 t = 0.06600000000000004

</pre>

t = 0.05600000000000004 t = 0.057000000000000044 t = 0.058000000000000045 t = 0.059000000000000045 t = 0.060000000000000046 t = 0.06100000000000005 t = 0.06200000000000005 t = 0.06300000000000004 t = 0.06400000000000004 t = 0.06500000000000004 t = 0.06600000000000004

end{sphinxVerbatim}

t = 0.05600000000000004 t = 0.057000000000000044 t = 0.058000000000000045 t = 0.059000000000000045 t = 0.060000000000000046 t = 0.06100000000000005 t = 0.06200000000000005 t = 0.06300000000000004 t = 0.06400000000000004 t = 0.06500000000000004 t = 0.06600000000000004

more-to-come:

t = 0.06700000000000005 t = 0.06800000000000005 t = 0.06900000000000005 t = 0.07000000000000005 t = 0.07100000000000005 t = 0.07200000000000005 t = 0.07300000000000005 t = 0.07400000000000005 t = 0.07500000000000005 t = 0.07600000000000005 t = 0.07700000000000005

</pre>

t = 0.06700000000000005 t = 0.06800000000000005 t = 0.06900000000000005 t = 0.07000000000000005 t = 0.07100000000000005 t = 0.07200000000000005 t = 0.07300000000000005 t = 0.07400000000000005 t = 0.07500000000000005 t = 0.07600000000000005 t = 0.07700000000000005

end{sphinxVerbatim}

t = 0.06700000000000005 t = 0.06800000000000005 t = 0.06900000000000005 t = 0.07000000000000005 t = 0.07100000000000005 t = 0.07200000000000005 t = 0.07300000000000005 t = 0.07400000000000005 t = 0.07500000000000005 t = 0.07600000000000005 t = 0.07700000000000005

more-to-come:

t = 0.07800000000000006 t = 0.07900000000000006 t = 0.08000000000000006 t = 0.08100000000000006 t = 0.08200000000000006 t = 0.08300000000000006 t = 0.08400000000000006 t = 0.08500000000000006 t = 0.08600000000000006 t = 0.08700000000000006

</pre>

t = 0.07800000000000006 t = 0.07900000000000006 t = 0.08000000000000006 t = 0.08100000000000006 t = 0.08200000000000006 t = 0.08300000000000006 t = 0.08400000000000006 t = 0.08500000000000006 t = 0.08600000000000006 t = 0.08700000000000006

end{sphinxVerbatim}

t = 0.07800000000000006 t = 0.07900000000000006 t = 0.08000000000000006 t = 0.08100000000000006 t = 0.08200000000000006 t = 0.08300000000000006 t = 0.08400000000000006 t = 0.08500000000000006 t = 0.08600000000000006 t = 0.08700000000000006

more-to-come:

t = 0.08800000000000006 t = 0.08900000000000007 t = 0.09000000000000007 t = 0.09100000000000007 t = 0.09200000000000007 t = 0.09300000000000007 t = 0.09400000000000007 t = 0.09500000000000007 t = 0.09600000000000007 t = 0.09700000000000007 t = 0.09800000000000007 t = 0.09900000000000007

</pre>

t = 0.08800000000000006 t = 0.08900000000000007 t = 0.09000000000000007 t = 0.09100000000000007 t = 0.09200000000000007 t = 0.09300000000000007 t = 0.09400000000000007 t = 0.09500000000000007 t = 0.09600000000000007 t = 0.09700000000000007 t = 0.09800000000000007 t = 0.09900000000000007

end{sphinxVerbatim}

t = 0.08800000000000006 t = 0.08900000000000007 t = 0.09000000000000007 t = 0.09100000000000007 t = 0.09200000000000007 t = 0.09300000000000007 t = 0.09400000000000007 t = 0.09500000000000007 t = 0.09600000000000007 t = 0.09700000000000007 t = 0.09800000000000007 t = 0.09900000000000007

more-to-come:

t = 0.10000000000000007 t = 0.10100000000000008 t = 0.10200000000000008 t = 0.10300000000000008 t = 0.10400000000000008 t = 0.10500000000000008 t = 0.10600000000000008 t = 0.10700000000000008 t = 0.10800000000000008 t = 0.10900000000000008

</pre>

t = 0.10000000000000007 t = 0.10100000000000008 t = 0.10200000000000008 t = 0.10300000000000008 t = 0.10400000000000008 t = 0.10500000000000008 t = 0.10600000000000008 t = 0.10700000000000008 t = 0.10800000000000008 t = 0.10900000000000008

end{sphinxVerbatim}

t = 0.10000000000000007 t = 0.10100000000000008 t = 0.10200000000000008 t = 0.10300000000000008 t = 0.10400000000000008 t = 0.10500000000000008 t = 0.10600000000000008 t = 0.10700000000000008 t = 0.10800000000000008 t = 0.10900000000000008

more-to-come:

t = 0.11000000000000008 t = 0.11100000000000008 t = 0.11200000000000009 t = 0.11300000000000009 t = 0.11400000000000009 t = 0.11500000000000009 t = 0.11600000000000009 t = 0.11700000000000009

</pre>

t = 0.11000000000000008 t = 0.11100000000000008 t = 0.11200000000000009 t = 0.11300000000000009 t = 0.11400000000000009 t = 0.11500000000000009 t = 0.11600000000000009 t = 0.11700000000000009

end{sphinxVerbatim}

t = 0.11000000000000008 t = 0.11100000000000008 t = 0.11200000000000009 t = 0.11300000000000009 t = 0.11400000000000009 t = 0.11500000000000009 t = 0.11600000000000009 t = 0.11700000000000009

more-to-come:

t = 0.11800000000000009 t = 0.11900000000000009 t = 0.12000000000000009 t = 0.1210000000000001 t = 0.1220000000000001 t = 0.1230000000000001 t = 0.1240000000000001

</pre>

t = 0.11800000000000009 t = 0.11900000000000009 t = 0.12000000000000009 t = 0.1210000000000001 t = 0.1220000000000001 t = 0.1230000000000001 t = 0.1240000000000001

end{sphinxVerbatim}

t = 0.11800000000000009 t = 0.11900000000000009 t = 0.12000000000000009 t = 0.1210000000000001 t = 0.1220000000000001 t = 0.1230000000000001 t = 0.1240000000000001

more-to-come:

t = 0.12500000000000008 t = 0.12600000000000008 t = 0.12700000000000009 t = 0.12800000000000009 t = 0.1290000000000001 t = 0.1300000000000001 t = 0.1310000000000001 t = 0.1320000000000001 t = 0.1330000000000001 t = 0.1340000000000001

</pre>

t = 0.12500000000000008 t = 0.12600000000000008 t = 0.12700000000000009 t = 0.12800000000000009 t = 0.1290000000000001 t = 0.1300000000000001 t = 0.1310000000000001 t = 0.1320000000000001 t = 0.1330000000000001 t = 0.1340000000000001

end{sphinxVerbatim}

t = 0.12500000000000008 t = 0.12600000000000008 t = 0.12700000000000009 t = 0.12800000000000009 t = 0.1290000000000001 t = 0.1300000000000001 t = 0.1310000000000001 t = 0.1320000000000001 t = 0.1330000000000001 t = 0.1340000000000001

more-to-come:

t = 0.1350000000000001 t = 0.1360000000000001 t = 0.1370000000000001 t = 0.1380000000000001

</pre>

t = 0.1350000000000001 t = 0.1360000000000001 t = 0.1370000000000001 t = 0.1380000000000001

end{sphinxVerbatim}

t = 0.1350000000000001 t = 0.1360000000000001 t = 0.1370000000000001 t = 0.1380000000000001

more-to-come:

t = 0.1390000000000001 t = 0.1400000000000001 t = 0.1410000000000001 t = 0.1420000000000001 t = 0.1430000000000001 t = 0.1440000000000001 t = 0.1450000000000001 t = 0.1460000000000001 t = 0.1470000000000001 t = 0.1480000000000001

</pre>

t = 0.1390000000000001 t = 0.1400000000000001 t = 0.1410000000000001 t = 0.1420000000000001 t = 0.1430000000000001 t = 0.1440000000000001 t = 0.1450000000000001 t = 0.1460000000000001 t = 0.1470000000000001 t = 0.1480000000000001

end{sphinxVerbatim}

t = 0.1390000000000001 t = 0.1400000000000001 t = 0.1410000000000001 t = 0.1420000000000001 t = 0.1430000000000001 t = 0.1440000000000001 t = 0.1450000000000001 t = 0.1460000000000001 t = 0.1470000000000001 t = 0.1480000000000001

more-to-come:

t = 0.1490000000000001 t = 0.1500000000000001 t = 0.1510000000000001 t = 0.1520000000000001 t = 0.1530000000000001 t = 0.1540000000000001 t = 0.1550000000000001 t = 0.1560000000000001 t = 0.1570000000000001 t = 0.1580000000000001 t = 0.1590000000000001 t = 0.16000000000000011

</pre>

t = 0.1490000000000001 t = 0.1500000000000001 t = 0.1510000000000001 t = 0.1520000000000001 t = 0.1530000000000001 t = 0.1540000000000001 t = 0.1550000000000001 t = 0.1560000000000001 t = 0.1570000000000001 t = 0.1580000000000001 t = 0.1590000000000001 t = 0.16000000000000011

end{sphinxVerbatim}

t = 0.1490000000000001 t = 0.1500000000000001 t = 0.1510000000000001 t = 0.1520000000000001 t = 0.1530000000000001 t = 0.1540000000000001 t = 0.1550000000000001 t = 0.1560000000000001 t = 0.1570000000000001 t = 0.1580000000000001 t = 0.1590000000000001 t = 0.16000000000000011

more-to-come:

t = 0.16100000000000012 t = 0.16200000000000012 t = 0.16300000000000012 t = 0.16400000000000012 t = 0.16500000000000012 t = 0.16600000000000012 t = 0.16700000000000012 t = 0.16800000000000012 t = 0.16900000000000012 t = 0.17000000000000012 t = 0.17100000000000012

</pre>

t = 0.16100000000000012 t = 0.16200000000000012 t = 0.16300000000000012 t = 0.16400000000000012 t = 0.16500000000000012 t = 0.16600000000000012 t = 0.16700000000000012 t = 0.16800000000000012 t = 0.16900000000000012 t = 0.17000000000000012 t = 0.17100000000000012

end{sphinxVerbatim}

t = 0.16100000000000012 t = 0.16200000000000012 t = 0.16300000000000012 t = 0.16400000000000012 t = 0.16500000000000012 t = 0.16600000000000012 t = 0.16700000000000012 t = 0.16800000000000012 t = 0.16900000000000012 t = 0.17000000000000012 t = 0.17100000000000012

more-to-come:

t = 0.17200000000000013 t = 0.17300000000000013 t = 0.17400000000000013 t = 0.17500000000000013 t = 0.17600000000000013 t = 0.17700000000000013 t = 0.17800000000000013 t = 0.17900000000000013 t = 0.18000000000000013 t = 0.18100000000000013 t = 0.18200000000000013

</pre>

t = 0.17200000000000013 t = 0.17300000000000013 t = 0.17400000000000013 t = 0.17500000000000013 t = 0.17600000000000013 t = 0.17700000000000013 t = 0.17800000000000013 t = 0.17900000000000013 t = 0.18000000000000013 t = 0.18100000000000013 t = 0.18200000000000013

end{sphinxVerbatim}

t = 0.17200000000000013 t = 0.17300000000000013 t = 0.17400000000000013 t = 0.17500000000000013 t = 0.17600000000000013 t = 0.17700000000000013 t = 0.17800000000000013 t = 0.17900000000000013 t = 0.18000000000000013 t = 0.18100000000000013 t = 0.18200000000000013

more-to-come:

t = 0.18300000000000013 t = 0.18400000000000014 t = 0.18500000000000014 t = 0.18600000000000014 t = 0.18700000000000014 t = 0.18800000000000014 t = 0.18900000000000014

</pre>

t = 0.18300000000000013 t = 0.18400000000000014 t = 0.18500000000000014 t = 0.18600000000000014 t = 0.18700000000000014 t = 0.18800000000000014 t = 0.18900000000000014

end{sphinxVerbatim}

t = 0.18300000000000013 t = 0.18400000000000014 t = 0.18500000000000014 t = 0.18600000000000014 t = 0.18700000000000014 t = 0.18800000000000014 t = 0.18900000000000014

more-to-come:

t = 0.19000000000000014 t = 0.19100000000000014 t = 0.19200000000000014 t = 0.19300000000000014 t = 0.19400000000000014 t = 0.19500000000000015 t = 0.19600000000000015 t = 0.19700000000000015 t = 0.19800000000000015 t = 0.19900000000000015 t = 0.20000000000000015 t = 0.20100000000000015

</pre>

t = 0.19000000000000014 t = 0.19100000000000014 t = 0.19200000000000014 t = 0.19300000000000014 t = 0.19400000000000014 t = 0.19500000000000015 t = 0.19600000000000015 t = 0.19700000000000015 t = 0.19800000000000015 t = 0.19900000000000015 t = 0.20000000000000015 t = 0.20100000000000015

end{sphinxVerbatim}

t = 0.19000000000000014 t = 0.19100000000000014 t = 0.19200000000000014 t = 0.19300000000000014 t = 0.19400000000000014 t = 0.19500000000000015 t = 0.19600000000000015 t = 0.19700000000000015 t = 0.19800000000000015 t = 0.19900000000000015 t = 0.20000000000000015 t = 0.20100000000000015

more-to-come:

t = 0.20200000000000015 t = 0.20300000000000015 t = 0.20400000000000015 t = 0.20500000000000015 t = 0.20600000000000016 t = 0.20700000000000016 t = 0.20800000000000016 t = 0.20900000000000016 t = 0.21000000000000016 t = 0.21100000000000016 t = 0.21200000000000016

</pre>

t = 0.20200000000000015 t = 0.20300000000000015 t = 0.20400000000000015 t = 0.20500000000000015 t = 0.20600000000000016 t = 0.20700000000000016 t = 0.20800000000000016 t = 0.20900000000000016 t = 0.21000000000000016 t = 0.21100000000000016 t = 0.21200000000000016

end{sphinxVerbatim}

t = 0.20200000000000015 t = 0.20300000000000015 t = 0.20400000000000015 t = 0.20500000000000015 t = 0.20600000000000016 t = 0.20700000000000016 t = 0.20800000000000016 t = 0.20900000000000016 t = 0.21000000000000016 t = 0.21100000000000016 t = 0.21200000000000016

more-to-come:

t = 0.21300000000000016 t = 0.21400000000000016 t = 0.21500000000000016 t = 0.21600000000000016 t = 0.21700000000000016 t = 0.21800000000000017 t = 0.21900000000000017 t = 0.22000000000000017 t = 0.22100000000000017 t = 0.22200000000000017 t = 0.22300000000000017

</pre>

t = 0.21300000000000016 t = 0.21400000000000016 t = 0.21500000000000016 t = 0.21600000000000016 t = 0.21700000000000016 t = 0.21800000000000017 t = 0.21900000000000017 t = 0.22000000000000017 t = 0.22100000000000017 t = 0.22200000000000017 t = 0.22300000000000017

end{sphinxVerbatim}

t = 0.21300000000000016 t = 0.21400000000000016 t = 0.21500000000000016 t = 0.21600000000000016 t = 0.21700000000000016 t = 0.21800000000000017 t = 0.21900000000000017 t = 0.22000000000000017 t = 0.22100000000000017 t = 0.22200000000000017 t = 0.22300000000000017

more-to-come:

t = 0.22400000000000017 t = 0.22500000000000017 t = 0.22600000000000017 t = 0.22700000000000017 t = 0.22800000000000017 t = 0.22900000000000018 t = 0.23000000000000018 t = 0.23100000000000018 t = 0.23200000000000018 t = 0.23300000000000018 t = 0.23400000000000018

</pre>

t = 0.22400000000000017 t = 0.22500000000000017 t = 0.22600000000000017 t = 0.22700000000000017 t = 0.22800000000000017 t = 0.22900000000000018 t = 0.23000000000000018 t = 0.23100000000000018 t = 0.23200000000000018 t = 0.23300000000000018 t = 0.23400000000000018

end{sphinxVerbatim}

t = 0.22400000000000017 t = 0.22500000000000017 t = 0.22600000000000017 t = 0.22700000000000017 t = 0.22800000000000017 t = 0.22900000000000018 t = 0.23000000000000018 t = 0.23100000000000018 t = 0.23200000000000018 t = 0.23300000000000018 t = 0.23400000000000018

more-to-come:

t = 0.23500000000000018 t = 0.23600000000000018 t = 0.23700000000000018 t = 0.23800000000000018 t = 0.23900000000000018 t = 0.24000000000000019 t = 0.2410000000000002 t = 0.2420000000000002 t = 0.2430000000000002 t = 0.2440000000000002

</pre>

t = 0.23500000000000018 t = 0.23600000000000018 t = 0.23700000000000018 t = 0.23800000000000018 t = 0.23900000000000018 t = 0.24000000000000019 t = 0.2410000000000002 t = 0.2420000000000002 t = 0.2430000000000002 t = 0.2440000000000002

end{sphinxVerbatim}

t = 0.23500000000000018 t = 0.23600000000000018 t = 0.23700000000000018 t = 0.23800000000000018 t = 0.23900000000000018 t = 0.24000000000000019 t = 0.2410000000000002 t = 0.2420000000000002 t = 0.2430000000000002 t = 0.2440000000000002

more-to-come:

t = 0.2450000000000002 t = 0.2460000000000002 t = 0.2470000000000002 t = 0.2480000000000002 t = 0.2490000000000002 t = 0.25000000000000017

CPU times: user 12.5 s, sys: 5.26 s, total: 17.8 s Wall time: 4.44 s

t = 0.249 t = 0.248 t = 0.247 t = 0.246 t = 0.245

</pre>

t = 0.2450000000000002 t = 0.2460000000000002 t = 0.2470000000000002 t = 0.2480000000000002 t = 0.2490000000000002 t = 0.25000000000000017

CPU times: user 12.5 s, sys: 5.26 s, total: 17.8 s Wall time: 4.44 s

t = 0.249 t = 0.248 t = 0.247 t = 0.246 t = 0.245

end{sphinxVerbatim}

t = 0.2450000000000002 t = 0.2460000000000002 t = 0.2470000000000002 t = 0.2480000000000002 t = 0.2490000000000002 t = 0.25000000000000017

CPU times: user 12.5 s, sys: 5.26 s, total: 17.8 s Wall time: 4.44 s

t = 0.249 t = 0.248 t = 0.247 t = 0.246 t = 0.245

more-to-come:

t = 0.244 t = 0.243 t = 0.242 t = 0.241 t = 0.24 t = 0.239 t = 0.238 t = 0.237 t = 0.236 t = 0.235

</pre>

t = 0.244 t = 0.243 t = 0.242 t = 0.241 t = 0.24 t = 0.239 t = 0.238 t = 0.237 t = 0.236 t = 0.235

end{sphinxVerbatim}

t = 0.244 t = 0.243 t = 0.242 t = 0.241 t = 0.24 t = 0.239 t = 0.238 t = 0.237 t = 0.236 t = 0.235

more-to-come:

t = 0.23399999999999999 t = 0.23299999999999998 t = 0.23199999999999998 t = 0.23099999999999998

</pre>

t = 0.23399999999999999 t = 0.23299999999999998 t = 0.23199999999999998 t = 0.23099999999999998

end{sphinxVerbatim}

t = 0.23399999999999999 t = 0.23299999999999998 t = 0.23199999999999998 t = 0.23099999999999998

more-to-come:

t = 0.22999999999999998 t = 0.22899999999999998 t = 0.22799999999999998 t = 0.22699999999999998 t = 0.22599999999999998 t = 0.22499999999999998 t = 0.22399999999999998 t = 0.22299999999999998 t = 0.22199999999999998 t = 0.22099999999999997 t = 0.21999999999999997 t = 0.21899999999999997

</pre>

t = 0.22999999999999998 t = 0.22899999999999998 t = 0.22799999999999998 t = 0.22699999999999998 t = 0.22599999999999998 t = 0.22499999999999998 t = 0.22399999999999998 t = 0.22299999999999998 t = 0.22199999999999998 t = 0.22099999999999997 t = 0.21999999999999997 t = 0.21899999999999997

end{sphinxVerbatim}

t = 0.22999999999999998 t = 0.22899999999999998 t = 0.22799999999999998 t = 0.22699999999999998 t = 0.22599999999999998 t = 0.22499999999999998 t = 0.22399999999999998 t = 0.22299999999999998 t = 0.22199999999999998 t = 0.22099999999999997 t = 0.21999999999999997 t = 0.21899999999999997

more-to-come:

t = 0.21799999999999997 t = 0.21699999999999997 t = 0.21599999999999997 t = 0.21499999999999997 t = 0.21399999999999997 t = 0.21299999999999997 t = 0.21199999999999997 t = 0.21099999999999997 t = 0.20999999999999996 t = 0.20899999999999996

</pre>

t = 0.21799999999999997 t = 0.21699999999999997 t = 0.21599999999999997 t = 0.21499999999999997 t = 0.21399999999999997 t = 0.21299999999999997 t = 0.21199999999999997 t = 0.21099999999999997 t = 0.20999999999999996 t = 0.20899999999999996

end{sphinxVerbatim}

t = 0.21799999999999997 t = 0.21699999999999997 t = 0.21599999999999997 t = 0.21499999999999997 t = 0.21399999999999997 t = 0.21299999999999997 t = 0.21199999999999997 t = 0.21099999999999997 t = 0.20999999999999996 t = 0.20899999999999996

more-to-come:

t = 0.20799999999999996 t = 0.20699999999999996 t = 0.20599999999999996 t = 0.20499999999999996 t = 0.20399999999999996 t = 0.20299999999999996 t = 0.20199999999999996 t = 0.20099999999999996 t = 0.19999999999999996 t = 0.19899999999999995

</pre>

t = 0.20799999999999996 t = 0.20699999999999996 t = 0.20599999999999996 t = 0.20499999999999996 t = 0.20399999999999996 t = 0.20299999999999996 t = 0.20199999999999996 t = 0.20099999999999996 t = 0.19999999999999996 t = 0.19899999999999995

end{sphinxVerbatim}

t = 0.20799999999999996 t = 0.20699999999999996 t = 0.20599999999999996 t = 0.20499999999999996 t = 0.20399999999999996 t = 0.20299999999999996 t = 0.20199999999999996 t = 0.20099999999999996 t = 0.19999999999999996 t = 0.19899999999999995

more-to-come:

t = 0.19799999999999995 t = 0.19699999999999995 t = 0.19599999999999995 t = 0.19499999999999995 t = 0.19399999999999995 t = 0.19299999999999995 t = 0.19199999999999995 t = 0.19099999999999995 t = 0.18999999999999995 t = 0.18899999999999995 t = 0.18799999999999994

</pre>

t = 0.19799999999999995 t = 0.19699999999999995 t = 0.19599999999999995 t = 0.19499999999999995 t = 0.19399999999999995 t = 0.19299999999999995 t = 0.19199999999999995 t = 0.19099999999999995 t = 0.18999999999999995 t = 0.18899999999999995 t = 0.18799999999999994

end{sphinxVerbatim}

t = 0.19799999999999995 t = 0.19699999999999995 t = 0.19599999999999995 t = 0.19499999999999995 t = 0.19399999999999995 t = 0.19299999999999995 t = 0.19199999999999995 t = 0.19099999999999995 t = 0.18999999999999995 t = 0.18899999999999995 t = 0.18799999999999994

more-to-come:

t = 0.18699999999999994 t = 0.18599999999999994 t = 0.18499999999999994 t = 0.18399999999999994 t = 0.18299999999999994 t = 0.18199999999999994 t = 0.18099999999999994 t = 0.17999999999999994 t = 0.17899999999999994 t = 0.17799999999999994 t = 0.17699999999999994

</pre>

t = 0.18699999999999994 t = 0.18599999999999994 t = 0.18499999999999994 t = 0.18399999999999994 t = 0.18299999999999994 t = 0.18199999999999994 t = 0.18099999999999994 t = 0.17999999999999994 t = 0.17899999999999994 t = 0.17799999999999994 t = 0.17699999999999994

end{sphinxVerbatim}

t = 0.18699999999999994 t = 0.18599999999999994 t = 0.18499999999999994 t = 0.18399999999999994 t = 0.18299999999999994 t = 0.18199999999999994 t = 0.18099999999999994 t = 0.17999999999999994 t = 0.17899999999999994 t = 0.17799999999999994 t = 0.17699999999999994

more-to-come:

t = 0.17599999999999993 t = 0.17499999999999993 t = 0.17399999999999993

</pre>

t = 0.17599999999999993 t = 0.17499999999999993 t = 0.17399999999999993

end{sphinxVerbatim}

t = 0.17599999999999993 t = 0.17499999999999993 t = 0.17399999999999993

more-to-come:

t = 0.17299999999999993 t = 0.17199999999999993 t = 0.17099999999999993 t = 0.16999999999999993 t = 0.16899999999999993 t = 0.16799999999999993 t = 0.16699999999999993 t = 0.16599999999999993 t = 0.16499999999999992 t = 0.16399999999999992 t = 0.16299999999999992

</pre>

t = 0.17299999999999993 t = 0.17199999999999993 t = 0.17099999999999993 t = 0.16999999999999993 t = 0.16899999999999993 t = 0.16799999999999993 t = 0.16699999999999993 t = 0.16599999999999993 t = 0.16499999999999992 t = 0.16399999999999992 t = 0.16299999999999992

end{sphinxVerbatim}

t = 0.17299999999999993 t = 0.17199999999999993 t = 0.17099999999999993 t = 0.16999999999999993 t = 0.16899999999999993 t = 0.16799999999999993 t = 0.16699999999999993 t = 0.16599999999999993 t = 0.16499999999999992 t = 0.16399999999999992 t = 0.16299999999999992

more-to-come:

t = 0.16199999999999992 t = 0.16099999999999992 t = 0.15999999999999992 t = 0.15899999999999992 t = 0.15799999999999992 t = 0.15699999999999992 t = 0.15599999999999992 t = 0.15499999999999992 t = 0.15399999999999991 t = 0.1529999999999999 t = 0.1519999999999999 t = 0.1509999999999999

</pre>

t = 0.16199999999999992 t = 0.16099999999999992 t = 0.15999999999999992 t = 0.15899999999999992 t = 0.15799999999999992 t = 0.15699999999999992 t = 0.15599999999999992 t = 0.15499999999999992 t = 0.15399999999999991 t = 0.1529999999999999 t = 0.1519999999999999 t = 0.1509999999999999

end{sphinxVerbatim}

t = 0.16199999999999992 t = 0.16099999999999992 t = 0.15999999999999992 t = 0.15899999999999992 t = 0.15799999999999992 t = 0.15699999999999992 t = 0.15599999999999992 t = 0.15499999999999992 t = 0.15399999999999991 t = 0.1529999999999999 t = 0.1519999999999999 t = 0.1509999999999999

more-to-come:

t = 0.1499999999999999 t = 0.1489999999999999 t = 0.1479999999999999 t = 0.1469999999999999 t = 0.1459999999999999 t = 0.1449999999999999 t = 0.1439999999999999 t = 0.1429999999999999 t = 0.1419999999999999 t = 0.1409999999999999 t = 0.1399999999999999 t = 0.1389999999999999

</pre>

t = 0.1499999999999999 t = 0.1489999999999999 t = 0.1479999999999999 t = 0.1469999999999999 t = 0.1459999999999999 t = 0.1449999999999999 t = 0.1439999999999999 t = 0.1429999999999999 t = 0.1419999999999999 t = 0.1409999999999999 t = 0.1399999999999999 t = 0.1389999999999999

end{sphinxVerbatim}

t = 0.1499999999999999 t = 0.1489999999999999 t = 0.1479999999999999 t = 0.1469999999999999 t = 0.1459999999999999 t = 0.1449999999999999 t = 0.1439999999999999 t = 0.1429999999999999 t = 0.1419999999999999 t = 0.1409999999999999 t = 0.1399999999999999 t = 0.1389999999999999

more-to-come:

t = 0.1379999999999999 t = 0.1369999999999999 t = 0.1359999999999999 t = 0.1349999999999999 t = 0.1339999999999999 t = 0.1329999999999999 t = 0.1319999999999999 t = 0.1309999999999999 t = 0.1299999999999999 t = 0.1289999999999999 t = 0.1279999999999999 t = 0.1269999999999999

</pre>

t = 0.1379999999999999 t = 0.1369999999999999 t = 0.1359999999999999 t = 0.1349999999999999 t = 0.1339999999999999 t = 0.1329999999999999 t = 0.1319999999999999 t = 0.1309999999999999 t = 0.1299999999999999 t = 0.1289999999999999 t = 0.1279999999999999 t = 0.1269999999999999

end{sphinxVerbatim}

t = 0.1379999999999999 t = 0.1369999999999999 t = 0.1359999999999999 t = 0.1349999999999999 t = 0.1339999999999999 t = 0.1329999999999999 t = 0.1319999999999999 t = 0.1309999999999999 t = 0.1299999999999999 t = 0.1289999999999999 t = 0.1279999999999999 t = 0.1269999999999999

more-to-come:

t = 0.1259999999999999 t = 0.12499999999999992 t = 0.12399999999999992 t = 0.12299999999999991 t = 0.12199999999999991 t = 0.12099999999999991 t = 0.11999999999999991 t = 0.11899999999999991 t = 0.11799999999999991 t = 0.11699999999999991 t = 0.11599999999999991

</pre>

t = 0.1259999999999999 t = 0.12499999999999992 t = 0.12399999999999992 t = 0.12299999999999991 t = 0.12199999999999991 t = 0.12099999999999991 t = 0.11999999999999991 t = 0.11899999999999991 t = 0.11799999999999991 t = 0.11699999999999991 t = 0.11599999999999991

end{sphinxVerbatim}

t = 0.1259999999999999 t = 0.12499999999999992 t = 0.12399999999999992 t = 0.12299999999999991 t = 0.12199999999999991 t = 0.12099999999999991 t = 0.11999999999999991 t = 0.11899999999999991 t = 0.11799999999999991 t = 0.11699999999999991 t = 0.11599999999999991

more-to-come:

t = 0.11499999999999991 t = 0.11399999999999991 t = 0.1129999999999999 t = 0.1119999999999999 t = 0.1109999999999999 t = 0.1099999999999999 t = 0.1089999999999999 t = 0.1079999999999999 t = 0.1069999999999999 t = 0.1059999999999999 t = 0.1049999999999999 t = 0.1039999999999999

</pre>

t = 0.11499999999999991 t = 0.11399999999999991 t = 0.1129999999999999 t = 0.1119999999999999 t = 0.1109999999999999 t = 0.1099999999999999 t = 0.1089999999999999 t = 0.1079999999999999 t = 0.1069999999999999 t = 0.1059999999999999 t = 0.1049999999999999 t = 0.1039999999999999

end{sphinxVerbatim}

t = 0.11499999999999991 t = 0.11399999999999991 t = 0.1129999999999999 t = 0.1119999999999999 t = 0.1109999999999999 t = 0.1099999999999999 t = 0.1089999999999999 t = 0.1079999999999999 t = 0.1069999999999999 t = 0.1059999999999999 t = 0.1049999999999999 t = 0.1039999999999999

more-to-come:

t = 0.1029999999999999 t = 0.1019999999999999 t = 0.1009999999999999 t = 0.0999999999999999 t = 0.0989999999999999 t = 0.09799999999999989 t = 0.09699999999999989 t = 0.09599999999999989 t = 0.09499999999999989 t = 0.09399999999999989 t = 0.09299999999999989 t = 0.09199999999999989

</pre>

t = 0.1029999999999999 t = 0.1019999999999999 t = 0.1009999999999999 t = 0.0999999999999999 t = 0.0989999999999999 t = 0.09799999999999989 t = 0.09699999999999989 t = 0.09599999999999989 t = 0.09499999999999989 t = 0.09399999999999989 t = 0.09299999999999989 t = 0.09199999999999989

end{sphinxVerbatim}

t = 0.1029999999999999 t = 0.1019999999999999 t = 0.1009999999999999 t = 0.0999999999999999 t = 0.0989999999999999 t = 0.09799999999999989 t = 0.09699999999999989 t = 0.09599999999999989 t = 0.09499999999999989 t = 0.09399999999999989 t = 0.09299999999999989 t = 0.09199999999999989

more-to-come:

t = 0.09099999999999989 t = 0.08999999999999989 t = 0.08899999999999988 t = 0.08799999999999988 t = 0.08699999999999988 t = 0.08599999999999988 t = 0.08499999999999988 t = 0.08399999999999988 t = 0.08299999999999988 t = 0.08199999999999988 t = 0.08099999999999988 t = 0.07999999999999988

</pre>

t = 0.09099999999999989 t = 0.08999999999999989 t = 0.08899999999999988 t = 0.08799999999999988 t = 0.08699999999999988 t = 0.08599999999999988 t = 0.08499999999999988 t = 0.08399999999999988 t = 0.08299999999999988 t = 0.08199999999999988 t = 0.08099999999999988 t = 0.07999999999999988

end{sphinxVerbatim}

t = 0.09099999999999989 t = 0.08999999999999989 t = 0.08899999999999988 t = 0.08799999999999988 t = 0.08699999999999988 t = 0.08599999999999988 t = 0.08499999999999988 t = 0.08399999999999988 t = 0.08299999999999988 t = 0.08199999999999988 t = 0.08099999999999988 t = 0.07999999999999988

more-to-come:

t = 0.07899999999999988 t = 0.07799999999999987 t = 0.07699999999999987 t = 0.07599999999999987 t = 0.07499999999999987 t = 0.07399999999999987 t = 0.07299999999999987 t = 0.07199999999999987 t = 0.07099999999999987 t = 0.06999999999999987

</pre>

t = 0.07899999999999988 t = 0.07799999999999987 t = 0.07699999999999987 t = 0.07599999999999987 t = 0.07499999999999987 t = 0.07399999999999987 t = 0.07299999999999987 t = 0.07199999999999987 t = 0.07099999999999987 t = 0.06999999999999987

end{sphinxVerbatim}

t = 0.07899999999999988 t = 0.07799999999999987 t = 0.07699999999999987 t = 0.07599999999999987 t = 0.07499999999999987 t = 0.07399999999999987 t = 0.07299999999999987 t = 0.07199999999999987 t = 0.07099999999999987 t = 0.06999999999999987

more-to-come:

t = 0.06899999999999987 t = 0.06799999999999987 t = 0.06699999999999987

</pre>

t = 0.06899999999999987 t = 0.06799999999999987 t = 0.06699999999999987

end{sphinxVerbatim}

t = 0.06899999999999987 t = 0.06799999999999987 t = 0.06699999999999987

more-to-come:

t = 0.06599999999999986 t = 0.06499999999999986 t = 0.06399999999999986 t = 0.06299999999999986 t = 0.06199999999999986 t = 0.06099999999999986 t = 0.05999999999999986 t = 0.05899999999999986 t = 0.05799999999999986 t = 0.056999999999999856 t = 0.055999999999999855

</pre>

t = 0.06599999999999986 t = 0.06499999999999986 t = 0.06399999999999986 t = 0.06299999999999986 t = 0.06199999999999986 t = 0.06099999999999986 t = 0.05999999999999986 t = 0.05899999999999986 t = 0.05799999999999986 t = 0.056999999999999856 t = 0.055999999999999855

end{sphinxVerbatim}

t = 0.06599999999999986 t = 0.06499999999999986 t = 0.06399999999999986 t = 0.06299999999999986 t = 0.06199999999999986 t = 0.06099999999999986 t = 0.05999999999999986 t = 0.05899999999999986 t = 0.05799999999999986 t = 0.056999999999999856 t = 0.055999999999999855

more-to-come:

t = 0.054999999999999855 t = 0.053999999999999854 t = 0.05299999999999985

</pre>

t = 0.054999999999999855 t = 0.053999999999999854 t = 0.05299999999999985

end{sphinxVerbatim}

t = 0.054999999999999855 t = 0.053999999999999854 t = 0.05299999999999985

more-to-come:

t = 0.05199999999999985 t = 0.05099999999999985 t = 0.04999999999999985 t = 0.04899999999999985 t = 0.04799999999999985 t = 0.04699999999999985 t = 0.04599999999999985 t = 0.044999999999999846 t = 0.043999999999999845 t = 0.042999999999999844 t = 0.04199999999999984 t = 0.04099999999999984

</pre>

t = 0.05199999999999985 t = 0.05099999999999985 t = 0.04999999999999985 t = 0.04899999999999985 t = 0.04799999999999985 t = 0.04699999999999985 t = 0.04599999999999985 t = 0.044999999999999846 t = 0.043999999999999845 t = 0.042999999999999844 t = 0.04199999999999984 t = 0.04099999999999984

end{sphinxVerbatim}

t = 0.05199999999999985 t = 0.05099999999999985 t = 0.04999999999999985 t = 0.04899999999999985 t = 0.04799999999999985 t = 0.04699999999999985 t = 0.04599999999999985 t = 0.044999999999999846 t = 0.043999999999999845 t = 0.042999999999999844 t = 0.04199999999999984 t = 0.04099999999999984

more-to-come:

t = 0.03999999999999984 t = 0.03899999999999984 t = 0.03799999999999984 t = 0.03699999999999984 t = 0.03599999999999984 t = 0.03499999999999984 t = 0.033999999999999836 t = 0.032999999999999835 t = 0.031999999999999834 t = 0.030999999999999833 t = 0.029999999999999832

</pre>

t = 0.03999999999999984 t = 0.03899999999999984 t = 0.03799999999999984 t = 0.03699999999999984 t = 0.03599999999999984 t = 0.03499999999999984 t = 0.033999999999999836 t = 0.032999999999999835 t = 0.031999999999999834 t = 0.030999999999999833 t = 0.029999999999999832

end{sphinxVerbatim}

t = 0.03999999999999984 t = 0.03899999999999984 t = 0.03799999999999984 t = 0.03699999999999984 t = 0.03599999999999984 t = 0.03499999999999984 t = 0.033999999999999836 t = 0.032999999999999835 t = 0.031999999999999834 t = 0.030999999999999833 t = 0.029999999999999832

more-to-come:

t = 0.02899999999999983 t = 0.02799999999999983 t = 0.02699999999999983 t = 0.02599999999999983 t = 0.024999999999999828 t = 0.023999999999999827 t = 0.022999999999999826 t = 0.021999999999999825 t = 0.020999999999999824 t = 0.019999999999999823 t = 0.018999999999999823

</pre>

t = 0.02899999999999983 t = 0.02799999999999983 t = 0.02699999999999983 t = 0.02599999999999983 t = 0.024999999999999828 t = 0.023999999999999827 t = 0.022999999999999826 t = 0.021999999999999825 t = 0.020999999999999824 t = 0.019999999999999823 t = 0.018999999999999823

end{sphinxVerbatim}

t = 0.02899999999999983 t = 0.02799999999999983 t = 0.02699999999999983 t = 0.02599999999999983 t = 0.024999999999999828 t = 0.023999999999999827 t = 0.022999999999999826 t = 0.021999999999999825 t = 0.020999999999999824 t = 0.019999999999999823 t = 0.018999999999999823

more-to-come:

t = 0.01799999999999982 t = 0.01699999999999982 t = 0.01599999999999982 t = 0.014999999999999819

</pre>

t = 0.01799999999999982 t = 0.01699999999999982 t = 0.01599999999999982 t = 0.014999999999999819

end{sphinxVerbatim}

t = 0.01799999999999982 t = 0.01699999999999982 t = 0.01599999999999982 t = 0.014999999999999819

more-to-come:

t = 0.013999999999999818 t = 0.012999999999999817 t = 0.011999999999999816 t = 0.010999999999999815 t = 0.009999999999999815 t = 0.008999999999999814 t = 0.007999999999999813 t = 0.006999999999999812 t = 0.005999999999999811 t = 0.00499999999999981 t = 0.003999999999999809

</pre>

t = 0.013999999999999818 t = 0.012999999999999817 t = 0.011999999999999816 t = 0.010999999999999815 t = 0.009999999999999815 t = 0.008999999999999814 t = 0.007999999999999813 t = 0.006999999999999812 t = 0.005999999999999811 t = 0.00499999999999981 t = 0.003999999999999809

end{sphinxVerbatim}

t = 0.013999999999999818 t = 0.012999999999999817 t = 0.011999999999999816 t = 0.010999999999999815 t = 0.009999999999999815 t = 0.008999999999999814 t = 0.007999999999999813 t = 0.006999999999999812 t = 0.005999999999999811 t = 0.00499999999999981 t = 0.003999999999999809

t = 0.0029999999999998084 t = 0.0019999999999998075 t = 0.0009999999999998066 t = -1.6653345369377348e-16

CPU times: user 12.7 s, sys: 5.04 s, total: 17.7 s Wall time: 4.44 s </pre>

t = 0.0029999999999998084 t = 0.0019999999999998075 t = 0.0009999999999998066 t = -1.6653345369377348e-16

CPU times: user 12.7 s, sys: 5.04 s, total: 17.7 s Wall time: 4.44 s end{sphinxVerbatim}

t = 0.0029999999999998084 t = 0.0019999999999998075 t = 0.0009999999999998066 t = -1.6653345369377348e-16

CPU times: user 12.7 s, sys: 5.04 s, total: 17.7 s Wall time: 4.44 s

Version 3:

Using the TraceOperator and geom_free=True with assembling

With \(p(x) = \hat{p}(\hat{x})\) and \(\mathbf{v}(x) = \frac{1}{|\operatorname{det}(F)|} F \cdot \hat{\mathbf{v}}(\hat{x})\) (Piola mapped) there holds

\begin{align} \int_T \mathbf{v} \cdot \nabla p = \int_{\hat{T}} \hat{\mathbf{v}} \cdot \hat{\nabla} \hat{p} \end{align}

and similarly for the facet integrals, cf. unit-2.11 and

Integrals are independent of the "physical element" (up to ordering)

\(\leadsto\) same element matrices for a large class of elements

Bel = BilinearForm(trialspace=fes_p, testspace=fes_u, geom_free = True)
Bel += grad(p)*v * dx -p*(v*n) * dx(element_boundary=True)
%time Bel.Assemble()

Btr = BilinearForm(trialspace=fes_tr, testspace=fes_u, geom_free = True)
Btr += 0.5 * phat * (v*n) *dx(element_boundary=True)
%time Btr.Assemble()

B = emb_u @ (Bel.mat + Btr.mat @ traceop) @ emb_p.T

CPU times: user 1.93 ms, sys: 419 µs, total: 2.35 ms
Wall time: 1.28 ms
CPU times: user 1.91 ms, sys: 0 ns, total: 1.91 ms
Wall time: 1.7 ms

Version 3 in action:

%time Run(B, B.T, backward=False)
%time Run(B, B.T, backward=True)

t = 0.001 t = 0.002 t = 0.003 t = 0.004 t = 0.005 t = 0.006 t = 0.007 t = 0.008 t = 0.009000000000000001 t = 0.010000000000000002 t = 0.011000000000000003 t = 0.012000000000000004 t = 0.013000000000000005 t = 0.014000000000000005 t = 0.015000000000000006 t = 0.016000000000000007 t = 0.017000000000000008 t = 0.01800000000000001 t = 0.01900000000000001 t = 0.02000000000000001 t = 0.02100000000000001 t = 0.022000000000000013 t = 0.023000000000000013 t = 0.024000000000000014 t = 0.025000000000000015 t = 0.026000000000000016 t = 0.027000000000000017 t = 0.028000000000000018 t = 0.02900000000000002 t = 0.03000000000000002 t = 0.03100000000000002 t = 0.03200000000000002 t = 0.03300000000000002 t = 0.03400000000000002 t = 0.035000000000000024 t = 0.036000000000000025 t = 0.037000000000000026 t = 0.03800000000000003 t = 0.03900000000000003 t = 0.04000000000000003 t = 0.04100000000000003 t = 0.04200000000000003 t = 0.04300000000000003 t = 0.04400000000000003 t = 0.04500000000000003 t = 0.046000000000000034 t = 0.047000000000000035 t = 0.048000000000000036 t = 0.04900000000000004 t = 0.05000000000000004 t = 0.05100000000000004 t = 0.05200000000000004 t = 0.05300000000000004 t = 0.05400000000000004 t = 0.05500000000000004 t = 0.05600000000000004 t = 0.057000000000000044 t = 0.058000000000000045 t = 0.059000000000000045 t = 0.060000000000000046 t = 0.06100000000000005 t = 0.06200000000000005 t = 0.06300000000000004 t = 0.06400000000000004 t = 0.06500000000000004 t = 0.06600000000000004 t = 0.06700000000000005 t = 0.06800000000000005 t = 0.06900000000000005 t = 0.07000000000000005 t = 0.07100000000000005 t = 0.07200000000000005 t = 0.07300000000000005 t = 0.07400000000000005 t = 0.07500000000000005 t = 0.07600000000000005 t = 0.07700000000000005 t = 0.07800000000000006 t = 0.07900000000000006 t = 0.08000000000000006 t = 0.08100000000000006 t = 0.08200000000000006 t = 0.08300000000000006 t = 0.08400000000000006 t = 0.08500000000000006 t = 0.08600000000000006 t = 0.08700000000000006 t = 0.08800000000000006 t = 0.08900000000000007 t = 0.09000000000000007 t = 0.09100000000000007 t = 0.09200000000000007 t = 0.09300000000000007 t = 0.09400000000000007 t = 0.09500000000000007 t = 0.09600000000000007 t = 0.09700000000000007 t = 0.09800000000000007 t = 0.09900000000000007 t = 0.10000000000000007 t = 0.10100000000000008 t = 0.10200000000000008 t = 0.10300000000000008 t = 0.10400000000000008 t = 0.10500000000000008 t = 0.10600000000000008 t = 0.10700000000000008 t = 0.10800000000000008 t = 0.10900000000000008 t = 0.11000000000000008 t = 0.11100000000000008 t = 0.11200000000000009 t = 0.11300000000000009 t = 0.11400000000000009 t = 0.11500000000000009 t = 0.11600000000000009 t = 0.11700000000000009 t = 0.11800000000000009 t = 0.11900000000000009 t = 0.12000000000000009 t = 0.1210000000000001 t = 0.1220000000000001 t = 0.1230000000000001 t = 0.1240000000000001 t = 0.12500000000000008 t = 0.12600000000000008 t = 0.12700000000000009 t = 0.12800000000000009 t = 0.1290000000000001 t = 0.1300000000000001 t = 0.1310000000000001 t = 0.1320000000000001 t = 0.1330000000000001 t = 0.1340000000000001 t = 0.1350000000000001 t = 0.1360000000000001 t = 0.1370000000000001 t = 0.1380000000000001 t = 0.1390000000000001 t = 0.1400000000000001 t = 0.1410000000000001 t = 0.1420000000000001 t = 0.1430000000000001 t = 0.1440000000000001 t = 0.1450000000000001 t = 0.1460000000000001 t = 0.1470000000000001 t = 0.1480000000000001 t = 0.1490000000000001 t = 0.1500000000000001 t = 0.1510000000000001 t = 0.1520000000000001 t = 0.1530000000000001 t = 0.1540000000000001 t = 0.1550000000000001 t = 0.1560000000000001 t = 0.1570000000000001 t = 0.1580000000000001 t = 0.1590000000000001 t = 0.16000000000000011 t = 0.16100000000000012 t = 0.16200000000000012 t = 0.16300000000000012 t = 0.16400000000000012 t = 0.16500000000000012 t = 0.16600000000000012 t = 0.16700000000000012 t = 0.16800000000000012 t = 0.16900000000000012 t = 0.17000000000000012 t = 0.17100000000000012 t = 0.17200000000000013 t = 0.17300000000000013 t = 0.17400000000000013 t = 0.17500000000000013 t = 0.17600000000000013 t = 0.17700000000000013 t = 0.17800000000000013 t = 0.17900000000000013 t = 0.18000000000000013 t = 0.18100000000000013 t = 0.18200000000000013 t = 0.18300000000000013 t = 0.18400000000000014 t = 0.18500000000000014 t = 0.18600000000000014 t = 0.18700000000000014 t = 0.18800000000000014 t = 0.18900000000000014 t = 0.19000000000000014 t = 0.19100000000000014 t = 0.19200000000000014 t = 0.19300000000000014 t = 0.19400000000000014 t = 0.19500000000000015 t = 0.19600000000000015 t = 0.19700000000000015 t = 0.19800000000000015 t = 0.19900000000000015 t = 0.20000000000000015 t = 0.20100000000000015 t = 0.20200000000000015 t = 0.20300000000000015 t = 0.20400000000000015 t = 0.20500000000000015 t = 0.20600000000000016 t = 0.20700000000000016 t = 0.20800000000000016 t = 0.20900000000000016 t = 0.21000000000000016 t = 0.21100000000000016 t = 0.21200000000000016 t = 0.21300000000000016 t = 0.21400000000000016 t = 0.21500000000000016 t = 0.21600000000000016 t = 0.21700000000000016 t = 0.21800000000000017 t = 0.21900000000000017 t = 0.22000000000000017 t = 0.22100000000000017 t = 0.22200000000000017 t = 0.22300000000000017 t = 0.22400000000000017 t = 0.22500000000000017 t = 0.22600000000000017 t = 0.22700000000000017 t = 0.22800000000000017 t = 0.22900000000000018 t = 0.23000000000000018 t = 0.23100000000000018 t = 0.23200000000000018 t = 0.23300000000000018 t = 0.23400000000000018 t = 0.23500000000000018 t = 0.23600000000000018 t = 0.23700000000000018 t = 0.23800000000000018 t = 0.23900000000000018 t = 0.24000000000000019 t = 0.2410000000000002 t = 0.2420000000000002 t = 0.2430000000000002 t = 0.2440000000000002 t = 0.2450000000000002 t = 0.2460000000000002 t = 0.2470000000000002 t = 0.2480000000000002 t = 0.2490000000000002 t = 0.25000000000000017

CPU times: user 3.2 s, sys: 132 ms, total: 3.33 s Wall time: 831 ms

t = 0.249 t = 0.248 t = 0.247 t = 0.246 t = 0.245 t = 0.244 t = 0.243 t = 0.242 t = 0.241 t = 0.24 t = 0.239 t = 0.238 t = 0.237 t = 0.236 t = 0.235 t = 0.23399999999999999 t = 0.23299999999999998 t = 0.23199999999999998 t = 0.23099999999999998 t = 0.22999999999999998 t = 0.22899999999999998 t = 0.22799999999999998 t = 0.22699999999999998 t = 0.22599999999999998 t = 0.22499999999999998 t = 0.22399999999999998 t = 0.22299999999999998 t = 0.22199999999999998 t = 0.22099999999999997 t = 0.21999999999999997 t = 0.21899999999999997 t = 0.21799999999999997 t = 0.21699999999999997 t = 0.21599999999999997 t = 0.21499999999999997 t = 0.21399999999999997 t = 0.21299999999999997 t = 0.21199999999999997 t = 0.21099999999999997 t = 0.20999999999999996 t = 0.20899999999999996 t = 0.20799999999999996 t = 0.20699999999999996 t = 0.20599999999999996 t = 0.20499999999999996 t = 0.20399999999999996 t = 0.20299999999999996 t = 0.20199999999999996 t = 0.20099999999999996 t = 0.19999999999999996 t = 0.19899999999999995 t = 0.19799999999999995 t = 0.19699999999999995 t = 0.19599999999999995 t = 0.19499999999999995 t = 0.19399999999999995 t = 0.19299999999999995 t = 0.19199999999999995 t = 0.19099999999999995 t = 0.18999999999999995 t = 0.18899999999999995 t = 0.18799999999999994 t = 0.18699999999999994 t = 0.18599999999999994 t = 0.18499999999999994 t = 0.18399999999999994 t = 0.18299999999999994 t = 0.18199999999999994 t = 0.18099999999999994 t = 0.17999999999999994 t = 0.17899999999999994 t = 0.17799999999999994 t = 0.17699999999999994 t = 0.17599999999999993 t = 0.17499999999999993 t = 0.17399999999999993 t = 0.17299999999999993 t = 0.17199999999999993 t = 0.17099999999999993 t = 0.16999999999999993 t = 0.16899999999999993 t = 0.16799999999999993 t = 0.16699999999999993 t = 0.16599999999999993 t = 0.16499999999999992 t = 0.16399999999999992 t = 0.16299999999999992 t = 0.16199999999999992 t = 0.16099999999999992 t = 0.15999999999999992 t = 0.15899999999999992 t = 0.15799999999999992 t = 0.15699999999999992 t = 0.15599999999999992 t = 0.15499999999999992 t = 0.15399999999999991 t = 0.1529999999999999 t = 0.1519999999999999 t = 0.1509999999999999 t = 0.1499999999999999 t = 0.1489999999999999 t = 0.1479999999999999 t = 0.1469999999999999 t = 0.1459999999999999 t = 0.1449999999999999 t = 0.1439999999999999 t = 0.1429999999999999 t = 0.1419999999999999 t = 0.1409999999999999 t = 0.1399999999999999 t = 0.1389999999999999 t = 0.1379999999999999 t = 0.1369999999999999 t = 0.1359999999999999 t = 0.1349999999999999 t = 0.1339999999999999 t = 0.1329999999999999 t = 0.1319999999999999 t = 0.1309999999999999 t = 0.1299999999999999 t = 0.1289999999999999 t = 0.1279999999999999 t = 0.1269999999999999 t = 0.1259999999999999 t = 0.12499999999999992 t = 0.12399999999999992 t = 0.12299999999999991 t = 0.12199999999999991 t = 0.12099999999999991 t = 0.11999999999999991 t = 0.11899999999999991 t = 0.11799999999999991 t = 0.11699999999999991 t = 0.11599999999999991 t = 0.11499999999999991 t = 0.11399999999999991 t = 0.1129999999999999 t = 0.1119999999999999 t = 0.1109999999999999 t = 0.1099999999999999 t = 0.1089999999999999 t = 0.1079999999999999 t = 0.1069999999999999 t = 0.1059999999999999 t = 0.1049999999999999 t = 0.1039999999999999 t = 0.1029999999999999 t = 0.1019999999999999 t = 0.1009999999999999 t = 0.0999999999999999 t = 0.0989999999999999 t = 0.09799999999999989 t = 0.09699999999999989 t = 0.09599999999999989 t = 0.09499999999999989 t = 0.09399999999999989 t = 0.09299999999999989 t = 0.09199999999999989 t = 0.09099999999999989 t = 0.08999999999999989 t = 0.08899999999999988 t = 0.08799999999999988 t = 0.08699999999999988 t = 0.08599999999999988 t = 0.08499999999999988 t = 0.08399999999999988 t = 0.08299999999999988 t = 0.08199999999999988 t = 0.08099999999999988 t = 0.07999999999999988 t = 0.07899999999999988 t = 0.07799999999999987 t = 0.07699999999999987 t = 0.07599999999999987 t = 0.07499999999999987 t = 0.07399999999999987 t = 0.07299999999999987 t = 0.07199999999999987 t = 0.07099999999999987 t = 0.06999999999999987 t = 0.06899999999999987 t = 0.06799999999999987 t = 0.06699999999999987 t = 0.06599999999999986 t = 0.06499999999999986 t = 0.06399999999999986 t = 0.06299999999999986 t = 0.06199999999999986 t = 0.06099999999999986 t = 0.05999999999999986 t = 0.05899999999999986 t = 0.05799999999999986 t = 0.056999999999999856 t = 0.055999999999999855 t = 0.054999999999999855 t = 0.053999999999999854 t = 0.05299999999999985 t = 0.05199999999999985 t = 0.05099999999999985 t = 0.04999999999999985 t = 0.04899999999999985 t = 0.04799999999999985 t = 0.04699999999999985 t = 0.04599999999999985 t = 0.044999999999999846 t = 0.043999999999999845 t = 0.042999999999999844 t = 0.04199999999999984 t = 0.04099999999999984 t = 0.03999999999999984 t = 0.03899999999999984 t = 0.03799999999999984 t = 0.03699999999999984 t = 0.03599999999999984 t = 0.03499999999999984 t = 0.033999999999999836 t = 0.032999999999999835 t = 0.031999999999999834 t = 0.030999999999999833 t = 0.029999999999999832 t = 0.02899999999999983 t = 0.02799999999999983 t = 0.02699999999999983 t = 0.02599999999999983 t = 0.024999999999999828 t = 0.023999999999999827 t = 0.022999999999999826 t = 0.021999999999999825 t = 0.020999999999999824 t = 0.019999999999999823 t = 0.018999999999999823 t = 0.01799999999999982 t = 0.01699999999999982 t = 0.01599999999999982 t = 0.014999999999999819 t = 0.013999999999999818 t = 0.012999999999999817 t = 0.011999999999999816 t = 0.010999999999999815 t = 0.009999999999999815 t = 0.008999999999999814 t = 0.007999999999999813 t = 0.006999999999999812 t = 0.005999999999999811 t = 0.00499999999999981 t = 0.003999999999999809 t = 0.0029999999999998084 t = 0.0019999999999998075 t = 0.0009999999999998066 t = -1.6653345369377348e-16

CPU times: user 3.23 s, sys: 82.9 ms, total: 3.31 s Wall time: 827 ms </pre>

t = 0.001 t = 0.002 t = 0.003 t = 0.004 t = 0.005 t = 0.006 t = 0.007 t = 0.008 t = 0.009000000000000001 t = 0.010000000000000002 t = 0.011000000000000003 t = 0.012000000000000004 t = 0.013000000000000005 t = 0.014000000000000005 t = 0.015000000000000006 t = 0.016000000000000007 t = 0.017000000000000008 t = 0.01800000000000001 t = 0.01900000000000001 t = 0.02000000000000001 t = 0.02100000000000001 t = 0.022000000000000013 t = 0.023000000000000013 t = 0.024000000000000014 t = 0.025000000000000015 t = 0.026000000000000016 t = 0.027000000000000017 t = 0.028000000000000018 t = 0.02900000000000002 t = 0.03000000000000002 t = 0.03100000000000002 t = 0.03200000000000002 t = 0.03300000000000002 t = 0.03400000000000002 t = 0.035000000000000024 t = 0.036000000000000025 t = 0.037000000000000026 t = 0.03800000000000003 t = 0.03900000000000003 t = 0.04000000000000003 t = 0.04100000000000003 t = 0.04200000000000003 t = 0.04300000000000003 t = 0.04400000000000003 t = 0.04500000000000003 t = 0.046000000000000034 t = 0.047000000000000035 t = 0.048000000000000036 t = 0.04900000000000004 t = 0.05000000000000004 t = 0.05100000000000004 t = 0.05200000000000004 t = 0.05300000000000004 t = 0.05400000000000004 t = 0.05500000000000004 t = 0.05600000000000004 t = 0.057000000000000044 t = 0.058000000000000045 t = 0.059000000000000045 t = 0.060000000000000046 t = 0.06100000000000005 t = 0.06200000000000005 t = 0.06300000000000004 t = 0.06400000000000004 t = 0.06500000000000004 t = 0.06600000000000004 t = 0.06700000000000005 t = 0.06800000000000005 t = 0.06900000000000005 t = 0.07000000000000005 t = 0.07100000000000005 t = 0.07200000000000005 t = 0.07300000000000005 t = 0.07400000000000005 t = 0.07500000000000005 t = 0.07600000000000005 t = 0.07700000000000005 t = 0.07800000000000006 t = 0.07900000000000006 t = 0.08000000000000006 t = 0.08100000000000006 t = 0.08200000000000006 t = 0.08300000000000006 t = 0.08400000000000006 t = 0.08500000000000006 t = 0.08600000000000006 t = 0.08700000000000006 t = 0.08800000000000006 t = 0.08900000000000007 t = 0.09000000000000007 t = 0.09100000000000007 t = 0.09200000000000007 t = 0.09300000000000007 t = 0.09400000000000007 t = 0.09500000000000007 t = 0.09600000000000007 t = 0.09700000000000007 t = 0.09800000000000007 t = 0.09900000000000007 t = 0.10000000000000007 t = 0.10100000000000008 t = 0.10200000000000008 t = 0.10300000000000008 t = 0.10400000000000008 t = 0.10500000000000008 t = 0.10600000000000008 t = 0.10700000000000008 t = 0.10800000000000008 t = 0.10900000000000008 t = 0.11000000000000008 t = 0.11100000000000008 t = 0.11200000000000009 t = 0.11300000000000009 t = 0.11400000000000009 t = 0.11500000000000009 t = 0.11600000000000009 t = 0.11700000000000009 t = 0.11800000000000009 t = 0.11900000000000009 t = 0.12000000000000009 t = 0.1210000000000001 t = 0.1220000000000001 t = 0.1230000000000001 t = 0.1240000000000001 t = 0.12500000000000008 t = 0.12600000000000008 t = 0.12700000000000009 t = 0.12800000000000009 t = 0.1290000000000001 t = 0.1300000000000001 t = 0.1310000000000001 t = 0.1320000000000001 t = 0.1330000000000001 t = 0.1340000000000001 t = 0.1350000000000001 t = 0.1360000000000001 t = 0.1370000000000001 t = 0.1380000000000001 t = 0.1390000000000001 t = 0.1400000000000001 t = 0.1410000000000001 t = 0.1420000000000001 t = 0.1430000000000001 t = 0.1440000000000001 t = 0.1450000000000001 t = 0.1460000000000001 t = 0.1470000000000001 t = 0.1480000000000001 t = 0.1490000000000001 t = 0.1500000000000001 t = 0.1510000000000001 t = 0.1520000000000001 t = 0.1530000000000001 t = 0.1540000000000001 t = 0.1550000000000001 t = 0.1560000000000001 t = 0.1570000000000001 t = 0.1580000000000001 t = 0.1590000000000001 t = 0.16000000000000011 t = 0.16100000000000012 t = 0.16200000000000012 t = 0.16300000000000012 t = 0.16400000000000012 t = 0.16500000000000012 t = 0.16600000000000012 t = 0.16700000000000012 t = 0.16800000000000012 t = 0.16900000000000012 t = 0.17000000000000012 t = 0.17100000000000012 t = 0.17200000000000013 t = 0.17300000000000013 t = 0.17400000000000013 t = 0.17500000000000013 t = 0.17600000000000013 t = 0.17700000000000013 t = 0.17800000000000013 t = 0.17900000000000013 t = 0.18000000000000013 t = 0.18100000000000013 t = 0.18200000000000013 t = 0.18300000000000013 t = 0.18400000000000014 t = 0.18500000000000014 t = 0.18600000000000014 t = 0.18700000000000014 t = 0.18800000000000014 t = 0.18900000000000014 t = 0.19000000000000014 t = 0.19100000000000014 t = 0.19200000000000014 t = 0.19300000000000014 t = 0.19400000000000014 t = 0.19500000000000015 t = 0.19600000000000015 t = 0.19700000000000015 t = 0.19800000000000015 t = 0.19900000000000015 t = 0.20000000000000015 t = 0.20100000000000015 t = 0.20200000000000015 t = 0.20300000000000015 t = 0.20400000000000015 t = 0.20500000000000015 t = 0.20600000000000016 t = 0.20700000000000016 t = 0.20800000000000016 t = 0.20900000000000016 t = 0.21000000000000016 t = 0.21100000000000016 t = 0.21200000000000016 t = 0.21300000000000016 t = 0.21400000000000016 t = 0.21500000000000016 t = 0.21600000000000016 t = 0.21700000000000016 t = 0.21800000000000017 t = 0.21900000000000017 t = 0.22000000000000017 t = 0.22100000000000017 t = 0.22200000000000017 t = 0.22300000000000017 t = 0.22400000000000017 t = 0.22500000000000017 t = 0.22600000000000017 t = 0.22700000000000017 t = 0.22800000000000017 t = 0.22900000000000018 t = 0.23000000000000018 t = 0.23100000000000018 t = 0.23200000000000018 t = 0.23300000000000018 t = 0.23400000000000018 t = 0.23500000000000018 t = 0.23600000000000018 t = 0.23700000000000018 t = 0.23800000000000018 t = 0.23900000000000018 t = 0.24000000000000019 t = 0.2410000000000002 t = 0.2420000000000002 t = 0.2430000000000002 t = 0.2440000000000002 t = 0.2450000000000002 t = 0.2460000000000002 t = 0.2470000000000002 t = 0.2480000000000002 t = 0.2490000000000002 t = 0.25000000000000017

CPU times: user 3.2 s, sys: 132 ms, total: 3.33 s Wall time: 831 ms

t = 0.249 t = 0.248 t = 0.247 t = 0.246 t = 0.245 t = 0.244 t = 0.243 t = 0.242 t = 0.241 t = 0.24 t = 0.239 t = 0.238 t = 0.237 t = 0.236 t = 0.235 t = 0.23399999999999999 t = 0.23299999999999998 t = 0.23199999999999998 t = 0.23099999999999998 t = 0.22999999999999998 t = 0.22899999999999998 t = 0.22799999999999998 t = 0.22699999999999998 t = 0.22599999999999998 t = 0.22499999999999998 t = 0.22399999999999998 t = 0.22299999999999998 t = 0.22199999999999998 t = 0.22099999999999997 t = 0.21999999999999997 t = 0.21899999999999997 t = 0.21799999999999997 t = 0.21699999999999997 t = 0.21599999999999997 t = 0.21499999999999997 t = 0.21399999999999997 t = 0.21299999999999997 t = 0.21199999999999997 t = 0.21099999999999997 t = 0.20999999999999996 t = 0.20899999999999996 t = 0.20799999999999996 t = 0.20699999999999996 t = 0.20599999999999996 t = 0.20499999999999996 t = 0.20399999999999996 t = 0.20299999999999996 t = 0.20199999999999996 t = 0.20099999999999996 t = 0.19999999999999996 t = 0.19899999999999995 t = 0.19799999999999995 t = 0.19699999999999995 t = 0.19599999999999995 t = 0.19499999999999995 t = 0.19399999999999995 t = 0.19299999999999995 t = 0.19199999999999995 t = 0.19099999999999995 t = 0.18999999999999995 t = 0.18899999999999995 t = 0.18799999999999994 t = 0.18699999999999994 t = 0.18599999999999994 t = 0.18499999999999994 t = 0.18399999999999994 t = 0.18299999999999994 t = 0.18199999999999994 t = 0.18099999999999994 t = 0.17999999999999994 t = 0.17899999999999994 t = 0.17799999999999994 t = 0.17699999999999994 t = 0.17599999999999993 t = 0.17499999999999993 t = 0.17399999999999993 t = 0.17299999999999993 t = 0.17199999999999993 t = 0.17099999999999993 t = 0.16999999999999993 t = 0.16899999999999993 t = 0.16799999999999993 t = 0.16699999999999993 t = 0.16599999999999993 t = 0.16499999999999992 t = 0.16399999999999992 t = 0.16299999999999992 t = 0.16199999999999992 t = 0.16099999999999992 t = 0.15999999999999992 t = 0.15899999999999992 t = 0.15799999999999992 t = 0.15699999999999992 t = 0.15599999999999992 t = 0.15499999999999992 t = 0.15399999999999991 t = 0.1529999999999999 t = 0.1519999999999999 t = 0.1509999999999999 t = 0.1499999999999999 t = 0.1489999999999999 t = 0.1479999999999999 t = 0.1469999999999999 t = 0.1459999999999999 t = 0.1449999999999999 t = 0.1439999999999999 t = 0.1429999999999999 t = 0.1419999999999999 t = 0.1409999999999999 t = 0.1399999999999999 t = 0.1389999999999999 t = 0.1379999999999999 t = 0.1369999999999999 t = 0.1359999999999999 t = 0.1349999999999999 t = 0.1339999999999999 t = 0.1329999999999999 t = 0.1319999999999999 t = 0.1309999999999999 t = 0.1299999999999999 t = 0.1289999999999999 t = 0.1279999999999999 t = 0.1269999999999999 t = 0.1259999999999999 t = 0.12499999999999992 t = 0.12399999999999992 t = 0.12299999999999991 t = 0.12199999999999991 t = 0.12099999999999991 t = 0.11999999999999991 t = 0.11899999999999991 t = 0.11799999999999991 t = 0.11699999999999991 t = 0.11599999999999991 t = 0.11499999999999991 t = 0.11399999999999991 t = 0.1129999999999999 t = 0.1119999999999999 t = 0.1109999999999999 t = 0.1099999999999999 t = 0.1089999999999999 t = 0.1079999999999999 t = 0.1069999999999999 t = 0.1059999999999999 t = 0.1049999999999999 t = 0.1039999999999999 t = 0.1029999999999999 t = 0.1019999999999999 t = 0.1009999999999999 t = 0.0999999999999999 t = 0.0989999999999999 t = 0.09799999999999989 t = 0.09699999999999989 t = 0.09599999999999989 t = 0.09499999999999989 t = 0.09399999999999989 t = 0.09299999999999989 t = 0.09199999999999989 t = 0.09099999999999989 t = 0.08999999999999989 t = 0.08899999999999988 t = 0.08799999999999988 t = 0.08699999999999988 t = 0.08599999999999988 t = 0.08499999999999988 t = 0.08399999999999988 t = 0.08299999999999988 t = 0.08199999999999988 t = 0.08099999999999988 t = 0.07999999999999988 t = 0.07899999999999988 t = 0.07799999999999987 t = 0.07699999999999987 t = 0.07599999999999987 t = 0.07499999999999987 t = 0.07399999999999987 t = 0.07299999999999987 t = 0.07199999999999987 t = 0.07099999999999987 t = 0.06999999999999987 t = 0.06899999999999987 t = 0.06799999999999987 t = 0.06699999999999987 t = 0.06599999999999986 t = 0.06499999999999986 t = 0.06399999999999986 t = 0.06299999999999986 t = 0.06199999999999986 t = 0.06099999999999986 t = 0.05999999999999986 t = 0.05899999999999986 t = 0.05799999999999986 t = 0.056999999999999856 t = 0.055999999999999855 t = 0.054999999999999855 t = 0.053999999999999854 t = 0.05299999999999985 t = 0.05199999999999985 t = 0.05099999999999985 t = 0.04999999999999985 t = 0.04899999999999985 t = 0.04799999999999985 t = 0.04699999999999985 t = 0.04599999999999985 t = 0.044999999999999846 t = 0.043999999999999845 t = 0.042999999999999844 t = 0.04199999999999984 t = 0.04099999999999984 t = 0.03999999999999984 t = 0.03899999999999984 t = 0.03799999999999984 t = 0.03699999999999984 t = 0.03599999999999984 t = 0.03499999999999984 t = 0.033999999999999836 t = 0.032999999999999835 t = 0.031999999999999834 t = 0.030999999999999833 t = 0.029999999999999832 t = 0.02899999999999983 t = 0.02799999999999983 t = 0.02699999999999983 t = 0.02599999999999983 t = 0.024999999999999828 t = 0.023999999999999827 t = 0.022999999999999826 t = 0.021999999999999825 t = 0.020999999999999824 t = 0.019999999999999823 t = 0.018999999999999823 t = 0.01799999999999982 t = 0.01699999999999982 t = 0.01599999999999982 t = 0.014999999999999819 t = 0.013999999999999818 t = 0.012999999999999817 t = 0.011999999999999816 t = 0.010999999999999815 t = 0.009999999999999815 t = 0.008999999999999814 t = 0.007999999999999813 t = 0.006999999999999812 t = 0.005999999999999811 t = 0.00499999999999981 t = 0.003999999999999809 t = 0.0029999999999998084 t = 0.0019999999999998075 t = 0.0009999999999998066 t = -1.6653345369377348e-16

CPU times: user 3.23 s, sys: 82.9 ms, total: 3.31 s Wall time: 827 ms end{sphinxVerbatim}

t = 0.001 t = 0.002 t = 0.003 t = 0.004 t = 0.005 t = 0.006 t = 0.007 t = 0.008 t = 0.009000000000000001 t = 0.010000000000000002 t = 0.011000000000000003 t = 0.012000000000000004 t = 0.013000000000000005 t = 0.014000000000000005 t = 0.015000000000000006 t = 0.016000000000000007 t = 0.017000000000000008 t = 0.01800000000000001 t = 0.01900000000000001 t = 0.02000000000000001 t = 0.02100000000000001 t = 0.022000000000000013 t = 0.023000000000000013 t = 0.024000000000000014 t = 0.025000000000000015 t = 0.026000000000000016 t = 0.027000000000000017 t = 0.028000000000000018 t = 0.02900000000000002 t = 0.03000000000000002 t = 0.03100000000000002 t = 0.03200000000000002 t = 0.03300000000000002 t = 0.03400000000000002 t = 0.035000000000000024 t = 0.036000000000000025 t = 0.037000000000000026 t = 0.03800000000000003 t = 0.03900000000000003 t = 0.04000000000000003 t = 0.04100000000000003 t = 0.04200000000000003 t = 0.04300000000000003 t = 0.04400000000000003 t = 0.04500000000000003 t = 0.046000000000000034 t = 0.047000000000000035 t = 0.048000000000000036 t = 0.04900000000000004 t = 0.05000000000000004 t = 0.05100000000000004 t = 0.05200000000000004 t = 0.05300000000000004 t = 0.05400000000000004 t = 0.05500000000000004 t = 0.05600000000000004 t = 0.057000000000000044 t = 0.058000000000000045 t = 0.059000000000000045 t = 0.060000000000000046 t = 0.06100000000000005 t = 0.06200000000000005 t = 0.06300000000000004 t = 0.06400000000000004 t = 0.06500000000000004 t = 0.06600000000000004 t = 0.06700000000000005 t = 0.06800000000000005 t = 0.06900000000000005 t = 0.07000000000000005 t = 0.07100000000000005 t = 0.07200000000000005 t = 0.07300000000000005 t = 0.07400000000000005 t = 0.07500000000000005 t = 0.07600000000000005 t = 0.07700000000000005 t = 0.07800000000000006 t = 0.07900000000000006 t = 0.08000000000000006 t = 0.08100000000000006 t = 0.08200000000000006 t = 0.08300000000000006 t = 0.08400000000000006 t = 0.08500000000000006 t = 0.08600000000000006 t = 0.08700000000000006 t = 0.08800000000000006 t = 0.08900000000000007 t = 0.09000000000000007 t = 0.09100000000000007 t = 0.09200000000000007 t = 0.09300000000000007 t = 0.09400000000000007 t = 0.09500000000000007 t = 0.09600000000000007 t = 0.09700000000000007 t = 0.09800000000000007 t = 0.09900000000000007 t = 0.10000000000000007 t = 0.10100000000000008 t = 0.10200000000000008 t = 0.10300000000000008 t = 0.10400000000000008 t = 0.10500000000000008 t = 0.10600000000000008 t = 0.10700000000000008 t = 0.10800000000000008 t = 0.10900000000000008 t = 0.11000000000000008 t = 0.11100000000000008 t = 0.11200000000000009 t = 0.11300000000000009 t = 0.11400000000000009 t = 0.11500000000000009 t = 0.11600000000000009 t = 0.11700000000000009 t = 0.11800000000000009 t = 0.11900000000000009 t = 0.12000000000000009 t = 0.1210000000000001 t = 0.1220000000000001 t = 0.1230000000000001 t = 0.1240000000000001 t = 0.12500000000000008 t = 0.12600000000000008 t = 0.12700000000000009 t = 0.12800000000000009 t = 0.1290000000000001 t = 0.1300000000000001 t = 0.1310000000000001 t = 0.1320000000000001 t = 0.1330000000000001 t = 0.1340000000000001 t = 0.1350000000000001 t = 0.1360000000000001 t = 0.1370000000000001 t = 0.1380000000000001 t = 0.1390000000000001 t = 0.1400000000000001 t = 0.1410000000000001 t = 0.1420000000000001 t = 0.1430000000000001 t = 0.1440000000000001 t = 0.1450000000000001 t = 0.1460000000000001 t = 0.1470000000000001 t = 0.1480000000000001 t = 0.1490000000000001 t = 0.1500000000000001 t = 0.1510000000000001 t = 0.1520000000000001 t = 0.1530000000000001 t = 0.1540000000000001 t = 0.1550000000000001 t = 0.1560000000000001 t = 0.1570000000000001 t = 0.1580000000000001 t = 0.1590000000000001 t = 0.16000000000000011 t = 0.16100000000000012 t = 0.16200000000000012 t = 0.16300000000000012 t = 0.16400000000000012 t = 0.16500000000000012 t = 0.16600000000000012 t = 0.16700000000000012 t = 0.16800000000000012 t = 0.16900000000000012 t = 0.17000000000000012 t = 0.17100000000000012 t = 0.17200000000000013 t = 0.17300000000000013 t = 0.17400000000000013 t = 0.17500000000000013 t = 0.17600000000000013 t = 0.17700000000000013 t = 0.17800000000000013 t = 0.17900000000000013 t = 0.18000000000000013 t = 0.18100000000000013 t = 0.18200000000000013 t = 0.18300000000000013 t = 0.18400000000000014 t = 0.18500000000000014 t = 0.18600000000000014 t = 0.18700000000000014 t = 0.18800000000000014 t = 0.18900000000000014 t = 0.19000000000000014 t = 0.19100000000000014 t = 0.19200000000000014 t = 0.19300000000000014 t = 0.19400000000000014 t = 0.19500000000000015 t = 0.19600000000000015 t = 0.19700000000000015 t = 0.19800000000000015 t = 0.19900000000000015 t = 0.20000000000000015 t = 0.20100000000000015 t = 0.20200000000000015 t = 0.20300000000000015 t = 0.20400000000000015 t = 0.20500000000000015 t = 0.20600000000000016 t = 0.20700000000000016 t = 0.20800000000000016 t = 0.20900000000000016 t = 0.21000000000000016 t = 0.21100000000000016 t = 0.21200000000000016 t = 0.21300000000000016 t = 0.21400000000000016 t = 0.21500000000000016 t = 0.21600000000000016 t = 0.21700000000000016 t = 0.21800000000000017 t = 0.21900000000000017 t = 0.22000000000000017 t = 0.22100000000000017 t = 0.22200000000000017 t = 0.22300000000000017 t = 0.22400000000000017 t = 0.22500000000000017 t = 0.22600000000000017 t = 0.22700000000000017 t = 0.22800000000000017 t = 0.22900000000000018 t = 0.23000000000000018 t = 0.23100000000000018 t = 0.23200000000000018 t = 0.23300000000000018 t = 0.23400000000000018 t = 0.23500000000000018 t = 0.23600000000000018 t = 0.23700000000000018 t = 0.23800000000000018 t = 0.23900000000000018 t = 0.24000000000000019 t = 0.2410000000000002 t = 0.2420000000000002 t = 0.2430000000000002 t = 0.2440000000000002 t = 0.2450000000000002 t = 0.2460000000000002 t = 0.2470000000000002 t = 0.2480000000000002 t = 0.2490000000000002 t = 0.25000000000000017

CPU times: user 3.2 s, sys: 132 ms, total: 3.33 s Wall time: 831 ms

t = 0.249 t = 0.248 t = 0.247 t = 0.246 t = 0.245 t = 0.244 t = 0.243 t = 0.242 t = 0.241 t = 0.24 t = 0.239 t = 0.238 t = 0.237 t = 0.236 t = 0.235 t = 0.23399999999999999 t = 0.23299999999999998 t = 0.23199999999999998 t = 0.23099999999999998 t = 0.22999999999999998 t = 0.22899999999999998 t = 0.22799999999999998 t = 0.22699999999999998 t = 0.22599999999999998 t = 0.22499999999999998 t = 0.22399999999999998 t = 0.22299999999999998 t = 0.22199999999999998 t = 0.22099999999999997 t = 0.21999999999999997 t = 0.21899999999999997 t = 0.21799999999999997 t = 0.21699999999999997 t = 0.21599999999999997 t = 0.21499999999999997 t = 0.21399999999999997 t = 0.21299999999999997 t = 0.21199999999999997 t = 0.21099999999999997 t = 0.20999999999999996 t = 0.20899999999999996 t = 0.20799999999999996 t = 0.20699999999999996 t = 0.20599999999999996 t = 0.20499999999999996 t = 0.20399999999999996 t = 0.20299999999999996 t = 0.20199999999999996 t = 0.20099999999999996 t = 0.19999999999999996 t = 0.19899999999999995 t = 0.19799999999999995 t = 0.19699999999999995 t = 0.19599999999999995 t = 0.19499999999999995 t = 0.19399999999999995 t = 0.19299999999999995 t = 0.19199999999999995 t = 0.19099999999999995 t = 0.18999999999999995 t = 0.18899999999999995 t = 0.18799999999999994 t = 0.18699999999999994 t = 0.18599999999999994 t = 0.18499999999999994 t = 0.18399999999999994 t = 0.18299999999999994 t = 0.18199999999999994 t = 0.18099999999999994 t = 0.17999999999999994 t = 0.17899999999999994 t = 0.17799999999999994 t = 0.17699999999999994 t = 0.17599999999999993 t = 0.17499999999999993 t = 0.17399999999999993 t = 0.17299999999999993 t = 0.17199999999999993 t = 0.17099999999999993 t = 0.16999999999999993 t = 0.16899999999999993 t = 0.16799999999999993 t = 0.16699999999999993 t = 0.16599999999999993 t = 0.16499999999999992 t = 0.16399999999999992 t = 0.16299999999999992 t = 0.16199999999999992 t = 0.16099999999999992 t = 0.15999999999999992 t = 0.15899999999999992 t = 0.15799999999999992 t = 0.15699999999999992 t = 0.15599999999999992 t = 0.15499999999999992 t = 0.15399999999999991 t = 0.1529999999999999 t = 0.1519999999999999 t = 0.1509999999999999 t = 0.1499999999999999 t = 0.1489999999999999 t = 0.1479999999999999 t = 0.1469999999999999 t = 0.1459999999999999 t = 0.1449999999999999 t = 0.1439999999999999 t = 0.1429999999999999 t = 0.1419999999999999 t = 0.1409999999999999 t = 0.1399999999999999 t = 0.1389999999999999 t = 0.1379999999999999 t = 0.1369999999999999 t = 0.1359999999999999 t = 0.1349999999999999 t = 0.1339999999999999 t = 0.1329999999999999 t = 0.1319999999999999 t = 0.1309999999999999 t = 0.1299999999999999 t = 0.1289999999999999 t = 0.1279999999999999 t = 0.1269999999999999 t = 0.1259999999999999 t = 0.12499999999999992 t = 0.12399999999999992 t = 0.12299999999999991 t = 0.12199999999999991 t = 0.12099999999999991 t = 0.11999999999999991 t = 0.11899999999999991 t = 0.11799999999999991 t = 0.11699999999999991 t = 0.11599999999999991 t = 0.11499999999999991 t = 0.11399999999999991 t = 0.1129999999999999 t = 0.1119999999999999 t = 0.1109999999999999 t = 0.1099999999999999 t = 0.1089999999999999 t = 0.1079999999999999 t = 0.1069999999999999 t = 0.1059999999999999 t = 0.1049999999999999 t = 0.1039999999999999 t = 0.1029999999999999 t = 0.1019999999999999 t = 0.1009999999999999 t = 0.0999999999999999 t = 0.0989999999999999 t = 0.09799999999999989 t = 0.09699999999999989 t = 0.09599999999999989 t = 0.09499999999999989 t = 0.09399999999999989 t = 0.09299999999999989 t = 0.09199999999999989 t = 0.09099999999999989 t = 0.08999999999999989 t = 0.08899999999999988 t = 0.08799999999999988 t = 0.08699999999999988 t = 0.08599999999999988 t = 0.08499999999999988 t = 0.08399999999999988 t = 0.08299999999999988 t = 0.08199999999999988 t = 0.08099999999999988 t = 0.07999999999999988 t = 0.07899999999999988 t = 0.07799999999999987 t = 0.07699999999999987 t = 0.07599999999999987 t = 0.07499999999999987 t = 0.07399999999999987 t = 0.07299999999999987 t = 0.07199999999999987 t = 0.07099999999999987 t = 0.06999999999999987 t = 0.06899999999999987 t = 0.06799999999999987 t = 0.06699999999999987 t = 0.06599999999999986 t = 0.06499999999999986 t = 0.06399999999999986 t = 0.06299999999999986 t = 0.06199999999999986 t = 0.06099999999999986 t = 0.05999999999999986 t = 0.05899999999999986 t = 0.05799999999999986 t = 0.056999999999999856 t = 0.055999999999999855 t = 0.054999999999999855 t = 0.053999999999999854 t = 0.05299999999999985 t = 0.05199999999999985 t = 0.05099999999999985 t = 0.04999999999999985 t = 0.04899999999999985 t = 0.04799999999999985 t = 0.04699999999999985 t = 0.04599999999999985 t = 0.044999999999999846 t = 0.043999999999999845 t = 0.042999999999999844 t = 0.04199999999999984 t = 0.04099999999999984 t = 0.03999999999999984 t = 0.03899999999999984 t = 0.03799999999999984 t = 0.03699999999999984 t = 0.03599999999999984 t = 0.03499999999999984 t = 0.033999999999999836 t = 0.032999999999999835 t = 0.031999999999999834 t = 0.030999999999999833 t = 0.029999999999999832 t = 0.02899999999999983 t = 0.02799999999999983 t = 0.02699999999999983 t = 0.02599999999999983 t = 0.024999999999999828 t = 0.023999999999999827 t = 0.022999999999999826 t = 0.021999999999999825 t = 0.020999999999999824 t = 0.019999999999999823 t = 0.018999999999999823 t = 0.01799999999999982 t = 0.01699999999999982 t = 0.01599999999999982 t = 0.014999999999999819 t = 0.013999999999999818 t = 0.012999999999999817 t = 0.011999999999999816 t = 0.010999999999999815 t = 0.009999999999999815 t = 0.008999999999999814 t = 0.007999999999999813 t = 0.006999999999999812 t = 0.005999999999999811 t = 0.00499999999999981 t = 0.003999999999999809 t = 0.0029999999999998084 t = 0.0019999999999998075 t = 0.0009999999999998066 t = -1.6653345369377348e-16

CPU times: user 3.23 s, sys: 82.9 ms, total: 3.31 s Wall time: 827 ms

References:

• [Hesthaven, Warburton. Nodal Discontinuous Galerkin Methods—Algorithms, Analysis and Applications, 2007]

• [Koutschan, Lehrenfeld, Schöberl. Computer algebra meets finite elements: An efficient implementation for Maxwells equations. , 2012]