This page was generated from unit-2.6-stokes/stokes.ipynb.

2.6 Stokes equation¶

Find \(u \in [H^1_D]^2\) and \(p \in L_2\) such that

\[\begin{split}\DeclareMathOperator{\Div}{div} \begin{array}{ccccll} \int \nabla u : \nabla v & + & \int \Div v \, p & = & \int f v & \forall \, v \\ \int \Div u \, q & & & = & 0 & \forall \, q \end{array}\end{split}\]

Define channel geometry and mesh it:

[1]:

from ngsolve import *
from ngsolve.webgui import Draw
from netgen.occ import *

shape = Rectangle(2,0.41).Circle(0.2,0.2,0.05).Reverse().Face()
shape.edges.name="wall"
shape.edges.Min(X).name="inlet"
shape.edges.Max(X).name="outlet"
Draw (shape);

[2]:

geo = OCCGeometry(shape, dim=2)
mesh = Mesh(geo.GenerateMesh(maxh=0.05))
mesh.Curve(3)
Draw (mesh);

Use Taylor Hood finite element pairing: Vector-valued continuous \(P^2\) elements for velocity, and continuous \(P^1\) for pressure:

[3]:

V = VectorH1(mesh, order=2, dirichlet="wall|inlet|cyl")
Q = H1(mesh, order=1)
X = V*Q

Setup bilinear-form for Stokes. We give names for all scalar field components. The divergence is constructed from partial derivatives of the velocity components.

[4]:

(u,p),(v,q) = X.TnT()

stokes = InnerProduct(Grad(u), Grad(v))*dx + div(u)*q*dx + div(v)*p*dx
a = BilinearForm(stokes).Assemble()

Set inhomogeneous Dirichlet boundary condition only on inlet boundary:

[5]:

gf = GridFunction(X)
gfu, gfp = gf.components

uin = CF ( (1.5*4*y*(0.41-y)/(0.41*0.41), 0) )
gfu.Set(uin, definedon=mesh.Boundaries("inlet"))
Draw(gfu, mesh, min=0, max=2)
SetVisualization(max=2)

Solve equation:

[6]:

res = -a.mat * gf.vec
inv = a.mat.Inverse(freedofs=X.FreeDofs(), inverse="umfpack")
gf.vec.data += inv * res
Draw(gfu, mesh);

Testing different velocity-pressure pairs¶

Now we define a Stokes setup function to test different spaces:

[7]:

def SolveStokes(X):
    (u,p),(v,q) = X.TnT()

    stokes = InnerProduct(Grad(u), Grad(v))*dx + div(u)*q*dx + div(v)*p*dx
    a = BilinearForm(stokes).Assemble()

    gf = GridFunction(X)
    gfu, gfp = gf.components

    uin = CF ( (1.5*4*y*(0.41-y)/(0.41*0.41), 0) )
    gfu.Set(uin, definedon=mesh.Boundaries("inlet"))

    res = -a.mat * gf.vec
    inv = a.mat.Inverse(freedofs=X.FreeDofs(), inverse="umfpack")
    gf.vec.data += inv * res

    Draw(gfu, mesh)

    return gfu

Higher order Taylor-Hood elements:

[8]:

V = VectorH1(mesh, order=4, dirichlet="wall|inlet|cyl")
Q = H1(mesh, order=3)
X = V*Q

gfu = SolveStokes(X)

With discontinuous pressure elements P2-P1 is unstable:

[9]:

V = VectorH1(mesh, order=2, dirichlet="wall|inlet|cyl")
Q = L2(mesh, order=1)
print ("V.ndof =", V.ndof, ", Q.ndof =", Q.ndof)
X = V*Q

gfu = SolveStokes(X)

V.ndof = 3424 , Q.ndof = 2406

---------------------------------------------------------------------------
NgException                               Traceback (most recent call last)
Cell In[9], line 6
      3 print ("V.ndof =", V.ndof, ", Q.ndof =", Q.ndof)
      4 X = V*Q
----> 6 gfu = SolveStokes(X)

Cell In[7], line 14, in SolveStokes(X)
     11 gfu.Set(uin, definedon=mesh.Boundaries("inlet"))
     13 res = -a.mat * gf.vec
---> 14 inv = a.mat.Inverse(freedofs=X.FreeDofs(), inverse="umfpack")
     15 gf.vec.data += inv * res
     17 Draw(gfu, mesh)

NgException: UmfpackInverse: Numeric factorization failed.

However, P2-P1 elements work on sub-divided simplicial meshes using Alfeld splits:

[10]:

mesh2 = Mesh(geo.GenerateMesh(maxh=0.05)).Curve(3)
mesh2.SplitElements_Alfeld()
V = VectorH1(mesh2, order=2, dirichlet="wall|inlet|cyl")
Q = L2(mesh2, order=1)
print ("V.ndof =", V.ndof, ", Q.ndof =", Q.ndof)
X = V*Q

gfu = SolveStokes(X)

V.ndof = 9840 , Q.ndof = 7218

\(P^{2,+} \times P^{1,dc}\) elements:

[11]:

V = VectorH1(mesh, order=4, dirichlet="wall|inlet|cyl")
# V.SetOrder(TRIG,4)
# V.SetOrder(SEGM,4)
V.Update()
print (V.ndof)
Q = L2(mesh, order=1)
X = V*Q
print ("V.ndof =", V.ndof, ", Q.ndof =", Q.ndof)

gfu = SolveStokes(X)

13264
V.ndof = 13264 , Q.ndof = 2406

the mini element:

[12]:

V = VectorH1(mesh, order=1, dirichlet="wall|inlet|cyl")
V.SetOrder(TRIG,3)
V.Update()
Q = H1(mesh, order=1)
X = V*Q

gfu = SolveStokes(X)

Stokes as a block-system¶

We can now define separate bilinear-form and matrices for A and B, and combine them to a block-system:

[13]:

V = VectorH1(mesh, order=3, dirichlet="wall|inlet|cyl")
Q = H1(mesh, order=2)

u,v = V.TnT()
p,q = Q.TnT()

a = BilinearForm(InnerProduct(Grad(u),Grad(v))*dx).Assemble()
b = BilinearForm(div(u)*q*dx).Assemble()

Needed as preconditioner for the pressure:

[14]:

mp = BilinearForm(p*q*dx).Assemble()

Two right hand sides for the two spaces:

[15]:

f = LinearForm(V).Assemble()
g = LinearForm(Q).Assemble();

Two GridFunctions for velocity and pressure:

[16]:

gfu = GridFunction(V, name="u")
gfp = GridFunction(Q, name="p")
uin = CoefficientFunction( (1.5*4*y*(0.41-y)/(0.41*0.41), 0) )
gfu.Set(uin, definedon=mesh.Boundaries("inlet"))

Combine everything to a block-system. BlockMatrix and BlockVector store references to the original matrices and vectors, no new large matrices are allocated. The same for the transpose matrix b.mat.T. It stores a wrapper for the original matrix, and replaces the call of the Mult function by MultTrans.

[17]:

K = BlockMatrix( [ [a.mat, b.mat.T], [b.mat, None] ] )
C = BlockMatrix( [ [a.mat.Inverse(V.FreeDofs()), None], [None, mp.mat.Inverse()] ] )

rhs = BlockVector ( [f.vec, g.vec] )
sol = BlockVector( [gfu.vec, gfp.vec] )

solvers.MinRes (mat=K, pre=C, rhs=rhs, sol=sol, printrates='\r', initialize=False);

LinearSolver iteration 1, residual = 4.518937179332287

LinearSolver iteration 2, residual = 2.3020782066183405 LinearSolver iteration 3, residual = 1.9171806179628295 LinearSolver iteration 4, residual = 1.5318309921095468 LinearSolver iteration 5, residual = 1.4827182650518633 LinearSolver iteration 6, residual = 1.2760882423711855 LinearSolver iteration 7, residual = 1.2316328218317878 LinearSolver iteration 8, residual = 1.0758256973668074 LinearSolver iteration 9, residual = 0.9742346042735229 LinearSolver iteration 10, residual = 0.8116311768045531 LinearSolver iteration 11, residual = 0.7620621123363509 LinearSolver iteration 12, residual = 0.6711057838761307 LinearSolver iteration 13, residual = 0.6467734480369526 LinearSolver iteration 14, residual = 0.5963885550482779 LinearSolver iteration 15, residual = 0.5715760562984827 LinearSolver iteration 16, residual = 0.5314172872757961 LinearSolver iteration 17, residual = 0.523188572014509 LinearSolver iteration 18, residual = 0.48916660118256133 LinearSolver iteration 19, residual = 0.48026612300248067 LinearSolver iteration 20, residual = 0.45410105844800014 LinearSolver iteration 21, residual = 0.4339175906792935 LinearSolver iteration 22, residual = 0.38970115024248825 LinearSolver iteration 23, residual = 0.3665030310531088 LinearSolver iteration 24, residual = 0.29515807082938256 LinearSolver iteration 25, residual = 0.2840989870359845 LinearSolver iteration 26, residual = 0.17263506434502293 LinearSolver iteration 27, residual = 0.16533278871221124 LinearSolver iteration 28, residual = 0.09207002076023608 LinearSolver iteration 29, residual = 0.08806383521916354 LinearSolver iteration 30, residual = 0.045707773020785346 LinearSolver iteration 31, residual = 0.04558872526001847 LinearSolver iteration 32, residual = 0.022509806937679445 LinearSolver iteration 33, residual = 0.01911902981935344 LinearSolver iteration 34, residual = 0.012572671188669397 LinearSolver iteration 35, residual = 0.012232920688064624 LinearSolver iteration 36, residual = 0.008796244991863617 LinearSolver iteration 37, residual = 0.008540624451742073 LinearSolver iteration 38, residual = 0.006168751068326351 LinearSolver iteration 39, residual = 0.005859614741848614 LinearSolver iteration 40, residual = 0.003919721191004725 LinearSolver iteration 41, residual = 0.0037309231359978446 LinearSolver iteration 42, residual = 0.002409644869021594 LinearSolver iteration 43, residual = 0.002233246013776026 LinearSolver iteration 44, residual = 0.001440117277538793 LinearSolver iteration 45, residual = 0.0014169553919431872 LinearSolver iteration 46, residual = 0.0009214338668519829 LinearSolver iteration 47, residual = 0.0008998464054884886 LinearSolver iteration 48, residual = 0.0005762503566586648 LinearSolver iteration 49, residual = 0.0005472865054142066 LinearSolver iteration 50, residual = 0.0003758901019645419 LinearSolver iteration 51, residual = 0.0003524418202292021 LinearSolver iteration 52, residual = 0.00023429185631029594 LinearSolver iteration 53, residual = 0.000231577683020344 LinearSolver iteration 54, residual = 0.00013261675061463022 LinearSolver iteration 55, residual = 0.00012528252842354204 LinearSolver iteration 56, residual = 5.852579935273627e-05 LinearSolver iteration 57, residual = 5.220921278375006e-05

</pre>

LinearSolver iteration 2, residual = 2.3020782066183405 LinearSolver iteration 3, residual = 1.9171806179628295 LinearSolver iteration 4, residual = 1.5318309921095468 LinearSolver iteration 5, residual = 1.4827182650518633 LinearSolver iteration 6, residual = 1.2760882423711855 LinearSolver iteration 7, residual = 1.2316328218317878 LinearSolver iteration 8, residual = 1.0758256973668074 LinearSolver iteration 9, residual = 0.9742346042735229 LinearSolver iteration 10, residual = 0.8116311768045531 LinearSolver iteration 11, residual = 0.7620621123363509 LinearSolver iteration 12, residual = 0.6711057838761307 LinearSolver iteration 13, residual = 0.6467734480369526 LinearSolver iteration 14, residual = 0.5963885550482779 LinearSolver iteration 15, residual = 0.5715760562984827 LinearSolver iteration 16, residual = 0.5314172872757961 LinearSolver iteration 17, residual = 0.523188572014509 LinearSolver iteration 18, residual = 0.48916660118256133 LinearSolver iteration 19, residual = 0.48026612300248067 LinearSolver iteration 20, residual = 0.45410105844800014 LinearSolver iteration 21, residual = 0.4339175906792935 LinearSolver iteration 22, residual = 0.38970115024248825 LinearSolver iteration 23, residual = 0.3665030310531088 LinearSolver iteration 24, residual = 0.29515807082938256 LinearSolver iteration 25, residual = 0.2840989870359845 LinearSolver iteration 26, residual = 0.17263506434502293 LinearSolver iteration 27, residual = 0.16533278871221124 LinearSolver iteration 28, residual = 0.09207002076023608 LinearSolver iteration 29, residual = 0.08806383521916354 LinearSolver iteration 30, residual = 0.045707773020785346 LinearSolver iteration 31, residual = 0.04558872526001847 LinearSolver iteration 32, residual = 0.022509806937679445 LinearSolver iteration 33, residual = 0.01911902981935344 LinearSolver iteration 34, residual = 0.012572671188669397 LinearSolver iteration 35, residual = 0.012232920688064624 LinearSolver iteration 36, residual = 0.008796244991863617 LinearSolver iteration 37, residual = 0.008540624451742073 LinearSolver iteration 38, residual = 0.006168751068326351 LinearSolver iteration 39, residual = 0.005859614741848614 LinearSolver iteration 40, residual = 0.003919721191004725 LinearSolver iteration 41, residual = 0.0037309231359978446 LinearSolver iteration 42, residual = 0.002409644869021594 LinearSolver iteration 43, residual = 0.002233246013776026 LinearSolver iteration 44, residual = 0.001440117277538793 LinearSolver iteration 45, residual = 0.0014169553919431872 LinearSolver iteration 46, residual = 0.0009214338668519829 LinearSolver iteration 47, residual = 0.0008998464054884886 LinearSolver iteration 48, residual = 0.0005762503566586648 LinearSolver iteration 49, residual = 0.0005472865054142066 LinearSolver iteration 50, residual = 0.0003758901019645419 LinearSolver iteration 51, residual = 0.0003524418202292021 LinearSolver iteration 52, residual = 0.00023429185631029594 LinearSolver iteration 53, residual = 0.000231577683020344 LinearSolver iteration 54, residual = 0.00013261675061463022 LinearSolver iteration 55, residual = 0.00012528252842354204 LinearSolver iteration 56, residual = 5.852579935273627e-05 LinearSolver iteration 57, residual = 5.220921278375006e-05

end{sphinxVerbatim}

[2KLinearSolver iteration 2, residual = 2.3020782066183405 [2KLinearSolver iteration 3, residual = 1.9171806179628295 [2KLinearSolver iteration 4, residual = 1.5318309921095468 [2KLinearSolver iteration 5, residual = 1.4827182650518633 [2KLinearSolver iteration 6, residual = 1.2760882423711855 [2KLinearSolver iteration 7, residual = 1.2316328218317878 [2KLinearSolver iteration 8, residual = 1.0758256973668074 [2KLinearSolver iteration 9, residual = 0.9742346042735229 [2KLinearSolver iteration 10, residual = 0.8116311768045531 [2KLinearSolver iteration 11, residual = 0.7620621123363509 [2KLinearSolver iteration 12, residual = 0.6711057838761307 [2KLinearSolver iteration 13, residual = 0.6467734480369526 [2KLinearSolver iteration 14, residual = 0.5963885550482779 [2KLinearSolver iteration 15, residual = 0.5715760562984827 [2KLinearSolver iteration 16, residual = 0.5314172872757961 [2KLinearSolver iteration 17, residual = 0.523188572014509 [2KLinearSolver iteration 18, residual = 0.48916660118256133 [2KLinearSolver iteration 19, residual = 0.48026612300248067 [2KLinearSolver iteration 20, residual = 0.45410105844800014 [2KLinearSolver iteration 21, residual = 0.4339175906792935 [2KLinearSolver iteration 22, residual = 0.38970115024248825 [2KLinearSolver iteration 23, residual = 0.3665030310531088 [2KLinearSolver iteration 24, residual = 0.29515807082938256 [2KLinearSolver iteration 25, residual = 0.2840989870359845 [2KLinearSolver iteration 26, residual = 0.17263506434502293 [2KLinearSolver iteration 27, residual = 0.16533278871221124 [2KLinearSolver iteration 28, residual = 0.09207002076023608 [2KLinearSolver iteration 29, residual = 0.08806383521916354 [2KLinearSolver iteration 30, residual = 0.045707773020785346 [2KLinearSolver iteration 31, residual = 0.04558872526001847 [2KLinearSolver iteration 32, residual = 0.022509806937679445 [2KLinearSolver iteration 33, residual = 0.01911902981935344 [2KLinearSolver iteration 34, residual = 0.012572671188669397 [2KLinearSolver iteration 35, residual = 0.012232920688064624 [2KLinearSolver iteration 36, residual = 0.008796244991863617 [2KLinearSolver iteration 37, residual = 0.008540624451742073 [2KLinearSolver iteration 38, residual = 0.006168751068326351 [2KLinearSolver iteration 39, residual = 0.005859614741848614 [2KLinearSolver iteration 40, residual = 0.003919721191004725 [2KLinearSolver iteration 41, residual = 0.0037309231359978446 [2KLinearSolver iteration 42, residual = 0.002409644869021594 [2KLinearSolver iteration 43, residual = 0.002233246013776026 [2KLinearSolver iteration 44, residual = 0.001440117277538793 [2KLinearSolver iteration 45, residual = 0.0014169553919431872 [2KLinearSolver iteration 46, residual = 0.0009214338668519829 [2KLinearSolver iteration 47, residual = 0.0008998464054884886 [2KLinearSolver iteration 48, residual = 0.0005762503566586648 [2KLinearSolver iteration 49, residual = 0.0005472865054142066 [2KLinearSolver iteration 50, residual = 0.0003758901019645419 [2KLinearSolver iteration 51, residual = 0.0003524418202292021 [2KLinearSolver iteration 52, residual = 0.00023429185631029594 [2KLinearSolver iteration 53, residual = 0.000231577683020344 [2KLinearSolver iteration 54, residual = 0.00013261675061463022 [2KLinearSolver iteration 55, residual = 0.00012528252842354204 [2KLinearSolver iteration 56, residual = 5.852579935273627e-05 [2KLinearSolver iteration 57, residual = 5.220921278375006e-05

LinearSolver iteration 58, residual = 2.445981833622924e-05

LinearSolver iteration 59, residual = 2.1011457947562368e-05 LinearSolver iteration 60, residual = 9.894725207855162e-06 LinearSolver iteration 61, residual = 9.325347229145796e-06 LinearSolver iteration 62, residual = 4.39262521268901e-06 LinearSolver iteration 63, residual = 4.195388653320418e-06 LinearSolver iteration 64, residual = 1.7610904076427593e-06 LinearSolver iteration 65, residual = 1.487707702456541e-06 LinearSolver iteration 66, residual = 5.821123043092815e-07 LinearSolver iteration 67, residual = 5.153650968031574e-07 LinearSolver iteration 68, residual = 2.3534539744561125e-07 LinearSolver converged in 68 iterations to residual 2.3534539744561125e-07

</pre>

LinearSolver iteration 59, residual = 2.1011457947562368e-05 LinearSolver iteration 60, residual = 9.894725207855162e-06 LinearSolver iteration 61, residual = 9.325347229145796e-06 LinearSolver iteration 62, residual = 4.39262521268901e-06 LinearSolver iteration 63, residual = 4.195388653320418e-06 LinearSolver iteration 64, residual = 1.7610904076427593e-06 LinearSolver iteration 65, residual = 1.487707702456541e-06 LinearSolver iteration 66, residual = 5.821123043092815e-07 LinearSolver iteration 67, residual = 5.153650968031574e-07 LinearSolver iteration 68, residual = 2.3534539744561125e-07 LinearSolver converged in 68 iterations to residual 2.3534539744561125e-07

end{sphinxVerbatim}

[2KLinearSolver iteration 59, residual = 2.1011457947562368e-05 [2KLinearSolver iteration 60, residual = 9.894725207855162e-06 [2KLinearSolver iteration 61, residual = 9.325347229145796e-06 [2KLinearSolver iteration 62, residual = 4.39262521268901e-06 [2KLinearSolver iteration 63, residual = 4.195388653320418e-06 [2KLinearSolver iteration 64, residual = 1.7610904076427593e-06 [2KLinearSolver iteration 65, residual = 1.487707702456541e-06 [2KLinearSolver iteration 66, residual = 5.821123043092815e-07 [2KLinearSolver iteration 67, residual = 5.153650968031574e-07 [2KLinearSolver iteration 68, residual = 2.3534539744561125e-07 [2KLinearSolver converged in 68 iterations to residual 2.3534539744561125e-07

[18]:

Draw (gfu);

A stabilised lowest order formulation:¶

[19]:

mesh = Mesh( geo.GenerateMesh(maxh=0.02))

V = VectorH1(mesh, order=1, dirichlet="wall|inlet|cyl")
Q = H1(mesh, order=1)

u,v = V.TnT()
p,q = Q.TnT()

a = BilinearForm(InnerProduct(Grad(u),Grad(v))*dx).Assemble()
b = BilinearForm(div(u)*q*dx).Assemble()
h = specialcf.mesh_size
c = BilinearForm(-0.1*h*h*grad(p)*grad(q)*dx).Assemble()

[20]:

mp = BilinearForm(p*q*dx).Assemble()

f = LinearForm(V)
f.Assemble()

g = LinearForm(Q)
g.Assemble()

gfu = GridFunction(V, name="u")
gfp = GridFunction(Q, name="p")
uin = CF( (1.5*4*y*(0.41-y)/(0.41*0.41), 0) )
gfu.Set(uin, definedon=mesh.Boundaries("inlet"))

K = BlockMatrix( [ [a.mat, b.mat.T], [b.mat, c.mat] ] )
C = BlockMatrix( [ [a.mat.Inverse(V.FreeDofs()), None], [None, mp.mat.Inverse()] ] )

rhs = BlockVector ( [f.vec, g.vec] )
sol = BlockVector( [gfu.vec, gfp.vec] )

solvers.MinRes (mat=K, pre=C, rhs=rhs, sol=sol, printrates='\r', initialize=False, maxsteps=200);

LinearSolver iteration 1, residual = 7.103717015785737

LinearSolver iteration 2, residual = 3.1240529736545715 LinearSolver iteration 3, residual = 2.132679123744692 LinearSolver iteration 4, residual = 1.8523744559707935 LinearSolver iteration 5, residual = 1.5949503379856076 LinearSolver iteration 6, residual = 1.5480493182937696 LinearSolver iteration 7, residual = 1.4639703507160722 LinearSolver iteration 8, residual = 1.3511179026398876 LinearSolver iteration 9, residual = 1.3380334785486305 LinearSolver iteration 10, residual = 1.2598116080932027 LinearSolver iteration 11, residual = 1.1487182058160144 LinearSolver iteration 12, residual = 1.1487081802610593 LinearSolver iteration 13, residual = 1.0553572664456368 LinearSolver iteration 14, residual = 0.976593791636269 LinearSolver iteration 15, residual = 0.9718806996194856 LinearSolver iteration 16, residual = 0.8599661205136232 LinearSolver iteration 17, residual = 0.8005914699770541 LinearSolver iteration 18, residual = 0.7815317337264195 LinearSolver iteration 19, residual = 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LinearSolver iteration 88, residual = 3.849464222210089e-05 LinearSolver iteration 89, residual = 3.2605363484935703e-05 LinearSolver iteration 90, residual = 2.8322245608267862e-05 LinearSolver iteration 91, residual = 1.82690807881192e-05 LinearSolver iteration 92, residual = 1.8051591698283513e-05 LinearSolver iteration 93, residual = 1.1906749700372199e-05 LinearSolver iteration 94, residual = 8.918371402129329e-06 LinearSolver iteration 95, residual = 7.708397821999984e-06 LinearSolver iteration 96, residual = 4.360196894806923e-06 LinearSolver iteration 97, residual = 4.241381799516438e-06 LinearSolver iteration 98, residual = 2.799363495931647e-06 LinearSolver iteration 99, residual = 1.8482147124704675e-06 LinearSolver iteration 100, residual = 1.7777878936095626e-06 LinearSolver iteration 101, residual = 9.742813141308085e-07 LinearSolver iteration 102, residual = 8.494431817617426e-07 LinearSolver iteration 103, residual = 6.2071818814675e-07 LinearSolver converged in 103 iterations to residual 6.2071818814675e-07

</pre>

LinearSolver iteration 2, residual = 3.1240529736545715 LinearSolver iteration 3, residual = 2.132679123744692 LinearSolver iteration 4, residual = 1.8523744559707935 LinearSolver iteration 5, residual = 1.5949503379856076 LinearSolver iteration 6, residual = 1.5480493182937696 LinearSolver iteration 7, residual = 1.4639703507160722 LinearSolver iteration 8, residual = 1.3511179026398876 LinearSolver iteration 9, residual = 1.3380334785486305 LinearSolver iteration 10, residual = 1.2598116080932027 LinearSolver iteration 11, residual = 1.1487182058160144 LinearSolver iteration 12, residual = 1.1487081802610593 LinearSolver iteration 13, residual = 1.0553572664456368 LinearSolver iteration 14, residual = 0.976593791636269 LinearSolver iteration 15, residual = 0.9718806996194856 LinearSolver iteration 16, residual = 0.8599661205136232 LinearSolver iteration 17, residual = 0.8005914699770541 LinearSolver iteration 18, residual = 0.7815317337264195 LinearSolver iteration 19, residual = 0.695091815685787 LinearSolver iteration 20, residual = 0.681097261054522 LinearSolver iteration 21, residual = 0.6601652993538863 LinearSolver iteration 22, residual = 0.6238986254835396 LinearSolver iteration 23, residual = 0.6195450703831556 LinearSolver iteration 24, residual = 0.6055276228835791 LinearSolver iteration 25, residual = 0.5784318059307156 LinearSolver iteration 26, residual = 0.5763221221836843 LinearSolver iteration 27, residual = 0.5571394201839083 LinearSolver iteration 28, residual = 0.5339897335990003 LinearSolver iteration 29, residual = 0.5339897224407189 LinearSolver iteration 30, residual = 0.5122111603632727 LinearSolver iteration 31, residual = 0.49709386368401864 LinearSolver iteration 32, residual = 0.49522266954245237 LinearSolver iteration 33, residual = 0.4757510640129083 LinearSolver iteration 34, residual = 0.4731826388330148 LinearSolver iteration 35, residual = 0.4601388105401166 LinearSolver iteration 36, residual = 0.4412886936788078 LinearSolver iteration 37, residual = 0.44120267904223176 LinearSolver iteration 38, residual = 0.4187428770452127 LinearSolver iteration 39, residual = 0.39925869154183435 LinearSolver iteration 40, residual = 0.3965165497660975 LinearSolver iteration 41, residual = 0.35958302389869107 LinearSolver iteration 42, residual = 0.34918705384313015 LinearSolver iteration 43, residual = 0.32081334730538613 LinearSolver iteration 44, residual = 0.27496623675114984 LinearSolver iteration 45, residual = 0.2739395154602775 LinearSolver iteration 46, residual = 0.22065410237349553 LinearSolver iteration 47, residual = 0.19723595462027937 LinearSolver iteration 48, residual = 0.18105301698811066 LinearSolver iteration 49, residual = 0.13692381837465906 LinearSolver iteration 50, residual = 0.13287558442797537 LinearSolver iteration 51, residual = 0.10926674275131437 LinearSolver iteration 52, residual = 0.08098202942409063 LinearSolver iteration 53, residual = 0.08094489870875758 LinearSolver iteration 54, residual = 0.056389394203058316 LinearSolver iteration 55, residual = 0.04514989779565494 LinearSolver iteration 56, residual = 0.040324817474304074 LinearSolver iteration 57, residual = 0.02515404088463173 LinearSolver iteration 58, residual = 0.023960521236286676 LinearSolver iteration 59, residual = 0.018012559701725786 LinearSolver iteration 60, residual = 0.01300583626679384 LinearSolver iteration 61, residual = 0.012973772374450137 LinearSolver iteration 62, residual = 0.009169395609592765 LinearSolver iteration 63, residual = 0.00763667146865482 LinearSolver iteration 64, residual = 0.0068485873124133475 LinearSolver iteration 65, residual = 0.004943988286513263 LinearSolver iteration 66, residual = 0.004911909351469885 LinearSolver iteration 67, residual = 0.004081919199534889 LinearSolver iteration 68, residual = 0.0035790619585567927 LinearSolver iteration 69, residual = 0.0034957389244193965 LinearSolver iteration 70, residual = 0.002702516402196035 LinearSolver iteration 71, residual = 0.0026133921922323797 LinearSolver iteration 72, residual = 0.002044199670776769 LinearSolver iteration 73, residual = 0.0014952781332316647 LinearSolver iteration 74, residual = 0.0014711480663878792 LinearSolver iteration 75, residual = 0.000991617872172705 LinearSolver iteration 76, residual = 0.0008714858820776782 LinearSolver iteration 77, residual = 0.0007056267834200474 LinearSolver iteration 78, residual = 0.0004773645624879543 LinearSolver iteration 79, residual = 0.00046424789573374227 LinearSolver iteration 80, residual = 0.00028094430610380993 LinearSolver iteration 81, residual = 0.0002548453786863786 LinearSolver iteration 82, residual = 0.00020256676252789458 LinearSolver iteration 83, residual = 0.00013267945882768027 LinearSolver iteration 84, residual = 0.00013194521868198242 LinearSolver iteration 85, residual = 8.593924383152689e-05 LinearSolver iteration 86, residual = 6.239009989375969e-05 LinearSolver iteration 87, residual = 5.9815516703039264e-05 LinearSolver iteration 88, residual = 3.849464222210089e-05 LinearSolver iteration 89, residual = 3.2605363484935703e-05 LinearSolver iteration 90, residual = 2.8322245608267862e-05 LinearSolver iteration 91, residual = 1.82690807881192e-05 LinearSolver iteration 92, residual = 1.8051591698283513e-05 LinearSolver iteration 93, residual = 1.1906749700372199e-05 LinearSolver iteration 94, residual = 8.918371402129329e-06 LinearSolver iteration 95, residual = 7.708397821999984e-06 LinearSolver iteration 96, residual = 4.360196894806923e-06 LinearSolver iteration 97, residual = 4.241381799516438e-06 LinearSolver iteration 98, residual = 2.799363495931647e-06 LinearSolver iteration 99, residual = 1.8482147124704675e-06 LinearSolver iteration 100, residual = 1.7777878936095626e-06 LinearSolver iteration 101, residual = 9.742813141308085e-07 LinearSolver iteration 102, residual = 8.494431817617426e-07 LinearSolver iteration 103, residual = 6.2071818814675e-07 LinearSolver converged in 103 iterations to residual 6.2071818814675e-07

end{sphinxVerbatim}

[2KLinearSolver iteration 2, residual = 3.1240529736545715 [2KLinearSolver iteration 3, residual = 2.132679123744692 [2KLinearSolver iteration 4, residual = 1.8523744559707935 [2KLinearSolver iteration 5, residual = 1.5949503379856076 [2KLinearSolver iteration 6, residual = 1.5480493182937696 [2KLinearSolver iteration 7, residual = 1.4639703507160722 [2KLinearSolver iteration 8, residual = 1.3511179026398876 [2KLinearSolver iteration 9, residual = 1.3380334785486305 [2KLinearSolver iteration 10, residual = 1.2598116080932027 [2KLinearSolver iteration 11, residual = 1.1487182058160144 [2KLinearSolver iteration 12, residual = 1.1487081802610593 [2KLinearSolver iteration 13, residual = 1.0553572664456368 [2KLinearSolver iteration 14, residual = 0.976593791636269 [2KLinearSolver iteration 15, residual = 0.9718806996194856 [2KLinearSolver iteration 16, residual = 0.8599661205136232 [2KLinearSolver iteration 17, residual = 0.8005914699770541 [2KLinearSolver iteration 18, residual = 0.7815317337264195 [2KLinearSolver iteration 19, residual = 0.695091815685787 [2KLinearSolver iteration 20, residual = 0.681097261054522 [2KLinearSolver iteration 21, residual = 0.6601652993538863 [2KLinearSolver iteration 22, residual = 0.6238986254835396 [2KLinearSolver iteration 23, residual = 0.6195450703831556 [2KLinearSolver iteration 24, residual = 0.6055276228835791 [2KLinearSolver iteration 25, residual = 0.5784318059307156 [2KLinearSolver iteration 26, residual = 0.5763221221836843 [2KLinearSolver iteration 27, residual = 0.5571394201839083 [2KLinearSolver iteration 28, residual = 0.5339897335990003 [2KLinearSolver iteration 29, residual = 0.5339897224407189 [2KLinearSolver iteration 30, residual = 0.5122111603632727 [2KLinearSolver iteration 31, residual = 0.49709386368401864 [2KLinearSolver iteration 32, residual = 0.49522266954245237 [2KLinearSolver iteration 33, residual = 0.4757510640129083 [2KLinearSolver iteration 34, residual = 0.4731826388330148 [2KLinearSolver iteration 35, residual = 0.4601388105401166 [2KLinearSolver iteration 36, residual = 0.4412886936788078 [2KLinearSolver iteration 37, residual = 0.44120267904223176 [2KLinearSolver iteration 38, residual = 0.4187428770452127 [2KLinearSolver iteration 39, residual = 0.39925869154183435 [2KLinearSolver iteration 40, residual = 0.3965165497660975 [2KLinearSolver iteration 41, residual = 0.35958302389869107 [2KLinearSolver iteration 42, residual = 0.34918705384313015 [2KLinearSolver iteration 43, residual = 0.32081334730538613 [2KLinearSolver iteration 44, residual = 0.27496623675114984 [2KLinearSolver iteration 45, residual = 0.2739395154602775 [2KLinearSolver iteration 46, residual = 0.22065410237349553 [2KLinearSolver iteration 47, residual = 0.19723595462027937 [2KLinearSolver iteration 48, residual = 0.18105301698811066 [2KLinearSolver iteration 49, residual = 0.13692381837465906 [2KLinearSolver iteration 50, residual = 0.13287558442797537 [2KLinearSolver iteration 51, residual = 0.10926674275131437 [2KLinearSolver iteration 52, residual = 0.08098202942409063 [2KLinearSolver iteration 53, residual = 0.08094489870875758 [2KLinearSolver iteration 54, residual = 0.056389394203058316 [2KLinearSolver iteration 55, residual = 0.04514989779565494 [2KLinearSolver iteration 56, residual = 0.040324817474304074 [2KLinearSolver iteration 57, residual = 0.02515404088463173 [2KLinearSolver iteration 58, residual = 0.023960521236286676 [2KLinearSolver iteration 59, residual = 0.018012559701725786 [2KLinearSolver iteration 60, residual = 0.01300583626679384 [2KLinearSolver iteration 61, residual = 0.012973772374450137 [2KLinearSolver iteration 62, residual = 0.009169395609592765 [2KLinearSolver iteration 63, residual = 0.00763667146865482 [2KLinearSolver iteration 64, residual = 0.0068485873124133475 [2KLinearSolver iteration 65, residual = 0.004943988286513263 [2KLinearSolver iteration 66, residual = 0.004911909351469885 [2KLinearSolver iteration 67, residual = 0.004081919199534889 [2KLinearSolver iteration 68, residual = 0.0035790619585567927 [2KLinearSolver iteration 69, residual = 0.0034957389244193965 [2KLinearSolver iteration 70, residual = 0.002702516402196035 [2KLinearSolver iteration 71, residual = 0.0026133921922323797 [2KLinearSolver iteration 72, residual = 0.002044199670776769 [2KLinearSolver iteration 73, residual = 0.0014952781332316647 [2KLinearSolver iteration 74, residual = 0.0014711480663878792 [2KLinearSolver iteration 75, residual = 0.000991617872172705 [2KLinearSolver iteration 76, residual = 0.0008714858820776782 [2KLinearSolver iteration 77, residual = 0.0007056267834200474 [2KLinearSolver iteration 78, residual = 0.0004773645624879543 [2KLinearSolver iteration 79, residual = 0.00046424789573374227 [2KLinearSolver iteration 80, residual = 0.00028094430610380993 [2KLinearSolver iteration 81, residual = 0.0002548453786863786 [2KLinearSolver iteration 82, residual = 0.00020256676252789458 [2KLinearSolver iteration 83, residual = 0.00013267945882768027 [2KLinearSolver iteration 84, residual = 0.00013194521868198242 [2KLinearSolver iteration 85, residual = 8.593924383152689e-05 [2KLinearSolver iteration 86, residual = 6.239009989375969e-05 [2KLinearSolver iteration 87, residual = 5.9815516703039264e-05 [2KLinearSolver iteration 88, residual = 3.849464222210089e-05 [2KLinearSolver iteration 89, residual = 3.2605363484935703e-05 [2KLinearSolver iteration 90, residual = 2.8322245608267862e-05 [2KLinearSolver iteration 91, residual = 1.82690807881192e-05 [2KLinearSolver iteration 92, residual = 1.8051591698283513e-05 [2KLinearSolver iteration 93, residual = 1.1906749700372199e-05 [2KLinearSolver iteration 94, residual = 8.918371402129329e-06 [2KLinearSolver iteration 95, residual = 7.708397821999984e-06 [2KLinearSolver iteration 96, residual = 4.360196894806923e-06 [2KLinearSolver iteration 97, residual = 4.241381799516438e-06 [2KLinearSolver iteration 98, residual = 2.799363495931647e-06 [2KLinearSolver iteration 99, residual = 1.8482147124704675e-06 [2KLinearSolver iteration 100, residual = 1.7777878936095626e-06 [2KLinearSolver iteration 101, residual = 9.742813141308085e-07 [2KLinearSolver iteration 102, residual = 8.494431817617426e-07 [2KLinearSolver iteration 103, residual = 6.2071818814675e-07 [2KLinearSolver converged in 103 iterations to residual 6.2071818814675e-07

[21]:

Draw (gfu);

[22]:

Draw (gfp);

(P1-iso-P2)-P1¶

The mesh is uniformly refined once. The velocity space is \(P^1\) on the refined mesh, but the pressure is continuous \(P^1\) in the coarse space. We obtain this by representing the pressure in the range of the coarse-to-fine prolongation operator.

[23]:

mesh = Mesh( geo.GenerateMesh(maxh=0.02))

V = VectorH1(mesh, order=1, dirichlet="wall|inlet|cyl")
Q = H1(mesh, order=1)

u,v = V.TnT()
p,q = Q.TnT()

mp = BilinearForm(p*q*dx).Assemble()
invmp = mp.mat.Inverse(inverse="sparsecholesky")

mesh.ngmesh.Refine()
V.Update()
Q.Update()

[24]:

# prolongation operator from mesh-level 0 to level 1:
prol = Q.Prolongation().Operator(1)
invmp2 = prol @ invmp @ prol.T

a = BilinearForm(InnerProduct(Grad(u),Grad(v))*dx).Assemble()
b = BilinearForm(div(u)*q*dx).Assemble()

f = LinearForm(V)
f.Assemble()

g = LinearForm(Q)
g.Assemble()

gfu = GridFunction(V, name="u")
gfp = GridFunction(Q, name="p")
uin = CF( (1.5*4*y*(0.41-y)/(0.41*0.41), 0) )
gfu.Set(uin, definedon=mesh.Boundaries("inlet"))

K = BlockMatrix( [ [a.mat, b.mat.T], [b.mat, None] ] )
C = BlockMatrix( [ [a.mat.Inverse(V.FreeDofs()), None], [None, invmp2] ] )

rhs = BlockVector ( [f.vec, g.vec] )
sol = BlockVector( [gfu.vec, gfp.vec] )

solvers.MinRes (mat=K, pre=C, rhs=rhs, sol=sol, \
                printrates='\r', initialize=False, maxsteps=200);

LinearSolver iteration 1, residual = 10.21762787865306

LinearSolver iteration 2, residual = 3.5470429131109387 LinearSolver iteration 3, residual = 2.4903542579953064 LinearSolver iteration 4, residual = 1.6025072100687434 LinearSolver iteration 5, residual = 1.6024983010872687 LinearSolver iteration 6, residual = 1.333058137571719 LinearSolver iteration 7, residual = 1.2854650025118084 LinearSolver iteration 8, residual = 1.0977943618957227 LinearSolver iteration 9, residual = 1.0548387055316268 LinearSolver iteration 10, residual = 0.860129395740141 LinearSolver iteration 11, residual = 0.802487644233047 LinearSolver iteration 12, residual = 0.6849389452406723 LinearSolver iteration 13, residual = 0.6766104045843154 LinearSolver iteration 14, residual = 0.614778666409925 LinearSolver iteration 15, residual = 0.6025286681808675 LinearSolver iteration 16, residual = 0.5556519286013119 LinearSolver iteration 17, residual = 0.5433107795923451 LinearSolver iteration 18, residual = 0.5095539811129478 LinearSolver iteration 19, residual = 0.49972652714034577 LinearSolver iteration 20, residual = 0.4689365281768775 LinearSolver iteration 21, residual = 0.45629750981730127 LinearSolver iteration 22, residual = 0.4247374804729276 LinearSolver iteration 23, residual = 0.4047038646206252 LinearSolver iteration 24, residual = 0.346152345361016 LinearSolver iteration 25, residual = 0.3190728657926522 LinearSolver iteration 26, residual = 0.24068756246457923 LinearSolver iteration 27, residual = 0.2340879104748604 LinearSolver iteration 28, residual = 0.13210286673685911 LinearSolver iteration 29, residual = 0.12149542221443291

</pre>

LinearSolver iteration 2, residual = 3.5470429131109387 LinearSolver iteration 3, residual = 2.4903542579953064 LinearSolver iteration 4, residual = 1.6025072100687434 LinearSolver iteration 5, residual = 1.6024983010872687 LinearSolver iteration 6, residual = 1.333058137571719 LinearSolver iteration 7, residual = 1.2854650025118084 LinearSolver iteration 8, residual = 1.0977943618957227 LinearSolver iteration 9, residual = 1.0548387055316268 LinearSolver iteration 10, residual = 0.860129395740141 LinearSolver iteration 11, residual = 0.802487644233047 LinearSolver iteration 12, residual = 0.6849389452406723 LinearSolver iteration 13, residual = 0.6766104045843154 LinearSolver iteration 14, residual = 0.614778666409925 LinearSolver iteration 15, residual = 0.6025286681808675 LinearSolver iteration 16, residual = 0.5556519286013119 LinearSolver iteration 17, residual = 0.5433107795923451 LinearSolver iteration 18, residual = 0.5095539811129478 LinearSolver iteration 19, residual = 0.49972652714034577 LinearSolver iteration 20, residual = 0.4689365281768775 LinearSolver iteration 21, residual = 0.45629750981730127 LinearSolver iteration 22, residual = 0.4247374804729276 LinearSolver iteration 23, residual = 0.4047038646206252 LinearSolver iteration 24, residual = 0.346152345361016 LinearSolver iteration 25, residual = 0.3190728657926522 LinearSolver iteration 26, residual = 0.24068756246457923 LinearSolver iteration 27, residual = 0.2340879104748604 LinearSolver iteration 28, residual = 0.13210286673685911 LinearSolver iteration 29, residual = 0.12149542221443291

end{sphinxVerbatim}

[2KLinearSolver iteration 2, residual = 3.5470429131109387 [2KLinearSolver iteration 3, residual = 2.4903542579953064 [2KLinearSolver iteration 4, residual = 1.6025072100687434 [2KLinearSolver iteration 5, residual = 1.6024983010872687 [2KLinearSolver iteration 6, residual = 1.333058137571719 [2KLinearSolver iteration 7, residual = 1.2854650025118084 [2KLinearSolver iteration 8, residual = 1.0977943618957227 [2KLinearSolver iteration 9, residual = 1.0548387055316268 [2KLinearSolver iteration 10, residual = 0.860129395740141 [2KLinearSolver iteration 11, residual = 0.802487644233047 [2KLinearSolver iteration 12, residual = 0.6849389452406723 [2KLinearSolver iteration 13, residual = 0.6766104045843154 [2KLinearSolver iteration 14, residual = 0.614778666409925 [2KLinearSolver iteration 15, residual = 0.6025286681808675 [2KLinearSolver iteration 16, residual = 0.5556519286013119 [2KLinearSolver iteration 17, residual = 0.5433107795923451 [2KLinearSolver iteration 18, residual = 0.5095539811129478 [2KLinearSolver iteration 19, residual = 0.49972652714034577 [2KLinearSolver iteration 20, residual = 0.4689365281768775 [2KLinearSolver iteration 21, residual = 0.45629750981730127 [2KLinearSolver iteration 22, residual = 0.4247374804729276 [2KLinearSolver iteration 23, residual = 0.4047038646206252 [2KLinearSolver iteration 24, residual = 0.346152345361016 [2KLinearSolver iteration 25, residual = 0.3190728657926522 [2KLinearSolver iteration 26, residual = 0.24068756246457923 [2KLinearSolver iteration 27, residual = 0.2340879104748604 [2KLinearSolver iteration 28, residual = 0.13210286673685911 [2KLinearSolver iteration 29, residual = 0.12149542221443291

LinearSolver iteration 30, residual = 0.06395535062747568

LinearSolver iteration 31, residual = 0.06313158950831 LinearSolver iteration 32, residual = 0.030899302683804065 LinearSolver iteration 33, residual = 0.029415713731240026 LinearSolver iteration 34, residual = 0.015690961721274318 LinearSolver iteration 35, residual = 0.014673468623521629 LinearSolver iteration 36, residual = 0.008379750118119063 LinearSolver iteration 37, residual = 0.008238251565404938 LinearSolver iteration 38, residual = 0.0057093459974702524 LinearSolver iteration 39, residual = 0.005684975230312444 LinearSolver iteration 40, residual = 0.0038358687668496633 LinearSolver iteration 41, residual = 0.003832185229124313 LinearSolver iteration 42, residual = 0.0021014564444746943 LinearSolver iteration 43, residual = 0.002079171810906782 LinearSolver iteration 44, residual = 0.0010349257957436345 LinearSolver iteration 45, residual = 0.0009452428556793068 LinearSolver iteration 46, residual = 0.000501830825401925 LinearSolver iteration 47, residual = 0.0004925087878099025 LinearSolver iteration 48, residual = 0.00026249207049366166 LinearSolver iteration 49, residual = 0.00025892211488896306 LinearSolver iteration 50, residual = 0.00014905477022946883 LinearSolver iteration 51, residual = 0.0001476873822812559 LinearSolver iteration 52, residual = 8.918410229414048e-05 LinearSolver iteration 53, residual = 8.884336870670624e-05 LinearSolver iteration 54, residual = 5.4437880694775415e-05 LinearSolver iteration 55, residual = 5.330171690493887e-05 LinearSolver iteration 56, residual = 3.3180348230693587e-05 LinearSolver iteration 57, residual = 3.259166338177849e-05 LinearSolver iteration 58, residual = 1.8456984186078352e-05

</pre>

LinearSolver iteration 31, residual = 0.06313158950831 LinearSolver iteration 32, residual = 0.030899302683804065 LinearSolver iteration 33, residual = 0.029415713731240026 LinearSolver iteration 34, residual = 0.015690961721274318 LinearSolver iteration 35, residual = 0.014673468623521629 LinearSolver iteration 36, residual = 0.008379750118119063 LinearSolver iteration 37, residual = 0.008238251565404938 LinearSolver iteration 38, residual = 0.0057093459974702524 LinearSolver iteration 39, residual = 0.005684975230312444 LinearSolver iteration 40, residual = 0.0038358687668496633 LinearSolver iteration 41, residual = 0.003832185229124313 LinearSolver iteration 42, residual = 0.0021014564444746943 LinearSolver iteration 43, residual = 0.002079171810906782 LinearSolver iteration 44, residual = 0.0010349257957436345 LinearSolver iteration 45, residual = 0.0009452428556793068 LinearSolver iteration 46, residual = 0.000501830825401925 LinearSolver iteration 47, residual = 0.0004925087878099025 LinearSolver iteration 48, residual = 0.00026249207049366166 LinearSolver iteration 49, residual = 0.00025892211488896306 LinearSolver iteration 50, residual = 0.00014905477022946883 LinearSolver iteration 51, residual = 0.0001476873822812559 LinearSolver iteration 52, residual = 8.918410229414048e-05 LinearSolver iteration 53, residual = 8.884336870670624e-05 LinearSolver iteration 54, residual = 5.4437880694775415e-05 LinearSolver iteration 55, residual = 5.330171690493887e-05 LinearSolver iteration 56, residual = 3.3180348230693587e-05 LinearSolver iteration 57, residual = 3.259166338177849e-05 LinearSolver iteration 58, residual = 1.8456984186078352e-05

end{sphinxVerbatim}

[2KLinearSolver iteration 31, residual = 0.06313158950831 [2KLinearSolver iteration 32, residual = 0.030899302683804065 [2KLinearSolver iteration 33, residual = 0.029415713731240026 [2KLinearSolver iteration 34, residual = 0.015690961721274318 [2KLinearSolver iteration 35, residual = 0.014673468623521629 [2KLinearSolver iteration 36, residual = 0.008379750118119063 [2KLinearSolver iteration 37, residual = 0.008238251565404938 [2KLinearSolver iteration 38, residual = 0.0057093459974702524 [2KLinearSolver iteration 39, residual = 0.005684975230312444 [2KLinearSolver iteration 40, residual = 0.0038358687668496633 [2KLinearSolver iteration 41, residual = 0.003832185229124313 [2KLinearSolver iteration 42, residual = 0.0021014564444746943 [2KLinearSolver iteration 43, residual = 0.002079171810906782 [2KLinearSolver iteration 44, residual = 0.0010349257957436345 [2KLinearSolver iteration 45, residual = 0.0009452428556793068 [2KLinearSolver iteration 46, residual = 0.000501830825401925 [2KLinearSolver iteration 47, residual = 0.0004925087878099025 [2KLinearSolver iteration 48, residual = 0.00026249207049366166 [2KLinearSolver iteration 49, residual = 0.00025892211488896306 [2KLinearSolver iteration 50, residual = 0.00014905477022946883 [2KLinearSolver iteration 51, residual = 0.0001476873822812559 [2KLinearSolver iteration 52, residual = 8.918410229414048e-05 [2KLinearSolver iteration 53, residual = 8.884336870670624e-05 [2KLinearSolver iteration 54, residual = 5.4437880694775415e-05 [2KLinearSolver iteration 55, residual = 5.330171690493887e-05 [2KLinearSolver iteration 56, residual = 3.3180348230693587e-05 [2KLinearSolver iteration 57, residual = 3.259166338177849e-05 [2KLinearSolver iteration 58, residual = 1.8456984186078352e-05

LinearSolver iteration 59, residual = 1.8306187978107568e-05

LinearSolver iteration 60, residual = 9.729978738601966e-06 LinearSolver iteration 61, residual = 9.554842425630255e-06 LinearSolver iteration 62, residual = 5.161704605552928e-06 LinearSolver iteration 63, residual = 5.149417860119836e-06 LinearSolver iteration 64, residual = 2.365836031715507e-06 LinearSolver iteration 65, residual = 2.2957609093109607e-06 LinearSolver iteration 66, residual = 8.861914669937115e-07 LinearSolver converged in 66 iterations to residual 8.861914669937115e-07

</pre>

LinearSolver iteration 60, residual = 9.729978738601966e-06 LinearSolver iteration 61, residual = 9.554842425630255e-06 LinearSolver iteration 62, residual = 5.161704605552928e-06 LinearSolver iteration 63, residual = 5.149417860119836e-06 LinearSolver iteration 64, residual = 2.365836031715507e-06 LinearSolver iteration 65, residual = 2.2957609093109607e-06 LinearSolver iteration 66, residual = 8.861914669937115e-07 LinearSolver converged in 66 iterations to residual 8.861914669937115e-07

end{sphinxVerbatim}

[2KLinearSolver iteration 60, residual = 9.729978738601966e-06 [2KLinearSolver iteration 61, residual = 9.554842425630255e-06 [2KLinearSolver iteration 62, residual = 5.161704605552928e-06 [2KLinearSolver iteration 63, residual = 5.149417860119836e-06 [2KLinearSolver iteration 64, residual = 2.365836031715507e-06 [2KLinearSolver iteration 65, residual = 2.2957609093109607e-06 [2KLinearSolver iteration 66, residual = 8.861914669937115e-07 [2KLinearSolver converged in 66 iterations to residual 8.861914669937115e-07

[25]:

Draw (gfu)
Draw (gfp);

[ ]: