2.10 Dual basis functions¶

We use dual basis functions to define interpolation operators, define transfer operators between different finite element spaces, and auxiliary space preconditioners.

Canonical interpolation¶

The canonical finite element interpolation operator is defined by specifying the degrees of freedom. For low order methods these are typically nodal values, while for high order methods these are most often moments. For example, the interpolation of a function $$u$$ onto the $$p^{th}$$ order triangle is given by: find $$u_{hp} \in V_{hp}$$ such that

\begin{eqnarray*} u_{hp} (V) & = & u(V) \quad \forall \text{ vertices } V \\ \int_E u_{hp} q & = & \int_E u q \quad \forall q \in P^{p-2}(E) \; \forall \text{ edges } E \\ \int_T u_{hp} q & = & \int_T u q \quad \forall q \in P^{p-3}(T) \; \forall \text{ triangles } T \end{eqnarray*}

[1]:

from ngsolve import *
from ngsolve.webgui import Draw

import matplotlib.pylab as plt
mesh = Mesh(unit_square.GenerateMesh(maxh=2))


The NGSolve 'Set' function does local projection, and simple averaging. In particular, this does not respect point values in mesh vertices.

[2]:

fes = H1(mesh, order=3, low_order_space=False)

func = x*x*x*x
gfu = GridFunction(fes)
gfu.Set(func)
Draw (gfu)
print (gfu.vec)

 -0.0223792
0.991843
0.977621
-0.00815655
3.34984
2.19912
3.34984
2
0.00660131
1.39013e-14
0.00660131
-3.19744e-14
3.34984
-1.80088
-2.17601
1.82399



Most NGSolve finite element spaces provide now a "dual" operator, which delivers the moments (i.e. the dual space basis functions) instead of function values. The integrals over faces, edges and also vertices are defined by co-dimension 1 (=BND), co-dimension 2 (=BBND) or co-dimension 3 (=BBBND) integrals over the volume elements. We define a variational problem for canonical interpolation:

[3]:

u,v = fes.TnT()
vdual = v.Operator("dual")

a = BilinearForm(fes)
a += u*vdual*dx + u*vdual*dx(element_vb=BND) + \
u*vdual*dx(element_vb=BBND)
a.Assemble()

f = LinearForm(fes)
f += func*vdual*dx + func*vdual*dx(element_vb=BND) + \
func*vdual*dx(element_vb=BBND)
f.Assemble()

# interpolation in vertices preserves values 0 and 1
gfu.vec.data = a.mat.Inverse() * f.vec
print (gfu.vec)
Draw (gfu);

       0
1
1
0
3.6
2
3.6
2
0
0
-6.93335e-32
-1.66533e-15
3.6
-2
-2
2



The vertex degrees of freedom vanish for edge and element basis functions, and the edge degrees of freedom vanish for element basis functions, but not vice-versa. Thus, the obtained matrix A is block-triangular:

[4]:

import scipy.sparse as sp
A = sp.csr_matrix(a.mat.CSR())
plt.rcParams['figure.figsize'] = (4,4)
plt.spy(A)
plt.show()

/usr/lib/python3/dist-packages/scipy/__init__.py:146: UserWarning: A NumPy version >=1.17.3 and <1.25.0 is required for this version of SciPy (detected version 1.26.4
warnings.warn(f"A NumPy version >={np_minversion} and <{np_maxversion}"


We can use proper block Gauss-Seidel smoothing for solving with that block triangular matrix by blocking the dofs for the individual vertices, edges and elements. Since the NGSolve Gauss-Seidel smoother reorders the order of smoothing blocks for parallelization, we have to take care to first compute vertex values, then edge values, and finally element values by running three different Gauss-Seidel sweeps.

[5]:

vblocks = [fes.GetDofNrs(vertex) for vertex in mesh.vertices]
eblocks = [fes.GetDofNrs(edge) for edge in mesh.edges]
fblocks = [fes.GetDofNrs(face) for face in mesh.faces]

print (vblocks)
print (eblocks)
print (fblocks)

vinv = a.mat.CreateBlockSmoother(vblocks)
einv = a.mat.CreateBlockSmoother(eblocks)
finv = a.mat.CreateBlockSmoother(fblocks)

vinv.Smooth(gfu.vec, f.vec)
einv.Smooth(gfu.vec, f.vec)
finv.Smooth(gfu.vec, f.vec)
print (gfu.vec)

[(0,), (1,), (2,), (3,)]
[(4, 5), (6, 7), (8, 9), (10, 11), (12, 13)]
[(14,), (15,)]
0
1
1
0
3.6
2
3.6
2
0
0
-6.93335e-32
-1.66533e-15
3.6
-2
-2
2



Embedding Finite Element Spaces¶

This interpolation can be used to transform functions from one finite element space $$V_{src}$$ to another one $$V_{dst}$$. We use the dual space of the destination space:

$\int_{node} u_{dst} v_{dual} = \int_{node} u_{src} v_{dual} \qquad \forall \, v_{dual} \; \forall \, \text{nodes}$

The left hand side leads to a non-symmetric square matrix, the right hand side to a rectangular matrix.

As an example we implement the transformation from an vector valued $$H^1$$ space into $$H(\operatorname{div})$$:

[6]:

from ngsolve import *
from ngsolve.webgui import Draw
from netgen.geom2d import unit_square
mesh = Mesh(unit_square.GenerateMesh(maxh=0.2))

fesh1 = VectorH1(mesh, order=2)
feshdiv = HDiv(mesh, order=2)

gfuh1 = GridFunction(fesh1)
gfuh1.Set ( (x*x,y*y) )

gfuhdiv = GridFunction(feshdiv, name="uhdiv")


Build the matrices, and use a direct solver:

[7]:

amixed = BilinearForm(trialspace=fesh1, testspace=feshdiv)
ahdiv = BilinearForm(feshdiv)

u,v = feshdiv.TnT()
vdual = v.Operator("dual")
uh1 = fesh1.TrialFunction()

dS = dx(element_boundary=True)
ahdiv += u*vdual * dx + u*vdual * dS
ahdiv.Assemble()

amixed += uh1*vdual*dx + uh1*vdual*dS
amixed.Assemble()

# build transformation operator:
transform = ahdiv.mat.Inverse() @ amixed.mat
gfuhdiv.vec.data = transform * gfuh1.vec

Draw (gfuh1)
Draw (gfuhdiv)

[7]:

BaseWebGuiScene


We implement a linear operator performing the fast conversion by Gauss-Seidel smoothing:

[8]:

class MyBlockInverse(BaseMatrix):
def __init__ (self, mat, eblocks, fblocks):
super(MyBlockInverse, self).__init__()
self.mat = mat
self.einv = mat.CreateBlockSmoother(eblocks)
self.finv = mat.CreateBlockSmoother(fblocks)
self.res = self.mat.CreateColVector()

def CreateRowVector(self):
return self.mat.CreateColVector()
def CreateColVector(self):
return self.mat.CreateRowVector()

def Mult(self, x, y):
# y[:] = 0
# self.einv.Smooth(y,x)    #   y = y +  A_E^-1  (x - A y)
# self.finv.Smooth(y,x)    #   y = y +  A_E^-1  (x - A y)

# the same, but we see how to transpose that
y.data = self.einv * x
self.res.data = x - self.mat * y
y.data += finv * self.res

def MultTrans(self, x, y):
y.data = self.finv.T * x
self.res.data = x - self.mat.T * y
y.data += einv.T * self.res

eblocks = [feshdiv.GetDofNrs(edge) for edge in mesh.edges]
fblocks = [feshdiv.GetDofNrs(face) for face in mesh.faces]

transform = MyBlockInverse(ahdiv.mat, eblocks, fblocks) @ amixed.mat
gfuhdiv.vec.data = transform * gfuh1.vec


Auxiliary Space Preconditioning¶

Nepomnyaschikh 91, Hiptmair-Xu 07, ….

Assume we have a complicated problem with some complicated discretization, and we have good preconditioners for a nodal finite element discretization for the Laplace operator. By auxiliary space preconditioning we can reuse the simple preconditioners for the complicated problems. It is simple, and works well in many cases.

As a simple example, we precondition a DG discretization by an $$H^1$$ conforming method.

[9]:

mesh = Mesh(unit_square.GenerateMesh(maxh=0.1))


The DG discretization:

[10]:

order=4
fesDG = L2(mesh, order=order, dgjumps=True)
u,v = fesDG.TnT()
jump_u = u-u.Other()
jump_v = v-v.Other()
n = specialcf.normal(2)
alpha = 4
h = specialcf.mesh_size
aDG += (-mean_dudn*jump_v -mean_dvdn*jump_u) * dx(skeleton=True)

fDG = LinearForm(fesDG)
fDG += 1*v * dx
fDG.Assemble()
gfuDG = GridFunction(fesDG)


The auxiliary $$H^1$$ discretization:

[11]:

fesH1 = H1(mesh, order=2, dirichlet=".*")
u,v = fesH1.TnT()
aH1 = BilinearForm(fesH1)
preH1 = Preconditioner(aH1, "bddc")
aH1.Assemble()

[11]:

<ngsolve.comp.BilinearForm at 0x7f9c040fa5f0>

[12]:

transform = fesH1.ConvertL2Operator(fesDG)
pre = transform @ preH1.mat @ transform.T + aDG.mat.CreateSmoother()

maxsteps=200)

Draw (gfuDG)

CG iteration 1, residual = 0.2532119733219237
CG iteration 2, residual = 0.032832421483037784
CG iteration 3, residual = 0.008130862525680703
CG iteration 4, residual = 0.003434248811060224
CG iteration 5, residual = 0.0011783456811530358
CG iteration 6, residual = 0.0009046659695335675
CG iteration 7, residual = 0.0009161002088522189
CG iteration 8, residual = 0.001069755888060449
CG iteration 9, residual = 0.0007971399933259396
CG iteration 10, residual = 0.0006627896366674116
CG iteration 11, residual = 0.0005095344857658828
CG iteration 12, residual = 0.0004246644493713289
CG iteration 13, residual = 0.0003261702302927999
CG iteration 14, residual = 0.0003439552949808912
CG iteration 15, residual = 0.0002710525126777649
CG iteration 16, residual = 0.00023336881656180788
CG iteration 17, residual = 0.000179837716816404
CG iteration 18, residual = 0.00013566763832044914
CG iteration 19, residual = 8.969965836308253e-05
CG iteration 20, residual = 7.588906438782425e-05
CG iteration 21, residual = 6.938043743084379e-05
CG iteration 22, residual = 6.381887596483378e-05
CG iteration 23, residual = 5.18020801910846e-05
CG iteration 24, residual = 4.3817631293676924e-05
CG iteration 25, residual = 3.64072498767944e-05
CG iteration 26, residual = 2.9808360832852204e-05
CG iteration 27, residual = 2.9223985181307486e-05
CG iteration 28, residual = 2.908952039243141e-05
CG iteration 29, residual = 2.6390001567258757e-05
CG iteration 30, residual = 2.0442446246668173e-05
CG iteration 31, residual = 1.7799471861373437e-05
CG iteration 32, residual = 1.4280353976732309e-05
CG iteration 33, residual = 1.1724086434440664e-05
CG iteration 34, residual = 1.0399082092347844e-05
CG iteration 35, residual = 9.357450254587507e-06
CG iteration 36, residual = 8.046132496256373e-06
CG iteration 37, residual = 6.4349727480618834e-06
CG iteration 38, residual = 4.977423074804303e-06
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CG iteration 40, residual = 3.879825859447484e-06
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CG iteration 48, residual = 1.412018002762827e-06
CG iteration 49, residual = 1.1860668787092792e-06
CG iteration 50, residual = 9.38436483757002e-07
CG iteration 51, residual = 7.855316166902485e-07
CG iteration 52, residual = 6.514966222954388e-07
CG iteration 53, residual = 5.847843124222835e-07
CG iteration 54, residual = 5.642748143010176e-07
CG iteration 55, residual = 4.856169352654835e-07
CG iteration 56, residual = 3.6490360544844576e-07
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CG iteration 60, residual = 1.81579453718241e-07
CG iteration 61, residual = 1.676579256739345e-07
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CG iteration 64, residual = 8.69249933766017e-08
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CG iteration 69, residual = 3.427094557093547e-08
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CG iteration 140, residual = 2.111743198953073e-13

[12]:

BaseWebGuiScene

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