Static condensation refers to the process of eliminating unknowns that are internal to elements from the global linear system. They are useful in standard methods and critical in methods like the HDG method. NGSolve automates static condensation across a variety of methods via a classification of degrees of freedom.
import netgen.gui
%gui tk
from ngsolve import *
from netgen.geom2d import unit_square
mesh = Mesh(unit_square.GenerateMesh(maxh=0.4))
fes = H1(mesh, order=4, dirichlet='bottom|right')
u = fes.TestFunction()
v = fes.TrialFunction()
a = BilinearForm(fes, flags={"eliminate_internal":True})
a += SymbolicBFI (grad(u) * grad(v))
a.Assemble()
f = LinearForm(fes)
f += SymbolicLFI (1 * v)
f.Assemble()
u = GridFunction(fes)
The assembled matrix $A=$ a.mat can be block partitioned into
$$
A =
\left(
\begin{array}{cc}
A_{LL} & A_{LE}
\\
A_{EL} & A_{EE}
\end{array}
\right)
$$
where
$L$ denotes the set of local or internal degrees of freedom and $E$ denotes the set of interface degrees of freedom.
In our current example $E$ consists of edge and vertex dofs while $L$ consists of triangle dofs. (Note that in practice, $L$ and $E$ may not be ordered contiguously and $L$ need not appear before $E$, but such details are immaterial for our discussion here.)
The condensed system is known as the Schur complement: $$ S = A_{EL} - A_{EL} A_{LL}^{-1} A_{LE}. $$
When eliminate_internal is set to True in a, the statement a.Assemble actually assembles $S$.
NGSolve provides
a.harmonic_extension_trans
$=
\left(\begin{array}{cc}
0 & 0 \\
-A_{EI} A_{LL}^{-1} & 0
\end{array}\right)
$
a.harmonic_extension
$=
\left(\begin{array}{cc}
0 & -A_{LL}^{-1} A_{LE}
\\
0 & 0
\end{array}\right)
$
a.inner_solve
$=\left(\begin{array}{cc}
A_{LL}^{-1} & 0
\\
0 & 0
\end{array}\right)$.
To solve $$ \left(\begin{array}{cc} A_{LL} & A_{LE}\\ A_{EL} & A_{EE} \end{array}\right) \left(\begin{array}{c} u_L \\ u_E \end{array}\right)= \left(\begin{array}{c} f_L \\ f_E \end{array}\right) $$
we use a factorization of $A^{-1}$ that uses $S^{-1}$. Namely, we use the following identity:
$$ \left(\begin{array}{c} u_L \\ u_E \end{array}\right) = \left(\begin{array}{cc} I & -A_{LL}^{-1} A_{LE} \\ 0 & I \end{array}\right) \left(\begin{array}{cc} A_{LL}^{-1} & 0 \\ 0 & S^{-1} \end{array}\right) \underbrace{ \left(\begin{array}{cc} I & 0 \\ -A_{EL} A_{LL}^{-1} & I \end{array}\right) \left(\begin{array}{c} f_L \\ f_E \end{array}\right)}_{ \left(\begin{array}{c} f'_L \\ f'_E \end{array}\right) } $$We implement this formula step by step, starting with the computation of $f_L'$ and $f_E'$.
f.vec.data += a.harmonic_extension_trans * f.vec
u.vec.data = \
a.mat.Inverse(freedofs=fes.FreeDofs(coupling=True)) * f.vec
Note:
Because we set eliminate_internal in a, the inverse a.mat.Inverse actually computes $S^{-1}$.
Note that instead of the usual fes.FreeDofs(), we have used fes.FreeDofs(coupling=True) or simply fes.FreeDofs(True) to specify that only the degrees of freedom that are not local and not Dirichlet should participate in the inverse computations. (The underlying assumption is that Dirichlet dofs cannot be local dofs.)
u.vec.data += a.inner_solve * f.vec
u.vec.data += a.harmonic_extension * u.vec
Draw(u)
CouplingType¶How does NGSolve know what is in the index sets $L$ and $E$?
fes.CouplingType to see a classification of degrees of freedom.for i in range(fes.ndof):
print(fes.CouplingType(i))
dof_types = {}
for i in range(fes.ndof):
type = fes.CouplingType(i)
if type in dof_types.keys():
dof_types[type] += 1
else:
dof_types[type] = 1
dof_types
The LOCAL_DOF forms the set $L$ and the remainder forms the set $E$. All finite element spaces in NGSolve have such dof classification.
Through this classification a bilinear form is able to automatically compute the Schur complement and the accompanying extension operators. Users need only specify the flag eliminate_internal. (Of course users should also make sure their method has an invertible $A_{LL}$!)
In case of inhomogeneous Dirichlet boundary conditions we must combine the technique of Dirichlet data extension in a previous tutorial with static condensation:
U = x*x*(1-y)*(1-y) # U = manufactured solution
DeltaU = 2*((1-y)*(1-y)+x*x) # Source: DeltaU = ∆U
f = LinearForm(fes)
f += SymbolicLFI(-DeltaU*v)
f.Assemble()
u = GridFunction(fes)
u.Set(U, BND) # Dirichlet b.c: u = U
# Modify source per Dirichlet extension technique:
r = f.vec.CreateVector()
r.data = f.vec - a.mat * u.vec
r.data += a.harmonic_extension_trans * r
# Apply the static condensation technique:
u.vec.data += a.mat.Inverse(fes.FreeDofs(True)) * r
u.vec.data += a.harmonic_extension * u.vec
u.vec.data += a.inner_solve * r
sqrt(Integrate((U-u)*(U-u),mesh)) # Compute error