This page was generated from unit-2.7-hybrid/hybrid.ipynb.

2.7 Facet spaces and hybrid methods

Mixed methods for second order problems lead to saddle point problems, and indefinite matrices. By hybridization one obtains a positive definite system again. It’s structure is similar to the non-conforming \(P^1\) method, but hybridization works for any order. See text-book by Brezzi and Fortin.

One skips the normal-continuity of the \(H(div)\) variable, and reinforces it by a Lagrange parameter. This leads to the following discrete system:

Find \(\sigma, u, \widehat u \in \Sigma_h \times V_h \times F_h\):

\[\begin{split}\DeclareMathOperator{\Div}{div} \begin{array}{ccccccll} \int \lambda^{-1} \sigma \tau & + & \sum_T \int_T \Div \tau \, u & + & \sum_F \int_F [\tau_n] \widehat u & = & 0 & \forall \, \tau \in \Sigma \\ \int \Div \sigma \, v &&&&& = & \int f v & \forall \, v \in V_h \\ \int [ \sigma_n ] \, \widehat v &&&&& = & \int_{\Gamma_n} g \widehat v & \forall \, \widehat v \in F_h \end{array}\end{split}\]

where \(\Sigma_h\) is an discontinuous \(H(div)\) finite element space, \(V_h\) a sub-space of \(L_2\), and \(F_h\) consists of polynomials on every edge.

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

same example as in 'mixed':

source = sin(3.14*x)
ud = CoefficientFunction(5)
g = CoefficientFunction([y*(1-y) if bc=="left" else 0 for bc in mesh.GetBoundaries()])
lam = 10

define spaces:

• The discontinuous flag generates an element-wise \(H(Div)\)-space

• FacetFESpace lives only on facets (i.e. faces in 3D, edges in 2D, points in 1D)

Boundary conditions are now posed for the facet-space

order = 3
V = HDiv(mesh, order=order, discontinuous=True)
# V = Discontinuous(HDiv(mesh, order=order))
Q = L2(mesh, order=order-1)
F = FacetFESpace(mesh, order=order, dirichlet="bottom")
X = FESpace([V,Q,F])
print ("sigmadofs:", X.Range(0))
print ("udofs:    ", X.Range(1))
print ("uhatdofs: ", X.Range(2))

sigmadofs: [0,1080)
udofs:     [1080,1404)
uhatdofs:  [1404,1768)

Assemble forms. The jump-term is rewritten as

\[\sum_F \int_F [\sigma_n] v = \sum_T \int_{\partial T} \sigma_n v\]

sigma,u,uhat = X.TrialFunction()
tau,v,vhat = X.TestFunction()

a = BilinearForm(X, condense=False)
a += (1/lam * sigma*tau + div(sigma)*v + div(tau)*u) * dx
n = specialcf.normal(mesh.dim)
a += (-sigma*n*vhat-tau*n*uhat) * dx(element_boundary=True)

c = Preconditioner(a, "bddc")

f = LinearForm(X)
f += -source*v * dx - g*vhat.Trace() * ds

a.Assemble()
print ("A non-zero", a.mat.nze)

gfu = GridFunction(X)

A non-zero 76840

Solve system. Either we leave everything to the sparse direct solver, or use CG

f.Assemble()
gfu.components[2].Set(ud, BND)

if a.condense:

    f.vec.data += a.harmonic_extension_trans * f.vec

    solvers.CG(mat=a.mat, pre=c.mat, rhs=f.vec, sol=gfu.vec, initialize=False)

    gfu.vec.data += a.harmonic_extension * gfu.vec
    gfu.vec.data += a.inner_solve * f.vec

else:

    r = f.vec.CreateVector()
    r.data = f.vec - a.mat * gfu.vec
    inv = a.mat.Inverse(freedofs=X.FreeDofs())
    gfu.vec.data += inv * r

Draw (gfu.components[0], mesh, "sigma")
Draw (gfu.components[1], mesh, "u")