A FEniCS Tutorial - FEniCS Project

The weak form then becomes
∫ ∫ ∫ ∫
∇u·∇vdx+ gvds+ p(u−q)vds =
Ω Γ N Γ R
Ω
fvdx,
We want to write this weak form in the standard notation a(u,v) = L(v), which
requires that we identify all integrals with both u and v, and collect these in
a(u,v), whiletheremainingintegralswithv andnotugointoL(v). Theintegral
from the Robin condition must of this reason be split in two parts:
∫ ∫ ∫
p(u−q)vds = puvds− pqvds.
Γ R Γ R Γ R
We then have
∫
a(u,v) =
∫
L(v) =
Ω
Ω
∫
∇u·∇vdx+ puvds, (95)
Γ
∫
R
∫
fvdx− gvds+ pqvds. (96)
Γ N Γ R
A natural starting point for implementation is the dn2_p2D.py program in
the directory stationary/poisson. The new aspects are
• definition of a mesh function over the boundary,
• marking each side as a subdomain, using the mesh function,
• splitting a boundary integral into parts.
Task 1 makes use of the MeshFunction object, but contrary to Section 5.2, this
is not a function over cells, but a function over cell facets. The topological dimension
of cell facets is one lower than the cell interiors, so in a two-dimensional
problem the dimension becomes 1. In general, the facet dimension is given as
mesh.topology().dim()-1, which we use in the code for ease of direct reuse in
other problems. The construction of a MeshFunction object to mark boundary
parts now reads
boundary_parts = \
MeshFunction("uint", mesh, mesh.topology().dim()-1)
As in Section 5.2 we use a subclass of SubDomain to identify the various parts of
the mesh function. Problems with domains of more complicated geometries may
set the mesh function for marking boundaries as part of the mesh generation.
In our case, the y = 0 boundary can be marked by
class LowerRobinBoundary(SubDomain):
def inside(self, x, on_boundary):
tol = 1E-14 # tolerance for coordinate comparisons
return on_boundary and abs(x[1]) < tol
Gamma_R = LowerRobinBoundary()
Gamma_R.mark(boundary_parts, 0)
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