A FEniCS Tutorial - FEniCS Project

accordingly:
f = −6,
{ −4, y = 1
g =
0, y = 0
u 0 = 1+x 2 +2y 2 .
For ease of programming we may introduce a g function defined over the whole
of Ω such that g takes on the right values at y = 0 and y = 1. One possible
extension is
g(x,y) = −4y.
The first task is to derive the variational problem. This time we cannot omit
the boundary term arising from the integration by parts, because v is only zero
on Γ D . We have
∫ ∫ ∫
− (∇ 2 ∂u
u)vdx = ∇u·∇vdx−
Ω Ω ∂Ω ∂n vds,
and since v = 0 on Γ D ,
∫
−
∂Ω
∫
∂u
∂n vds = −
Γ N
∂u
∂n vds = ∫
Γ N
gvds,
by applying the boundary condition on Γ N . The resulting weak form reads
∫ ∫ ∫
∇u·∇vdx+ gvds = fvdx. (33)
Ω Γ N
Expressing this equation in the standard notation a(u,v) = L(v) is straightforward
with
∫
a(u,v) = ∇u·∇vdx, (34)
Ω
∫ ∫
L(v) = fvdx− gvds. (35)
Γ N
Ω
How does the Neumann condition impact the implementation? Starting
with any of the previous files d*_p2D.py, say d4_p2D.py, we realize that the
statements remain almost the same. Only two adjustments are necessary:
• The function describing the boundary where Dirichlet conditions apply
must be modified.
• The new boundary term must be added to the expression in L.
Step 1 can be coded as
Ω
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