A FEniCS Tutorial - FEniCS Project

Let us continue to use our favorite solution u(x,y) = 1 + x 2 + 2y 2 and
then prescribe p(x,y) = x + y. It follows that u 0 (x,y) = 1 + x 2 + 2y 2 and
f(x,y) = −8x−10y.
What are the modifications we need to do in the d4_p2D.py program from
Section 1.6?
• f must be an Expression since it is no longer a constant,
• a new Expression ‘p‘ must be defined for the variable coefficient,
• the variational problem is slightly changed.
First we address the modified variational problem. Multiplying the PDE by a
test function v and integrating by parts now results in
∫ ∫
p∇u·∇vdx− p ∂u
∂n
∫Ω
vds = fvdx.
Ω
∂Ω
The function spaces for u and v are the same as in Section 1.2, implying that
the boundary integral vanishes since v = 0 on ∂Ω where we have Dirichlet
conditions. The weak form a(u,v) = L(v) then has
∫
a(u,v) = p∇u·∇vdx, (22)
Ω
∫
L(v) = fvdx. (23)
In the code from Section 1.3 we must replace
a = inner(nabla_grad(u), nabla_grad(v))*dx
by
a = p*inner(nabla_grad(u), nabla_grad(v))*dx
The definitions of p and f read
p = Expression(’x[0] + x[1]’)
f = Expression(’-8*x[0] - 10*x[1]’)
Ω
No additional modifications are necessary. The complete code can be found in
in the file vcp2D.py (variable-coefficient Poisson problem in 2D). You can run
it and confirm that it recovers the exact u at the nodes.
The flux −p∇u may be of particular interest in variable-coefficient Poisson
problems as it often has an interesting physical significance. As explained in
Section 1.9, we normally want the piecewise discontinuous flux or gradient to be
approximated by a continuous vector field, using the same elements as used for
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