A FEniCS Tutorial - FEniCS Project

1.11 Computing Functionals
After the solution u of a PDE is computed, we occasionally want to compute
functionals of u, for example,
1
2 ||∇u||2 ≡ 1 ∫
∇u·∇udx, (27)
2
which often reflects some energy quantity. Another frequently occurring functional
is the error
(∫ 1/2
||ue −u|| = (ue −u) dx) 2 , (28)
Ω
where u e is the exact solution. The error is of particular interest when studying
convergence properties. Sometimes the interest concerns the flux out of a part
Γ of the boundary ∂Ω, ∫
F = − p∇u·nds, (29)
Γ
where n is an outward unit normal at Γ and p is a coefficient (see the problem in
Section 1.10 for a specific example). All these functionals are easy to compute
with FEniCS, and this section describes how it can be done.
Energy Functional. The integrand of the energy functional (27) is described
in the UFL language in the same manner as we describe weak forms:
energy = 0.5*inner(grad(u), grad(u))*dx
E = assemble(energy)
The assemble call performs the integration. It is possible to restrict the integration
to subdomains, or parts of the boundary, by using a mesh function to
mark the subdomains as explained in Section 5.3. The program membrane2.py
carries out the computation of the elastic energy
1
2 ||T∇D||2 = 1 2
Ω
( AR
8πσ
in the membrane problem from Section 1.7.
) 2
||∇w|| 2
Convergence Estimation. Toillustrateerrorcomputationsandconvergence
of finite element solutions, we modify the d5_p2D.py program from Section 1.9
and specify a more complicated solution,
u(x,y) = sin(ωπx)sin(ωπy)
on the unit square. This choice implies f(x,y) = 2ω 2 π 2 u(x,y). With ω restricted
to an integer it follows that u 0 = 0.
We need to define the appropriate boundary conditions, the exact solution,
and the f function in the code:
31