A FEniCS Tutorial - FEniCS Project

The variational formulation of our model problem reads: Find u ∈ V such
that
F(u;v) = 0 ∀v ∈ ˆV, (40)
where
and
∫
F(u;v) =
ˆV = {v ∈ H 1 (Ω) : v = 0 on x 0 = 0 and x 0 = 1},
Ω
q(u)∇u·∇vdx, (41)
V = {v ∈ H 1 (Ω) : v = 0 on x 0 = 0 and v = 1 on x 0 = 1}.
The discrete problem arises as usual by restricting V and ˆV to a pair of discrete
spaces. As usual, we omit any subscript on discrete spaces and simply say V and
ˆV arechosenfinitedimensionalaccordingtosomemeshwithsomeelementtype.
Similarly, we let u be the discrete solution and use ue for the exact solution if
it becomes necessary to distinguish between the two.
The discrete nonlinear problem is then wirtten as: find u ∈ V such that
F(u;v) = 0 ∀v ∈ ˆV, (42)
with u = ∑ N
j=1 U jφ j . Since F is a nonlinear function of u, the variational
statement gives rise to a system of nonlinear algebraic equations. [[[ FEniCS
can be used in alternative ways for solving a nonlinear PDE problem. We shall
in the following subsections go through four solution strategies:
1. a simple Picard-type iteration,
2. a Newton method at the algebraic level,
3. a Newton method at the PDE level, and
4. an automatic approach where FEniCS attacks the nonlinear variational
problem directly.
The ”black box” strategy 4 is definitely the simplest one from a programmer’s
point of view, but the others give more manual control of the solution process
for nonlinear equations (which also has some pedagogical advantages, especially
for newcomers to nonlinear finite element problems).
2.1 Picard Iteration
Picard iteration is an easy way of handling nonlinear PDEs: we simply use
a known, previous solution in the nonlinear terms so that these terms become
linear in the unknown u. The strategy is also known as the method of successive
substitutions. For our particular problem, we use a known, previous solution
in the coefficient q(u). More precisely, given a solution u k from iteration k, we
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