A FEniCS Tutorial - FEniCS Project

We may collect the terms with the unknown δu on the left-hand side,
∇·(q(u k )∇δu ) +∇·(q ′ (u k )δu∇u k) = −∇·(q(u k )∇u k) , (58)
The weak form of this PDE is derived by multiplying by a test function v and
integrating over Ω, integrating as usual the second-order derivatives by parts:
∫
(
q(u k )∇δu·∇v +q ′ (u k )δu∇u k ·∇v ) ∫
dx = − q(u k )∇u k ·∇vdx. (59)
Ω
The variational problem reads: find δu ∈ V such that a(δu,v) = L(v) for all
v ∈ ˆV, where
∫
(
a(δu,v) = q(u k )∇δu·∇v +q ′ (u k )δu∇u k ·∇v ) dx, (60)
Ω∫
L(v) = − q(u k )∇u k ·∇vdx. (61)
Ω
The function spaces V and ˆV, being continuous or discrete, are as in the linear
Poisson problem from Section 1.2.
We must provide some initial guess, e.g., the solution of the PDE with
q(u) = 1. The corresponding weak form a 0 (u 0 ,v) = L 0 (v) has
∫
a 0 (u,v) = ∇u·∇vdx, L 0 (v) = 0.
Ω
Thereafter, we enter a loop and solve a(δu,v) = L(v) for δu and compute a new
approximation u k+1 = u k + δu. Note that δu is a correction, so if u 0 satisfies
the prescribed Dirichlet conditions on some part Γ D of the boundary, we must
demand δu = 0 on Γ D .
Looking at (60) and (61), we see that the variational form is the same as for
the Newton method at thealgebraic level inSection 2.2. SinceNewton’s method
at the algebraic level required some ”backward” construction of the underlying
weak forms, FEniCS users may prefer Newton’s method at the PDE level, which
this author finds more straightforward, although not so commonly documented
in the literature on numerical methods for PDEs. There is seemingly no need
for differentiations to derive a Jacobian matrix, but a mathematically equivalent
derivation is done when nonlinear terms are linearized using the first two Taylor
series terms and when products in the perturbation δu are neglected.
The implementation is identical to the one in Section 2.2 and is found in the
file pde_newton_np.py. The reader is encouraged to go through this code to be
convinced that the present method actually ends up with the same program as
needed for the Newton method at the linear algebra level in Section 2.2.
2.4 Solving the Nonlinear Variational Problem Directly
The previous hand-calculations and manual implementation of Picard or Newton
methods can be automated by tools in FEniCS. In a nutshell, one can just
write
Ω
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