A FEniCS Tutorial - FEniCS Project

an implementation where u is the TrialFunction. We have implemented both
approaches in two files: vp1_np.py with u as TrialFunction, and vp2_np.py
with du as TrialFunction. The directory stationary/nonlinear_poisson
contains both files. The first command-line argument determines if the Jacobian
is to be automatically derived or computed from the hand-derived formula.
The following code defines the nonlinear variational problem and an associated
solver based on Newton’s method. We here demonstrate how key parameters
in Newton’s method can be set, as well as the choice of solver and
preconditioner, and associated parameters, for the linear system occurring in
the Newton iteration.
problem = NonlinearVariationalProblem(F, u_, bcs, J)
solver = NonlinearVariationalSolver(problem)
prm = solver.parameters
prm[’newton_solver’][’absolute_tolerance’] = 1E-8
prm[’newton_solver’][’relative_tolerance’] = 1E-7
prm[’newton_solver’][’maximum_iterations’] = 25
prm[’newton_solver’][’relaxation_parameter’] = 1.0
if iterative_solver:
prm[’linear_solver’] = ’gmres’
prm[’preconditioner’] = ’ilu’
prm[’krylov_solver’][’absolute_tolerance’] = 1E-9
prm[’krylov_solver’][’relative_tolerance’] = 1E-7
prm[’krylov_solver’][’maximum_iterations’] = 1000
prm[’krylov_solver’][’gmres’][’restart’] = 40
prm[’krylov_solver’][’preconditioner’][’ilu’][’fill_level’] = 0
set_log_level(PROGRESS)
solver.solve()
A list of available parameters and their default values can as usual be printed by
calling info(prm, True). The u_ we feed to the nonlinear variational problem
object is filled with the solution by the call solver.solve().
3 Time-Dependent Problems
The examples in Section 1 illustrate that solving linear, stationary PDE problems
with the aid of FEniCS is easy and requires little programming. That is,
FEniCS automates the spatial discretization by the finite element method. The
solutionofnonlinearproblems, asweshowedinSection2, canalsobeautomated
(cf. Section 2.4), but many scientists will prefer to code the solution strategy of
the nonlinear problem themselves and experiment with various combinations of
strategies in difficult problems. Time-dependent problems are somewhat similar
in this respect: we have to add a time discretization scheme, which is often quite
simple, making it natural to explicitly code the details of the scheme so that the
programmer has full control. We shall explain how easily this is accomplished
through examples.
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