A FEniCS Tutorial - FEniCS Project

The demo program d2_p2D.py in the stationary/poisson directory incorporates
the above shown control of the linear solver and precnditioner, but is
otherwise similar to the previous d1_p2D.py program.
We remark that default values for the global parameter database can be
defined in an XML file, see the example file dolfin_parameters.xml in the
directory stationary/poisson. If such a file is found in the directory where a
FEniCS program is run, this file is read and used to initialize the parameters
object. Otherwise, the file .config/fenics/dolfin_parameters.xml in the
user’s home directory is read, if it exists. The XML file can also be in gzip’ed
form with the extension .xml.gz.
1.5 Linear Variational Problem and Solver Objects
The solve(a == L, u, bc) call is just a compact syntax alternative to a
slightly more comprehensive specification of the variational equation and the
solution of the associated linear system. This alternative syntax is used in a lot
of FEniCS applications and will also be used later in this tutorial, so we show
it already now:
u = Function(V)
problem = LinearVariationalProblem(a, L, u, bc)
solver = LinearVariationalSolver(problem)
solver.solve()
Many objects have an attribute parameters corresponding to a parameter
set in the global parameters database, but local to the object. Here,
solver.parameters play that role. Setting the CG method with ILU preconditiong
as solution method and specifying solver-specific parameters can be
done like this:
solver.parameters[’linear_solver’] = ’cg’
solver.parameters[’preconditioner’] = ’ilu’
cg_prm = solver.parameters[’krylov_solver’] # short form
cg_prm[’absolute_tolerance’] = 1E-7
cg_prm[’relative_tolerance’] = 1E-4
cg_prm[’maximum_iterations’] = 1000
Calling info(solver.parameters, True) lists all the available parameter sets
with default values for each parameter. Settings in the global parameters
database are propagated to parameter sets in individual objects, with the possibility
of being overwritten as done above.
The d3_p2D.py program modifies the d2_p2D.py file to incorporate objects
for the variational problem and solver.
1.6 Examining the Discrete Solution
Weknowthat, intheparticularboundary-valueproblemofSection1.3, thecomputed
solution u should equal the exact solution at the vertices of the cells. An
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