A FEniCS Tutorial - FEniCS Project

where backendname is a string, either ’PETSc’, ’uBLAS’, ’Epetra’, or ’MTL4’.
All these backends offer high-quality implementations of both iterative and direct
solvers for linear systems of equations.
A common platform for FEniCS users is Ubuntu Linux. The FEniCS distribution
for Ubuntu contains PETSc, making this package the default linear
algebra backend. The default solver is sparse LU decomposition (’lu’), and
the actual software that is called is then the sparse LU solver from UMFPACK
(which PETSc has an interface to).
We will normally like to control the tolerance in the stopping criterion and
the maximum number of iterations when running an iterative method. Such
parameters can be set by accessing the global parameter database, which is called
parameters and which behaves as a nested dictionary. Write
info(parameters, True)
to list all parameters and their default values in the database. The nesting of
parameter sets is indicated through indentation in the output from info. According
to this output, the relevant parameter set is named ’krylov_solver’,
and the parameters are set like this:
prm = parameters[’krylov_solver’] # short form
prm[’absolute_tolerance’] = 1E-10
prm[’relative_tolerance’] = 1E-6
prm[’maximum_iterations’] = 1000
Stopping criteria for Krylov solvers usually involve the norm of the residual,
which must be smaller than the absolute tolerance parameter or smaller than
the relative tolerance parameter times the initial residual.
To see the number of actual iterations to reach the stopping criterion, we
can insert
set_log_level(PROGRESS)
# or
set_log_level(DEBUG)
A message with the equation system size, solver type, and number of iterations
arises from specifying the argument PROGRESS, while DEBUG results in more information,
includingCPUtimespentinthevariouspartsofthematrixassembly
and solve process.
The complete solution process with control of the solver parameters now
contains the statements
prm = parameters[’krylov_solver’] # short form
prm[’absolute_tolerance’] = 1E-10
prm[’relative_tolerance’] = 1E-6
prm[’maximum_iterations’] = 1000
set_log_level(PROGRESS)
solve(a == L, u, bc,
solver_parameters={’linear_solver’: ’cg’,
’preconditioner’: ’ilu’})
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