# Stokes Equations
# ================
#
# A simple example of a saddle-point system, we will use the Stokes
# equations to demonstrate some of the ways we can do field-splitting
# with matrix-free operators. We set up the problem as a lid-driven
# cavity.
#
# As ever, we import firedrake and define a mesh.::
from firedrake import *
N = 64
M = UnitSquareMesh(N, N)
V = VectorFunctionSpace(M, "CG", 2)
W = FunctionSpace(M, "CG", 1)
Z = V * W
u, p = TrialFunctions(Z)
v, q = TestFunctions(Z)
a = (inner(grad(u), grad(v)) - inner(p, div(v)) + inner(div(u), q))*dx
L = inner(Constant((0, 0)), v) * dx
# The boundary conditions are defined on the velocity space. Zero
# Dirichlet conditions on the bottom and side walls, a constant :math:`u
# = (1, 0)` condition on the lid.::
bcs = [DirichletBC(Z.sub(0), Constant((1, 0)), (4,)),
DirichletBC(Z.sub(0), Constant((0, 0)), (1, 2, 3))]
up = Function(Z)
# Since we do not specify boundary conditions on the pressure space, it
# is only defined up to a constant. We will remove this component of
# the solution in the solver by providing the appropriate nullspace.::
nullspace = MixedVectorSpaceBasis(
Z, [Z.sub(0), VectorSpaceBasis(constant=True)])
# First up, we will solve the problem directly. For this to work, the
# sparse direct solver MUMPS must be installed. Hence this solve is
# wrapped in a ``try/except`` block so that an error is not raised in
# the case that it is not, to do this we must import ``PETSc``::
from firedrake.petsc import PETSc
# To factor the matrix from this mixed system, we must specify
# a ``mat_type`` of ``aij`` to the solve call.::
try:
solve(a == L, up, bcs=bcs, nullspace=nullspace,
solver_parameters={"ksp_type": "gmres",
"mat_type": "aij",
"pc_type": "lu",
"pc_factor_mat_solver_type": "mumps"})
except PETSc.Error as e:
if e.ierr == 92:
warning("MUMPS not installed, skipping direct solve")
else:
raise e
# Now we'll use a Schur complement preconditioner using unassembled
# matrices. We can do all of this purely by changing the solver
# options. We'll define the parameters separately to run through the
# options.::
parameters = {
# First up we select the unassembled matrix type::
"mat_type": "matfree",
# Now we configure the solver, using GMRES using the diagonal part of
# the Schur complement factorisation to approximate the inverse. We'll
# also monitor the convergence of the residual, and ask PETSc to view
# the configured Krylov solver object.::
"ksp_type": "gmres",
"ksp_monitor_true_residual": None,
"ksp_view": None,
"pc_type": "fieldsplit",
"pc_fieldsplit_type": "schur",
"pc_fieldsplit_schur_fact_type": "diag",
# Next we configure the solvers for the blocks. For the velocity block,
# we use an :class:`.AssembledPC` and approximate the inverse of the
# vector laplacian using a single multigrid V-cycle.::
"fieldsplit_0_ksp_type": "preonly",
"fieldsplit_0_pc_type": "python",
"fieldsplit_0_pc_python_type": "firedrake.AssembledPC",
"fieldsplit_0_assembled_pc_type": "hypre",
# For the Schur complement block, we approximate the inverse of the
# schur complement with a pressure mass inverse. For constant viscosity
# this works well. For variable, but low-contrast viscosity, one should
# use a viscosity-weighted mass-matrix. This is achievable by passing a
# dictionary with "mu" associated with the viscosity into solve. The
# MassInvPC will choose a default value of 1.0 if not set. For high viscosity
# contrasts, this preconditioner is mesh-dependent and should be replaced
# by some form of approximate commutator.::
"fieldsplit_1_ksp_type": "preonly",
"fieldsplit_1_pc_type": "python",
"fieldsplit_1_pc_python_type": "firedrake.MassInvPC",
# The mass inverse is dense, and therefore approximated with a Krylov
# iteration, which we configure now::
"fieldsplit_1_Mp_ksp_type": "preonly",
"fieldsplit_1_Mp_pc_type": "ilu"
}
# Having set up the parameters, we can now go ahead and solve the
# problem.::
up.assign(0)
solve(a == L, up, bcs=bcs, nullspace=nullspace, solver_parameters=parameters)
# Last, but not least, we'll write the solution to a file for later
# visualisation. We split the function into its velocity and pressure
# parts and give them reasonable names, then write them to a paraview
# file.::
u, p = up.split()
u.rename("Velocity")
p.rename("Pressure")
File("stokes.pvd").write(u, p)
# By default, the mass matrix is assembled in the :class:`~.MassInvPC`
# preconditioner, however, this can be controlled using a ``mat_type``
# argument. To do this, we must specify the ``mat_type`` inside the
# preconditioner. We can use the previous set of parameters and just
# modify them slightly. ::
parameters["fieldsplit_1_Mp_mat_type"] = "matfree"
# With an unassembled matrix, of course, we are not able to use standard
# preconditioners, so for this example, we will just invert the mass
# matrix using unpreconditioned conjugate gradients. ::
parameters["fieldsplit_1_Mp_ksp_type"] = "cg"
parameters["fieldsplit_1_Mp_pc_type"] = "none"
up.assign(0)
solve(a == L, up, bcs=bcs, nullspace=nullspace, solver_parameters=parameters)
# A runnable python script implementing this demo file is available
# `here `__.