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 \(u = (1, 0)\) 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.:

    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")
        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 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 an incomplete LU factorization, which we configure now:

   "fieldsplit_1_Mp_mat_type": "aij",
   "fieldsplit_1_Mp_pc_type": "ilu"

Having set up the parameters, we can now go ahead and solve the problem.:

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.subfunctions

VTKFile("stokes.pvd").write(u, p)

By default, the mass matrix is assembled in the 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_pc_type"] = "ksp"
parameters["fieldsplit_1_Mp_ksp_ksp_type"] = "cg"
parameters["fieldsplit_1_Mp_ksp_pc_type"] = "none"

solve(a == L, up, bcs=bcs, nullspace=nullspace, solver_parameters=parameters)

A runnable python script implementing this demo file is available here.