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Navier-Stokes equations

We solve the Navier-Stokes equations using Taylor-Hood elements. The example is that of a lid-driven cavity.

from firedrake import *

N = 64

M = UnitSquareMesh(N, N)

V = VectorFunctionSpace(M, "CG", 2)
W = FunctionSpace(M, "CG", 1)
Z = V * W

up = Function(Z)
u, p = split(up)
v, q = TestFunctions(Z)

Re = Constant(100.0)

F = (
    1.0 / Re * inner(grad(u), grad(v)) * dx +
    inner(dot(grad(u), u), v) * dx -
    p * div(v) * dx +
    div(u) * q * dx
)

bcs = [DirichletBC(Z.sub(0), Constant((1, 0)), (4,)),
       DirichletBC(Z.sub(0), Constant((0, 0)), (1, 2, 3))]

nullspace = MixedVectorSpaceBasis(
    Z, [Z.sub(0), VectorSpaceBasis(constant=True)])

Having set up the problem, we now move on to solving it. Some preconditioners, for example pressure convection-diffusion (PCD), require information about the the problem that is not easily accessible from the bilinear form. In the case of PCD, we need the Reynolds number and additionally which part of the mixed velocity-pressure space the velocity corresponds to. We provide this information to preconditioners by passing in a dictionary context to the solver. This is propagated down through the matrix-free operators and is therefore accessible to custom preconditioners.

appctx = {"Re": Re, "velocity_space": 0}

Now we’ll solve the problem. First, using a direct solver. Again, if MUMPS is not installed, this solve will not work, so we wrap the solve in a try/except block.

from firedrake.petsc import PETSc

try:
    solve(F == 0, up, bcs=bcs, nullspace=nullspace,
          solver_parameters={"snes_monitor": None,
                             "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 show an example using the PCDPC preconditioner that implements the pressure convection-diffusion approximation to the pressure Schur complement. We’ll need more solver parameters this time, so again we’ll set those up in a dictionary.

parameters = {"mat_type": "matfree",
              "snes_monitor": None,

We’ll use a non-stationary Krylov solve for the Schur complement, so we need to use a flexible Krylov method on the outside.

"ksp_type": "fgmres",
"ksp_gmres_modifiedgramschmidt": None,
"ksp_monitor_true_residual": None,

Now to configure the preconditioner:

"pc_type": "fieldsplit",
"pc_fieldsplit_type": "schur",
"pc_fieldsplit_schur_fact_type": "lower",

we invert the velocity block with LU:

"fieldsplit_0_ksp_type": "preonly",
"fieldsplit_0_pc_type": "python",
"fieldsplit_0_pc_python_type": "firedrake.AssembledPC",
"fieldsplit_0_assembled_pc_type": "lu",

and invert the schur complement inexactly using GMRES, preconditioned with PCD.

"fieldsplit_1_ksp_type": "gmres",
"fieldsplit_1_ksp_rtol": 1e-4,
"fieldsplit_1_pc_type": "python",
"fieldsplit_1_pc_python_type": "firedrake.PCDPC",

We now need to configure the mass and stiffness solvers in the PCD preconditioner. For this example, we will just invert them with LU, although of course we can use a scalable method if we wish. First the mass solve:

"fieldsplit_1_pcd_Mp_ksp_type": "preonly",
"fieldsplit_1_pcd_Mp_pc_type": "lu",

and the stiffness solve.:

"fieldsplit_1_pcd_Kp_ksp_type": "preonly",
"fieldsplit_1_pcd_Kp_pc_type": "lu",

Finally, we just need to decide whether to apply the action of the pressure-space convection-diffusion operator with an assembled matrix or matrix free. Here we will use matrix-free:

"fieldsplit_1_pcd_Fp_mat_type": "matfree"}

With the parameters set up, we can solve the problem, remembering to pass in the application context so that the PCD preconditioner can find the Reynolds number.

up.assign(0)

solve(F == 0, up, bcs=bcs, nullspace=nullspace, solver_parameters=parameters,
      appctx=appctx)

And finally we write the results to a file for visualisation.

u, p = up.subfunctions
u.rename("Velocity")
p.rename("Pressure")

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

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