Source code for firedrake.preconditioners.pcd

from firedrake.preconditioners.base import PCBase
from firedrake.petsc import PETSc

__all__ = ("PCDPC", )

[docs]class PCDPC(PCBase): needs_python_pmat = True r"""A Pressure-Convection-Diffusion preconditioner for Navier-Stokes. This preconditioner approximates the inverse of the pressure schur complement for the Navier-Stokes equations by. .. math:: S^{-1} \sim K^{-1} F_p M^{-1} Where :math:`K = \\nabla^2`, :math:`M = \mathbb{I}` and :math:`F_p = 1/\mathrm{Re} \\nabla^2 + u\cdot\\nabla`. The inverse of :math:`K` is approximated by a KSP which can be controlled using the options prefix ``pcd_Kp_``. The inverse of :math:`M` is similarly approximated by a KSP which can be controlled using the options prefix ``pcd_Mp_``. :math:`F_p` requires both the Reynolds number and the current velocity. You must provide these with options using the glbaol option ``Re`` for the Reynolds number and the prefixed option ``pcd_velocity_space`` which should be the index into the full space that gives velocity field. .. note:: Currently, the boundary conditions applied to the PCD operator are correct for characteristic velocity boundary conditions, but sub-optimal for in and outflow boundaries. """
[docs] def initialize(self, pc): from firedrake import TrialFunction, TestFunction, dx, \ assemble, inner, grad, split, Constant, parameters from firedrake.assemble import allocate_matrix, create_assembly_callable if pc.getType() != "python": raise ValueError("Expecting PC type python") prefix = pc.getOptionsPrefix() + "pcd_" # we assume P has things stuffed inside of it _, P = pc.getOperators() context = P.getPythonContext() test, trial = context.a.arguments() if test.function_space() != trial.function_space(): raise ValueError("Pressure space test and trial space differ") Q = test.function_space() p = TrialFunction(Q) q = TestFunction(Q) mass = p*q*dx # Regularisation to avoid having to think about nullspaces. stiffness = inner(grad(p), grad(q))*dx + Constant(1e-6)*p*q*dx opts = PETSc.Options() # we're inverting Mp and Kp, so default them to assembled. # Fp only needs its action, so default it to mat-free. # These can of course be overridden. # only Fp is referred to in update, so that's the only # one we stash. default = parameters["default_matrix_type"] Mp_mat_type = opts.getString(prefix+"Mp_mat_type", default) Kp_mat_type = opts.getString(prefix+"Kp_mat_type", default) self.Fp_mat_type = opts.getString(prefix+"Fp_mat_type", "matfree") Mp = assemble(mass, form_compiler_parameters=context.fc_params, mat_type=Mp_mat_type, options_prefix=prefix + "Mp_") Kp = assemble(stiffness, form_compiler_parameters=context.fc_params, mat_type=Kp_mat_type, options_prefix=prefix + "Kp_") # FIXME: Should we transfer nullspaces over. I think not. Mksp = PETSc.KSP().create(comm=pc.comm) Mksp.incrementTabLevel(1, parent=pc) Mksp.setOptionsPrefix(prefix + "Mp_") Mksp.setOperators(Mp.petscmat) Mksp.setUp() Mksp.setFromOptions() self.Mksp = Mksp Kksp = PETSc.KSP().create(comm=pc.comm) Kksp.incrementTabLevel(1, parent=pc) Kksp.setOptionsPrefix(prefix + "Kp_") Kksp.setOperators(Kp.petscmat) Kksp.setUp() Kksp.setFromOptions() self.Kksp = Kksp state = context.appctx["state"] Re = context.appctx.get("Re", 1.0) velid = context.appctx["velocity_space"] u0 = split(state)[velid] fp = 1.0/Re * inner(grad(p), grad(q))*dx + inner(u0, grad(p))*q*dx self.Re = Re self.Fp = allocate_matrix(fp, form_compiler_parameters=context.fc_params, mat_type=self.Fp_mat_type, options_prefix=prefix + "Fp_") self._assemble_Fp = create_assembly_callable(fp, tensor=self.Fp, form_compiler_parameters=context.fc_params, mat_type=self.Fp_mat_type) self._assemble_Fp() Fpmat = self.Fp.petscmat self.workspace = [Fpmat.createVecLeft() for i in (0, 1)]
[docs] def update(self, pc): self._assemble_Fp()
[docs] def apply(self, pc, x, y): a, b = self.workspace self.Mksp.solve(x, a) self.Fp.petscmat.mult(a, b) self.Kksp.solve(b, y)
[docs] def applyTranspose(self, pc, x, y): a, b = self.workspace self.Kksp.solveTranspose(x, b) self.Fp.petscmat.multTranspose(b, a) self.Mksp.solveTranspose(y, a)
[docs] def view(self, pc, viewer=None): super(PCDPC, self).view(pc, viewer) viewer.printfASCII("Pressure-Convection-Diffusion inverse K^-1 F_p M^-1:\n") viewer.printfASCII("Reynolds number in F_p (applied matrix-free) is %s\n" % str(self.Re)) viewer.printfASCII("KSP solver for K^-1:\n") self.Kksp.view(viewer) viewer.printfASCII("KSP solver for M^-1:\n") self.Mksp.view(viewer)