def stokes_solve(
up_out,
mu,
u_bcs, p_bcs,
f,
dx=dx,
verbose=True,
tol=1.0e-10,
maxiter=1000
):
# Some initial sanity checks.
assert mu > 0.0
WP = up_out.function_space()
# Translate the boundary conditions into the product space.
new_bcs = []
for k, bcs in enumerate([u_bcs, p_bcs]):
for bc in bcs:
space = bc.function_space()
C = space.component()
if len(C) == 0:
new_bcs.append(DirichletBC(WP.sub(k),
bc.value(),
bc.domain_args[0]))
elif len(C) == 1:
new_bcs.append(DirichletBC(WP.sub(k).sub(int(C[0])),
bc.value(),
bc.domain_args[0]))
else:
raise RuntimeError('Illegal number of subspace components.')
# TODO define p*=-1 and reverse sign in the end to get symmetric system?
# Define variational problem
(u, p) = TrialFunctions(WP)
(v, q) = TestFunctions(WP)
r = Expression('x[0]', degree=1, domain=WP.mesh())
print("mu = %e" % mu)
# build system
a = mu * inner(r * grad(u), grad(v)) * 2 * pi * dx \
- ((r * v[0]).dx(0) + (r * v[1]).dx(1)) * p * 2 * pi * dx \
+ ((r * u[0]).dx(0) + (r * u[1]).dx(1)) * q * 2 * pi * dx
#- div(r*v)*p* 2*pi*dx \
#+ q*div(r*u)* 2*pi*dx
L = inner(f, v) * 2 * pi * r * dx
A, b = assemble_system(a, L, new_bcs)
mode = 'lu'
if mode == 'lu':
solve(A, up_out.vector(), b, 'lu')
elif mode == 'gmres':
# For preconditioners for the Stokes system, see
#
# Fast iterative solvers for discrete Stokes equations;
# J. Peters, V. Reichelt, A. Reusken.
#
prec = mu * inner(r * grad(u), grad(v)) * 2 * pi * dx \
- p * q * 2 * pi * r * dx
P, btmp = assemble_system(prec, L, new_bcs)
solver = KrylovSolver('tfqmr', 'amg')
#solver = KrylovSolver('gmres', 'amg')
solver.set_operators(A, P)
solver.parameters['monitor_convergence'] = verbose
solver.parameters['report'] = verbose
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['relative_tolerance'] = tol
solver.parameters['maximum_iterations'] = maxiter
# Solve
solver.solve(up_out.vector(), b)
elif mode == 'fieldsplit':
raise NotImplementedError('Fieldsplit solver not yet implemented.')
# For an assortment of preconditioners, see
#
# Performance and analysis of saddle point preconditioners
# for the discrete steady-state Navier-Stokes equations;
# H.C. Elman, D.J. Silvester, A.J. Wathen;
# Numer. Math. (2002) 90: 665-688;
# <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.145.3554>.
#
# Set up field split.
W = SubSpace(WP, 0)
P = SubSpace(WP, 1)
u_dofs = W.dofmap().dofs()
p_dofs = P.dofmap().dofs()
prec = PETScPreconditioner()
prec.set_fieldsplit([u_dofs, p_dofs], ['u', 'p'])
PETScOptions.set('pc_type', 'fieldsplit')
PETScOptions.set('pc_fieldsplit_type', 'additive')
PETScOptions.set('fieldsplit_u_pc_type', 'lu')
PETScOptions.set('fieldsplit_p_pc_type', 'jacobi')
# Create Krylov solver with custom preconditioner.
solver = PETScKrylovSolver('gmres', prec)
solver.set_operator(A)
return
def solve(W, P,
mu,
u_bcs, p_bcs,
f,
verbose=True,
tol=1.0e-10
):
# Some initial sanity checks.
assert mu > 0.0
WP = MixedFunctionSpace([W, P])
# Translate the boundary conditions into the product space.
# This conditional loop is able to deal with conditions of the kind
#
# DirichletBC(W.sub(1), 0.0, right_boundary)
#
new_bcs = []
for k, bcs in enumerate([u_bcs, p_bcs]):
for bc in bcs:
space = bc.function_space()
C = space.component()
if len(C) == 0:
new_bcs.append(DirichletBC(WP.sub(k),
bc.value(),
bc.domain_args[0]))
elif len(C) == 1:
new_bcs.append(DirichletBC(WP.sub(k).sub(int(C[0])),
bc.value(),
bc.domain_args[0]))
else:
raise RuntimeError('Illegal number of subspace components.')
# Define variational problem
(u, p) = TrialFunctions(WP)
(v, q) = TestFunctions(WP)
# Build system.
# The sign of the div(u)-term is somewhat arbitrary since the right-hand
# side is 0 here. We can either make the system symmetric or positive-
# definite.
# On a second note, we have
#
# \int grad(p).v = - \int p * div(v) + \int_\Gamma p n.v.
#
# Since, we have either p=0 or n.v=0 on the boundary, we could as well
# replace the term dot(grad(p), v) by -p*div(v).
#
a = mu * inner(grad(u), grad(v))*dx \
- p * div(v) * dx \
- q * div(u) * dx
#a = mu * inner(grad(u), grad(v))*dx + dot(grad(p), v) * dx \
# - div(u) * q * dx
L = dot(f, v)*dx
A, b = assemble_system(a, L, new_bcs)
if has_petsc():
# For an assortment of preconditioners, see
#
# Performance and analysis of saddle point preconditioners
# for the discrete steady-state Navier-Stokes equations;
# H.C. Elman, D.J. Silvester, A.J. Wathen;
# Numer. Math. (2002) 90: 665-688;
# <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.145.3554>.
#
# Set up field split.
W = SubSpace(WP, 0)
P = SubSpace(WP, 1)
u_dofs = W.dofmap().dofs()
p_dofs = P.dofmap().dofs()
prec = PETScPreconditioner()
prec.set_fieldsplit([u_dofs, p_dofs], ['u', 'p'])
PETScOptions.set('pc_type', 'fieldsplit')
PETScOptions.set('pc_fieldsplit_type', 'additive')
PETScOptions.set('fieldsplit_u_pc_type', 'lu')
PETScOptions.set('fieldsplit_p_pc_type', 'jacobi')
## <http://scicomp.stackexchange.com/questions/7288/which-preconditioners-and-solver-in-petsc-for-indefinite-symmetric-systems-sho>
#PETScOptions.set('pc_type', 'fieldsplit')
##PETScOptions.set('pc_fieldsplit_type', 'schur')
##PETScOptions.set('pc_fieldsplit_schur_fact_type', 'upper')
#PETScOptions.set('pc_fieldsplit_detect_saddle_point')
##PETScOptions.set('fieldsplit_u_pc_type', 'lsc')
##PETScOptions.set('fieldsplit_u_ksp_type', 'preonly')
#PETScOptions.set('pc_type', 'fieldsplit')
#PETScOptions.set('fieldsplit_u_pc_type', 'hypre')
#PETScOptions.set('fieldsplit_u_ksp_type', 'preonly')
#PETScOptions.set('fieldsplit_p_pc_type', 'jacobi')
#PETScOptions.set('fieldsplit_p_ksp_type', 'preonly')
## From PETSc/src/ksp/ksp/examples/tutorials/ex42-fsschur.opts:
#PETScOptions.set('pc_type', 'fieldsplit')
#PETScOptions.set('pc_fieldsplit_type', 'SCHUR')
#PETScOptions.set('pc_fieldsplit_schur_fact_type', 'UPPER')
#PETScOptions.set('fieldsplit_p_ksp_type', 'preonly')
#PETScOptions.set('fieldsplit_u_pc_type', 'bjacobi')
## From
##
## Composable Linear Solvers for Multiphysics;
## J. Brown, M. Knepley, D.A. May, L.C. McInnes, B. Smith;
## <http://www.computer.org/csdl/proceedings/ispdc/2012/4805/00/4805a055-abs.html>;
## <http://www.mcs.anl.gov/uploads/cels/papers/P2017-0112.pdf>.
##
#PETScOptions.set('pc_type', 'fieldsplit')
#PETScOptions.set('pc_fieldsplit_type', 'schur')
#PETScOptions.set('pc_fieldsplit_schur_factorization_type', 'upper')
##
#PETScOptions.set('fieldsplit_u_ksp_type', 'cg')
#PETScOptions.set('fieldsplit_u_ksp_rtol', 1.0e-6)
#PETScOptions.set('fieldsplit_u_pc_type', 'bjacobi')
#PETScOptions.set('fieldsplit_u_sub_pc_type', 'cholesky')
##
#PETScOptions.set('fieldsplit_p_ksp_type', 'fgmres')
#PETScOptions.set('fieldsplit_p_ksp_constant_null_space')
#PETScOptions.set('fieldsplit_p_pc_type', 'lsc')
##
#PETScOptions.set('fieldsplit_p_lsc_ksp_type', 'cg')
#PETScOptions.set('fieldsplit_p_lsc_ksp_rtol', 1.0e-2)
#PETScOptions.set('fieldsplit_p_lsc_ksp_constant_null_space')
##PETScOptions.set('fieldsplit_p_lsc_ksp_converged_reason')
#PETScOptions.set('fieldsplit_p_lsc_pc_type', 'bjacobi')
#PETScOptions.set('fieldsplit_p_lsc_sub_pc_type', 'icc')
# Create Krylov solver with custom preconditioner.
solver = PETScKrylovSolver('gmres', prec)
solver.set_operator(A)
else:
# Use the preconditioner as recommended in
# <http://fenicsproject.org/documentation/dolfin/dev/python/demo/pde/stokes-iterative/python/documentation.html>,
#
# prec = inner(grad(u), grad(v))*dx - p*q*dx
#
# although it doesn't seem to be too efficient.
# The sign on the last term doesn't matter.
prec = mu * inner(grad(u), grad(v))*dx \
- p*q*dx
M, _ = assemble_system(prec, L, new_bcs)
#solver = KrylovSolver('tfqmr', 'amg')
solver = KrylovSolver('gmres', 'amg')
solver.set_operators(A, M)
solver.parameters['monitor_convergence'] = verbose
solver.parameters['report'] = verbose
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['relative_tolerance'] = tol
solver.parameters['maximum_iterations'] = 500
# Solve
up = Function(WP)
solver.solve(up.vector(), b)
# Get sub-functions
u, p = up.split()
return u, p