def test_save_and_checkpoint_timeseries(tempdir, encoding):
mesh = UnitSquareMesh(MPI.comm_world, 16, 16)
filename = os.path.join(tempdir, "u2_checkpoint.xdmf")
FE = FiniteElement("CG", mesh.ufl_cell(), 2)
V = FunctionSpace(mesh, FE)
times = [0.5, 0.2, 0.1]
u_out = [None] * len(times)
u_in = [None] * len(times)
with XDMFFile(mesh.mpi_comm(), filename, encoding=encoding) as file:
for i, p in enumerate(times):
u_out[i] = interpolate(Expression("x[0]*p", p=p, degree=1), V)
file.write_checkpoint(u_out[i], "u_out", p)
with XDMFFile(mesh.mpi_comm(), filename) as file:
for i, p in enumerate(times):
u_in[i] = file.read_checkpoint(V, "u_out", i)
for i, p in enumerate(times):
u_in[i].vector().axpy(-1.0, u_out[i].vector())
assert u_in[i].vector().norm(cpp.la.Norm.l2) < 1.0e-12
# test reading last
with XDMFFile(mesh.mpi_comm(), filename) as file:
u_in_last = file.read_checkpoint(V, "u_out", -1)
u_out[-1].vector().axpy(-1.0, u_in_last.vector())
assert u_out[-1].vector().norm(cpp.la.Norm.l2) < 1.0e-12
def test_save_and_checkpoint_timeseries(tempdir, encoding):
mesh = UnitSquareMesh(MPI.comm_world, 16, 16)
filename = os.path.join(tempdir, "u2_checkpoint.xdmf")
FE = FiniteElement("CG", mesh.ufl_cell(), 2)
V = FunctionSpace(mesh, FE)
times = [0.5, 0.2, 0.1]
u_out = [None] * len(times)
u_in = [None] * len(times)
p = 0.0
def expr_eval(values, x):
values[:, 0] = x[:, 0] * p
with XDMFFile(mesh.mpi_comm(), filename, encoding=encoding) as file:
for i, p in enumerate(times):
u_out[i] = interpolate(expr_eval, V)
file.write_checkpoint(u_out[i], "u_out", p)
with XDMFFile(mesh.mpi_comm(), filename) as file:
for i, p in enumerate(times):
u_in[i] = file.read_checkpoint(V, "u_out", i)
for i, p in enumerate(times):
u_in[i].vector.axpy(-1.0, u_out[i].vector)
assert u_in[i].vector.norm() < 1.0e-12
# test reading last
with XDMFFile(mesh.mpi_comm(), filename) as file:
u_in_last = file.read_checkpoint(V, "u_out", -1)
u_out[-1].vector.axpy(-1.0, u_in_last.vector)
assert u_out[-1].vector.norm() < 1.0e-12
def _test_eigen_solver_sparse(callback_type):
from rbnics.backends.dolfin import EigenSolver
# Define mesh
mesh = UnitSquareMesh(10, 10)
# Define function space
V_element = VectorElement("Lagrange", mesh.ufl_cell(), 2)
Q_element = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
W_element = MixedElement(V_element, Q_element)
W = FunctionSpace(mesh, W_element)
# Create boundaries
class Wall(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and (x[1] < 0 + DOLFIN_EPS
or x[1] > 1 - DOLFIN_EPS)
boundaries = MeshFunction("size_t", mesh, mesh.topology().dim() - 1)
boundaries.set_all(0)
wall = Wall()
wall.mark(boundaries, 1)
# Define variational problem
vq = TestFunction(W)
(v, q) = split(vq)
up = TrialFunction(W)
(u, p) = split(up)
lhs = inner(grad(u), grad(v)) * dx - div(v) * p * dx - div(u) * q * dx
rhs = -inner(p, q) * dx
# Define boundary condition
bc = [DirichletBC(W.sub(0), Constant((0., 0.)), boundaries, 1)]
# Define eigensolver depending on callback type
assert callback_type in ("form callbacks", "tensor callbacks")
if callback_type == "form callbacks":
solver = EigenSolver(W, lhs, rhs, bc)
elif callback_type == "tensor callbacks":
LHS = assemble(lhs)
RHS = assemble(rhs)
solver = EigenSolver(W, LHS, RHS, bc)
# Solve the eigenproblem
solver.set_parameters({
"linear_solver": "mumps",
"problem_type": "gen_non_hermitian",
"spectrum": "target real",
"spectral_transform": "shift-and-invert",
"spectral_shift": 1.e-5
})
solver.solve(1)
r, c = solver.get_eigenvalue(0)
assert abs(c) < 1.e-10
assert r > 0., "r = " + str(r) + " is not positive"
print("Sparse inf-sup constant: ", sqrt(r))
return (sqrt(r), solver.condensed_A, solver.condensed_B)
def test_clear_sub_map_data_vector(mesh):
mesh = UnitSquareMesh(8, 8)
P1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
W = FunctionSpace(mesh, P1 * P1)
# Check block size
assert W.dofmap().index_map.block_size == 2
W.dofmap().clear_sub_map_data()
with pytest.raises(RuntimeError):
W0 = W.sub(0)
assert (W0)
with pytest.raises(RuntimeError):
W1 = W.sub(1)
assert (W1)
#
# Next, various model parameters are defined::
# Model parameters
lmbda = 1.0e-02 # surface parameter
dt = 5.0e-06 # time step
theta = 0.5 # time stepping family, e.g. theta=1 -> backward Euler, theta=0.5 -> Crank-Nicolson
# A unit square mesh with 97 (= 96 + 1) vertices in each direction is
# created, and on this mesh a
# :py:class:`FunctionSpace<dolfin.functions.functionspace.FunctionSpace>`
# ``ME`` is built using a pair of linear Lagrangian elements. ::
# Create mesh and build function space
mesh = UnitSquareMesh(MPI.comm_world, 96, 96, CellType.triangle)
P1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
ME = FunctionSpace(mesh, P1 * P1)
# Trial and test functions of the space ``ME`` are now defined::
# Define trial and test functions
du = TrialFunction(ME)
q, v = TestFunctions(ME)
# .. index:: split functions
#
# For the test functions,
# :py:func:`TestFunctions<dolfin.functions.function.TestFunctions>` (note
# the 's' at the end) is used to define the scalar test functions ``q``
# and ``v``. The
# :py:class:`TrialFunction<dolfin.functions.function.TrialFunction>`
def test_unsteady_stokes():
nx, ny = 15, 15
k = 1
nu = Constant(1.0e-0)
dt = Constant(2.5e-2)
num_steps = 20
theta0 = 1.0 # Initial theta value
theta1 = 0.5 # Theta after 1 step
theta = Constant(theta0)
mesh = UnitSquareMesh(nx, ny)
# The 'unsteady version' of the benchmark in the 2012 paper by Labeur&Wells
u_exact = Expression(
(
"sin(t) * x[0]*x[0]*(1.0 - x[0])*(1.0 - x[0])*(2.0*x[1] \
-6.0*x[1]*x[1] + 4.0*x[1]*x[1]*x[1])",
"-sin(t)* x[1]*x[1]*(1.0 - x[1])*(1.0 - x[1])*(2.0*x[0] \
- 6.0*x[0]*x[0] + 4.0*x[0]*x[0]*x[0])",
),
t=0,
degree=7,
domain=mesh,
)
p_exact = Expression("sin(t) * x[0]*(1.0 - x[0])",
t=0,
degree=7,
domain=mesh)
du_exact = Expression(
(
"cos(t) * x[0]*x[0]*(1.0 - x[0])*(1.0 - x[0])*(2.0*x[1] \
- 6.0*x[1]*x[1] + 4.0*x[1]*x[1]*x[1])",
"-cos(t)* x[1]*x[1]*(1.0 - x[1])*(1.0 - x[1])*(2.0*x[0] \
-6.0*x[0]*x[0] + 4.0*x[0]*x[0]*x[0])",
),
t=0,
degree=7,
domain=mesh,
)
ux_exact = Expression(
(
"x[0]*x[0]*(1.0 - x[0])*(1.0 - x[0])*(2.0*x[1] \
- 6.0*x[1]*x[1] + 4.0*x[1]*x[1]*x[1])",
"-x[1]*x[1]*(1.0 - x[1])*(1.0 - x[1])*(2.0*x[0] \
- 6.0*x[0]*x[0] + 4.0*x[0]*x[0]*x[0])",
),
degree=7,
domain=mesh,
)
px_exact = Expression("x[0]*(1.0 - x[0])", degree=7, domain=mesh)
sin_ext = Expression("sin(t)", t=0, degree=7, domain=mesh)
f = du_exact + sin_ext * div(px_exact * Identity(2) -
2 * sym(grad(ux_exact)))
Vhigh = VectorFunctionSpace(mesh, "DG", 7)
Phigh = FunctionSpace(mesh, "DG", 7)
# New syntax:
V = VectorElement("DG", mesh.ufl_cell(), k)
Q = FiniteElement("DG", mesh.ufl_cell(), k - 1)
Vbar = VectorElement("DGT", mesh.ufl_cell(), k)
Qbar = FiniteElement("DGT", mesh.ufl_cell(), k)
mixedL = FunctionSpace(mesh, MixedElement([V, Q]))
mixedG = FunctionSpace(mesh, MixedElement([Vbar, Qbar]))
V2 = FunctionSpace(mesh, V)
Uh = Function(mixedL)
Uhbar = Function(mixedG)
U0 = Function(mixedL)
Uhbar0 = Function(mixedG)
u0, p0 = split(U0)
ubar0, pbar0 = split(Uhbar0)
ustar = Function(V2)
# Then the boundary conditions
bc0 = DirichletBC(mixedG.sub(0), Constant((0, 0)), Gamma)
bc1 = DirichletBC(mixedG.sub(1), Constant(0), Corner, "pointwise")
bcs = [bc0, bc1]
alpha = Constant(6 * k * k)
forms_stokes = FormsStokes(mesh, mixedL, mixedG,
alpha).forms_unsteady(ustar, dt, nu, f)
ssc = StokesStaticCondensation(
mesh,
forms_stokes["A_S"],
forms_stokes["G_S"],
forms_stokes["G_ST"],
forms_stokes["B_S"],
forms_stokes["Q_S"],
forms_stokes["S_S"],
)
t = 0.0
step = 0
for step in range(num_steps):
step += 1
t += float(dt)
if comm.Get_rank() == 0:
print("Step " + str(step) + " Time " + str(t))
# Set time level in exact solution
u_exact.t = t
p_exact.t = t
du_exact.t = t - (1 - float(theta)) * float(dt)
sin_ext.t = t - (1 - float(theta)) * float(dt)
ssc.assemble_global_lhs()
ssc.assemble_global_rhs()
for bc in bcs:
ssc.apply_boundary(bc)
ssc.solve_problem(Uhbar, Uh, "none", "default")
assign(U0, Uh)
assign(ustar, U0.sub(0))
assign(Uhbar0, Uhbar)
if step == 1:
theta.assign(theta1)
udiv_e = sqrt(assemble(div(Uh.sub(0)) * div(Uh.sub(0)) * dx))
u_ex_h = interpolate(u_exact, Vhigh)
p_ex_h = interpolate(p_exact, Phigh)
u_error = sqrt(assemble(dot(Uh.sub(0) - u_ex_h, Uh.sub(0) - u_ex_h) * dx))
p_error = sqrt(assemble(dot(Uh.sub(1) - p_ex_h, Uh.sub(1) - p_ex_h) * dx))
assert udiv_e < 1e-12
assert u_error < 1.5e-4
assert p_error < 1e-2
def test_steady_stokes(k):
# Polynomial order and mesh resolution
nx_list = [4, 8, 16]
nu = Constant(1)
if comm.Get_rank() == 0:
print('{:=^72}'.format('Computing for polynomial order ' + str(k)))
# Error listst
error_u, error_p, error_div = [], [], []
for nx in nx_list:
if comm.Get_rank() == 0:
print('# Resolution ' + str(nx))
mesh = UnitSquareMesh(nx, nx)
# Get forcing from exact solutions
u_exact, p_exact = exact_solution(mesh)
f = div(p_exact * Identity(2) - 2 * nu * sym(grad(u_exact)))
# Define FunctionSpaces and functions
V = VectorElement("DG", mesh.ufl_cell(), k)
Q = FiniteElement("DG", mesh.ufl_cell(), k - 1)
Vbar = VectorElement("DGT", mesh.ufl_cell(), k)
Qbar = FiniteElement("DGT", mesh.ufl_cell(), k)
mixedL = FunctionSpace(mesh, MixedElement([V, Q]))
mixedG = FunctionSpace(mesh, MixedElement([Vbar, Qbar]))
Uh = Function(mixedL)
Uhbar = Function(mixedG)
# Set forms
alpha = Constant(6 * k * k)
forms_stokes = FormsStokes(mesh, mixedL, mixedG,
alpha).forms_steady(nu, f)
# No-slip boundary conditions, set pressure in one of the corners
bc0 = DirichletBC(mixedG.sub(0), Constant((0, 0)), Gamma)
bc1 = DirichletBC(mixedG.sub(1), Constant(0), Corner, "pointwise")
bcs = [bc0, bc1]
# Initialize static condensation class
ssc = StokesStaticCondensation(mesh, forms_stokes['A_S'],
forms_stokes['G_S'],
forms_stokes['B_S'],
forms_stokes['Q_S'],
forms_stokes['S_S'], bcs)
# Assemble global system and incorporates bcs
ssc.assemble_global_system(True)
# Solve using mumps
ssc.solve_problem(Uhbar, Uh, "mumps", "default")
# Compute velocity/pressure/local div error
uh, ph = Uh.split()
e_u = np.sqrt(np.abs(assemble(dot(uh - u_exact, uh - u_exact) * dx)))
e_p = np.sqrt(np.abs(assemble((ph - p_exact) * (ph - p_exact) * dx)))
e_d = np.sqrt(np.abs(assemble(div(uh) * div(uh) * dx)))
if comm.rank == 0:
error_u.append(e_u)
error_p.append(e_p)
error_div.append(e_d)
print('Error in velocity ' + str(error_u[-1]))
print('Error in pressure ' + str(error_p[-1]))
print('Local mass error ' + str(error_div[-1]))
if comm.rank == 0:
iterator_list = [1. / float(nx) for nx in nx_list]
conv_u = compute_convergence(iterator_list, error_u)
conv_p = compute_convergence(iterator_list, error_p)
assert any(conv > k + 0.75 for conv in conv_u)
assert any(conv > (k - 1) + 0.75 for conv in conv_p)
class Lid_driven_cavity(object):
def __init__(self):
n = 40
self.mesh = UnitSquareMesh(n, n, "crossed")
# Define mesh and boundaries.
class LeftBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[0] < DOLFIN_EPS
class RightBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[0] > 1.0 - DOLFIN_EPS
class LowerBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[1] < DOLFIN_EPS
class UpperBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[1] > 1.0 - DOLFIN_EPS
class RestrictedUpperBoundary(SubDomain):
def inside(self, x, on_boundary):
return (
on_boundary
and x[1] > 1.0 - DOLFIN_EPS
and DOLFIN_EPS < x[0]
and x[0] < 0.5 - DOLFIN_EPS
)
left = LeftBoundary()
right = RightBoundary()
lower = LowerBoundary()
upper = UpperBoundary()
# restricted_upper = RestrictedUpperBoundary()
# Be particularly careful with the boundary conditions.
# The main problem here is that the PPE system is consistent if and
# only if
#
# \int_\Omega div(u) = \int_\Gamma n.u = 0.
#
# This is exactly and even pointwise fulfilled for the continuous
# problem. In the discrete case, we can have to make sure that n.u is
# 0 all along the boundary.
# In the lid-driven cavity problem, of particular interest are the
# corner points at the lid. One has to assert that the z-component of u
# is 0 all across the lid, and the x-component of u is 0 everywhere but
# the lid. Since u is L2-"continuous", the lid condition on u_x must
# not be enforced in the corner points. The u_y component must be
# enforced all over the lid, including the end points.
V_element = FiniteElement("CG", self.mesh.ufl_cell(), 2)
self.W = FunctionSpace(self.mesh, V_element * V_element)
self.u_bcs = [
DirichletBC(self.W, (0.0, 0.0), left),
DirichletBC(self.W, (0.0, 0.0), right),
# DirichletBC(self.W.sub(0), Expression('x[0]'), restricted_upper),
DirichletBC(self.W, (0.0, 0.0), lower),
DirichletBC(self.W.sub(0), Constant("1.0"), upper),
DirichletBC(self.W.sub(1), 0.0, upper),
# DirichletBC(self.W.sub(0), Constant('-1.0'), lower),
# DirichletBC(self.W.sub(1), 0.0, lower),
# DirichletBC(self.W.sub(1), Constant('1.0'), left),
# DirichletBC(self.W.sub(0), 0.0, left),
# DirichletBC(self.W.sub(1), Constant('-1.0'), right),
# DirichletBC(self.W.sub(0), 0.0, right),
]
self.P = FunctionSpace(self.mesh, "CG", 1)
self.p_bcs = []
return
class Lid_driven_cavity(object):
def __init__(self):
n = 40
self.mesh = UnitSquareMesh(n, n, "crossed")
# Define mesh and boundaries.
class LeftBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[0] < DOLFIN_EPS
class RightBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[0] > 1.0 - DOLFIN_EPS
class LowerBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[1] < DOLFIN_EPS
class UpperBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[1] > 1.0 - DOLFIN_EPS
class RestrictedUpperBoundary(SubDomain):
def inside(self, x, on_boundary):
return (on_boundary and x[1] > 1.0 - DOLFIN_EPS
and DOLFIN_EPS < x[0] and x[0] < 0.5 - DOLFIN_EPS)
left = LeftBoundary()
right = RightBoundary()
lower = LowerBoundary()
upper = UpperBoundary()
# restricted_upper = RestrictedUpperBoundary()
# Be particularly careful with the boundary conditions.
# The main problem here is that the PPE system is consistent if and
# only if
#
# \int_\Omega div(u) = \int_\Gamma n.u = 0.
#
# This is exactly and even pointwise fulfilled for the continuous
# problem. In the discrete case, we can have to make sure that n.u is
# 0 all along the boundary.
# In the lid-driven cavity problem, of particular interest are the
# corner points at the lid. One has to assert that the z-component of u
# is 0 all across the lid, and the x-component of u is 0 everywhere but
# the lid. Since u is L2-"continuous", the lid condition on u_x must
# not be enforced in the corner points. The u_y component must be
# enforced all over the lid, including the end points.
V_element = FiniteElement("CG", self.mesh.ufl_cell(), 2)
self.W = FunctionSpace(self.mesh, V_element * V_element)
self.u_bcs = [
DirichletBC(self.W, (0.0, 0.0), left),
DirichletBC(self.W, (0.0, 0.0), right),
# DirichletBC(self.W.sub(0), Expression('x[0]'), restricted_upper),
DirichletBC(self.W, (0.0, 0.0), lower),
DirichletBC(self.W.sub(0), Constant("1.0"), upper),
DirichletBC(self.W.sub(1), 0.0, upper),
# DirichletBC(self.W.sub(0), Constant('-1.0'), lower),
# DirichletBC(self.W.sub(1), 0.0, lower),
# DirichletBC(self.W.sub(1), Constant('1.0'), left),
# DirichletBC(self.W.sub(0), 0.0, left),
# DirichletBC(self.W.sub(1), Constant('-1.0'), right),
# DirichletBC(self.W.sub(0), 0.0, right),
]
self.P = FunctionSpace(self.mesh, "CG", 1)
self.p_bcs = []
return
from dolfin import CellDiameter, Constant, det, dx, Expression, Function, FunctionSpace, grad, has_pybind11, inner, TensorFunctionSpace, TestFunction, TrialFunction, UnitSquareMesh, VectorFunctionSpace
if has_pybind11():
from dolfin.cpp.log import log, LogLevel, set_log_level
PROGRESS = LogLevel.PROGRESS
else:
from dolfin import log, PROGRESS, set_log_level
set_log_level(PROGRESS)
from rbnics.backends.dolfin import SeparatedParametrizedForm
from rbnics.eim.utils.decorators.store_map_from_solution_to_problem import _solution_to_problem_map
# Common variables
mesh = UnitSquareMesh(10, 10)
V = VectorFunctionSpace(mesh, "Lagrange", 2)
expr1 = Expression("x[0]", mu_0=0., degree=1, cell=mesh.ufl_cell()) # f_5
expr2 = Expression(("x[0]", "x[1]"), mu_0=0., degree=1, cell=mesh.ufl_cell()) # f_6
expr3 = Expression((("1*x[0]", "2*x[1]"), ("3*x[0]", "4*x[1]")), mu_0=0., degree=1, cell=mesh.ufl_cell()) # f_7
expr4 = Expression((("4*x[0]", "3*x[1]"), ("2*x[0]", "1*x[1]")), mu_0=0., degree=1, cell=mesh.ufl_cell()) # f_8
expr5 = Expression("x[0]", degree=1, cell=mesh.ufl_cell()) # f_9
expr6 = Expression(("x[0]", "x[1]"), degree=1, cell=mesh.ufl_cell()) # f_10
expr7 = Expression((("1*x[0]", "2*x[1]"), ("3*x[0]", "4*x[1]")), degree=1, cell=mesh.ufl_cell()) # f_11
expr8 = Expression((("4*x[0]", "3*x[1]"), ("2*x[0]", "1*x[1]")), degree=1, cell=mesh.ufl_cell()) # f_12
expr9 = Constant(((1, 2), (3, 4))) # f_13
scalar_V = FunctionSpace(mesh, "Lagrange", 3)
tensor_V = TensorFunctionSpace(mesh, "Lagrange", 1)
expr10 = Function(scalar_V) # f_18
expr11 = Function(V) # f_21
expr12 = Function(tensor_V) # f_24
expr13 = Function(scalar_V) # f_27
def test_poisson(k):
# Polynomial order and mesh resolution
nx_list = [4, 8, 16]
# Error list
error_u_l2, error_u_h1 = [], []
for nx in nx_list:
mesh = UnitSquareMesh(nx, nx)
# Define FunctionSpaces and functions
V = FunctionSpace(mesh, "DG", k)
Vbar = FunctionSpace(mesh,
FiniteElement("CG", mesh.ufl_cell(), k)["facet"])
u_soln = Expression("sin(pi*x[0])*sin(pi*x[1])",
degree=k + 1,
domain=mesh)
f = Expression("2*pi*pi*sin(pi*x[0])*sin(pi*x[1])", degree=k + 1)
u, v = Function(V), TestFunction(V)
ubar, vbar = Function(Vbar), TestFunction(Vbar)
n = FacetNormal(mesh)
h = CellDiameter(mesh)
alpha = Constant(6 * k * k)
penalty = alpha / h
def facet_integral(integrand):
return integrand('-') * dS + integrand('+') * dS + integrand * ds
u_flux = ubar
F_v_flux = grad(u) + penalty * outer(u_flux - u, n)
residual_local = inner(grad(u), grad(v)) * dx
residual_local += facet_integral(inner(outer(u_flux - u, n), grad(v)))
residual_local -= facet_integral(inner(F_v_flux, outer(v, n)))
residual_local -= f * v * dx
residual_global = facet_integral(inner(F_v_flux, outer(vbar, n)))
a_ll = derivative(residual_local, u)
a_lg = derivative(residual_local, ubar)
a_gl = derivative(residual_global, u)
a_gg = derivative(residual_global, ubar)
l_l = -residual_local
l_g = -residual_global
bcs = [DirichletBC(Vbar, u_soln, "on_boundary")]
# Initialize static condensation assembler
assembler = AssemblerStaticCondensation(a_ll, a_lg, a_gl, a_gg, l_l,
l_g, bcs)
A_g, b_g = PETScMatrix(), PETScVector()
assembler.assemble_global_lhs(A_g)
assembler.assemble_global_rhs(b_g)
for bc in bcs:
bc.apply(A_g, b_g)
solver = PETScKrylovSolver()
solver.set_operator(A_g)
PETScOptions.set("ksp_type", "preonly")
PETScOptions.set("pc_type", "lu")
PETScOptions.set("pc_factor_mat_solver_type", "mumps")
solver.set_from_options()
solver.solve(ubar.vector(), b_g)
assembler.backsubstitute(ubar._cpp_object, u._cpp_object)
# Compute L2 and H1 norms
e_u_l2 = assemble((u - u_soln)**2 * dx)**0.5
e_u_h1 = assemble(grad(u - u_soln)**2 * dx)**0.5
if mesh.mpi_comm().rank == 0:
error_u_l2.append(e_u_l2)
error_u_h1.append(e_u_h1)
if mesh.mpi_comm().rank == 0:
iterator_list = [1.0 / float(nx) for nx in nx_list]
conv_u_l2 = compute_convergence(iterator_list, error_u_l2)
conv_u_h1 = compute_convergence(iterator_list, error_u_h1)
# Optimal rate of k + 1 - tolerance
assert np.all(conv_u_l2 >= (k + 1.0 - 0.15))
# Optimal rate of k - tolerance
assert np.all(conv_u_h1 >= (k - 0.1))
from dolfin import CellDiameter, Constant, det, dx, Expression, Function, FunctionSpace, grad, has_pybind11, inner, TensorFunctionSpace, TestFunction, TrialFunction, UnitSquareMesh, VectorFunctionSpace
if has_pybind11():
from dolfin.cpp.log import log, LogLevel, set_log_level
PROGRESS = LogLevel.PROGRESS
else:
from dolfin import log, PROGRESS, set_log_level
set_log_level(PROGRESS)
from rbnics.backends.dolfin import SeparatedParametrizedForm
from rbnics.utils.decorators.store_map_from_solution_to_problem import _solution_to_problem_map
# Common variables
mesh = UnitSquareMesh(10, 10)
V = VectorFunctionSpace(mesh, "Lagrange", 2)
expr1 = Expression("x[0]", mu_0=0., degree=1, cell=mesh.ufl_cell()) # f_5
expr2 = Expression(("x[0]", "x[1]"), mu_0=0., degree=1,
cell=mesh.ufl_cell()) # f_6
expr3 = Expression((("1*x[0]", "2*x[1]"), ("3*x[0]", "4*x[1]")),
mu_0=0.,
degree=1,
cell=mesh.ufl_cell()) # f_7
expr4 = Expression((("4*x[0]", "3*x[1]"), ("2*x[0]", "1*x[1]")),
mu_0=0.,
degree=1,
cell=mesh.ufl_cell()) # f_8
expr5 = Expression("x[0]", degree=1, cell=mesh.ufl_cell()) # f_9
expr6 = Expression(("x[0]", "x[1]"), degree=1, cell=mesh.ufl_cell()) # f_10
expr7 = Expression((("1*x[0]", "2*x[1]"), ("3*x[0]", "4*x[1]")),
degree=1,
cell=mesh.ufl_cell()) # f_11
