def test_surf_approx(n, f, true=None):
'''Surface integral (for benchmark problem)'''
omega = UnitCubeMesh(n, n, 2 * n)
A = np.array([0.5, 0.5, 0.0])
B = np.array([0.5, 0.5, 1.0])
gamma = StraightLineMesh(A, B, 2 * n)
V3 = FunctionSpace(omega, 'CG', 1)
V1 = FunctionSpace(gamma, 'CG', 1)
f3 = interpolate(f, V3)
shape = SquareRim(P=lambda x: np.array([0.25, 0.25, x[2]]), degree=10)
Pi = average_matrix(V3, V1, shape=shape)
x = Pi * f3.vector()
f1 = Function(V1, x)
if true is None:
true = f
error = sqrt(abs(assemble(inner(true - f1, true - f1) * dx)))
return gamma.hmin(), error, f1
def poisson_1d(n, u_true, f):
'''1d part of benchmark'''
A, B = np.array([0.5, 0.5, 0]), np.array([0.5, 0.5, 1])
mesh = StraightLineMesh(A, B, n)
V = FunctionSpace(mesh, 'CG', 1)
bc = DirichletBC(V, u_true, 'on_boundary')
u, v = TrialFunction(V), TestFunction(V)
a = inner(grad(u), grad(v)) * dx
L = inner(f, v) * dx
A, b = assemble_system(a, L, bc)
solver = KrylovSolver('cg', 'amg')
solver.parameters['relative_tolerance'] = 1E-13
uh = Function(V)
solver.solve(A, uh.vector(), b)
return mesh.hmin(), errornorm(u_true, uh, 'H1'), uh
def test_Pi(n, f3, f1, Pi_f3):
'''(Pi u_3d, u_1d)_Gamma.'''
# True here is the reduced one
omega = UnitCubeMesh(n, n, 2*n)
A = np.array([0.5, 0.5, 0.0])
B = np.array([0.5, 0.5, 1.0])
gamma = StraightLineMesh(A, B, 2*n)
V3 = FunctionSpace(omega, 'CG', 1)
V1 = FunctionSpace(gamma, 'CG', 1)
u3 = TrialFunction(V3)
v1 = TestFunction(V1)
shape = SquareRim(P=lambda x: np.array([0.25, 0.25, x[2]]),
degree=10)
dx_ = Measure('dx', domain=gamma)
a = inner(Average(u3, gamma, shape=shape), v1)*dx_
A = ii_assemble(a)
# Now check action
f3h = interpolate(f3, V3)
x = A*f3h.vector()
f1h = interpolate(f1, V1)
Pi_f = Function(V1, x)
result = f1h.vector().inner(x)
true = assemble(inner(Pi_f3, f1)*dx_)
error = abs(true - result)
return gamma.hmin(), error, Pi_f