This tutorial is part of multiphenicsx.

Tutorial 01, case 1: weak imposition of Dirichlet BCs by a Lagrange multiplier (linear problem)

In this tutorial we solve the problem

$$\begin{cases} -\Delta u = f, & \text{in } \Omega,\\ u = g, & \text{on } \Gamma = \partial\Omega, \end{cases}$$

where $\Omega$ is a ball in 2D.

We compare the following two cases:

• strong imposition of Dirichlet BCs: the corresponding weak formulation is $$ \text{find } u \in V_g \text{ s.t. } \int_\Omega \nabla u \cdot \nabla v \ dx = \int_\Omega f v \ dx, \quad \forall v \in V_0\\ $$ where $$ V_g = \{v \in H^1(\Omega): v|_\Gamma = g\},\\ V_0 = \{v \in H^1(\Omega): v|_\Gamma = 0\}.\\ $$
• weak imposition of Dirichlet BCs: this requires an introduction of a multiplier $\lambda$ which is restricted to $\Gamma$, and solves $$ \text{find } w, \lambda \in V \times M \text{ s.t. }\\ \begin{align*} &\int_\Omega \nabla w \cdot \nabla v \ dx & &+ \int_\Gamma \lambda v \ ds & &= \int_\Omega f v \ dx, & \forall v \in V,\\ &\int_\Gamma w \mu \ ds & & & &= \int_\Gamma g \mu \ ds, & \forall \mu \in M \end{align*} $$ where $$ V = H^1(\Omega),\\ M = L^{2}(\Gamma).\\ $$

This example is a prototypical case of problems containing subdomain/boundary restricted variables (the Lagrange multiplier, in this case).

import dolfinx.fem
import dolfinx.fem.petsc
import dolfinx.io
import gmsh
import mpi4py.MPI
import numpy as np
import ufl
import viskex

import multiphenicsx.fem
import multiphenicsx.fem.petsc

Geometrical parameters

r = 3
mesh_size = 1. / 4.

Mesh

gmsh.initialize()
gmsh.model.add("mesh")

p0 = gmsh.model.geo.addPoint(0.0, 0.0, 0.0, mesh_size)
p1 = gmsh.model.geo.addPoint(0.0, +r, 0.0, mesh_size)
p2 = gmsh.model.geo.addPoint(0.0, -r, 0.0, mesh_size)
c0 = gmsh.model.geo.addCircleArc(p1, p0, p2)
c1 = gmsh.model.geo.addCircleArc(p2, p0, p1)
boundary = gmsh.model.geo.addCurveLoop([c0, c1])
domain = gmsh.model.geo.addPlaneSurface([boundary])

gmsh.model.geo.synchronize()
gmsh.model.addPhysicalGroup(1, [c0, c1], 1)
gmsh.model.addPhysicalGroup(2, [boundary], 0)
gmsh.model.mesh.generate(2)

Info    : Meshing 1D...
Info    : [  0%] Meshing curve 1 (Circle)
Info    : [ 60%] Meshing curve 2 (Circle)
Info    : Done meshing 1D (Wall 0.000305383s, CPU 0.000674s)
Info    : Meshing 2D...
Info    : Meshing surface 1 (Plane, Frontal-Delaunay)
Info    : Done meshing 2D (Wall 0.0121244s, CPU 0.012392s)
Info    : 589 nodes 1177 elements

mesh, subdomains, boundaries, *_ = dolfinx.io.gmshio.model_to_mesh(
    gmsh.model, comm=mpi4py.MPI.COMM_WORLD, rank=0, gdim=2)
gmsh.finalize()
assert subdomains is not None
assert boundaries is not None

# Create connectivities required by the rest of the code
mesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)

facets_Gamma = boundaries.indices[boundaries.values == 1]

viskex.dolfinx.plot_mesh(mesh)

viskex.dolfinx.plot_mesh_tags(mesh, boundaries, "boundaries")

Weak imposition of Dirichlet BCs

# Define a function space
V = dolfinx.fem.functionspace(mesh, ("Lagrange", 2))
M = V.clone()

# Define restrictions
dofs_V = np.arange(0, V.dofmap.index_map.size_local + V.dofmap.index_map.num_ghosts)
dofs_M_Gamma = dolfinx.fem.locate_dofs_topological(M, boundaries.dim, facets_Gamma)
restriction_V = multiphenicsx.fem.DofMapRestriction(V.dofmap, dofs_V)
restriction_M_Gamma = multiphenicsx.fem.DofMapRestriction(M.dofmap, dofs_M_Gamma)
restriction = [restriction_V, restriction_M_Gamma]

# Define trial and test functions
(u, l) = (ufl.TrialFunction(V), ufl.TrialFunction(M))
(v, m) = (ufl.TestFunction(V), ufl.TestFunction(M))

# Define problem block forms
g = dolfinx.fem.Function(V)
g.interpolate(lambda x: np.sin(3 * x[0] + 1) * np.sin(3 * x[1] + 1))
a = [[ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx, ufl.inner(l, v) * ufl.ds],
     [ufl.inner(u, m) * ufl.ds, None]]
f = [ufl.inner(1, v) * ufl.dx, ufl.inner(g, m) * ufl.ds]

# Solve
petsc_options = {
    "ksp_type": "preonly",
    "pc_type": "lu",
    "pc_factor_mat_solver_type": "mumps",
    "ksp_error_if_not_converged": True,
}
problem = multiphenicsx.fem.petsc.LinearProblem(
    a, f, petsc_options_prefix="tutorial_1_lagrange_multipliers_linear_", petsc_options=petsc_options,
    kind="mpi", restriction=restriction
)
u, l = problem.solve()
del problem

viskex.dolfinx.plot_scalar_field(u, "u")

viskex.dolfinx.plot_scalar_field(l, "l")

Strong imposition of Dirichlet BCs

# Define Dirichlet BC object on Gamma
dofs_V_Gamma = dolfinx.fem.locate_dofs_topological(V, boundaries.dim, facets_Gamma)
bc_ex = dolfinx.fem.dirichletbc(g, dofs_V_Gamma)

# Solve
problem_ex = dolfinx.fem.petsc.LinearProblem(
    a[0][0], f[0], bcs=[bc_ex],
    petsc_options_prefix="tutorial_1_lagrange_multipliers_linear_ex_", petsc_options=petsc_options
)
u_ex = problem_ex.solve()
assert isinstance(u_ex, dolfinx.fem.Function)
del problem_ex

viskex.dolfinx.plot_scalar_field(u_ex, "u")

Comparison and error computation

u_ex_norm = np.sqrt(mesh.comm.allreduce(
    dolfinx.fem.assemble_scalar(dolfinx.fem.form(ufl.inner(ufl.grad(u_ex), ufl.grad(u_ex)) * ufl.dx)),
    op=mpi4py.MPI.SUM))
err_norm = np.sqrt(mesh.comm.allreduce(
    dolfinx.fem.assemble_scalar(dolfinx.fem.form(ufl.inner(ufl.grad(u_ex - u), ufl.grad(u_ex - u)) * ufl.dx)),
    op=mpi4py.MPI.SUM))
print("Relative error is equal to", err_norm / u_ex_norm)
assert np.isclose(err_norm / u_ex_norm, 0., atol=1.e-10)

Relative error is equal to 5.277836536859761e-15