Poisson equation

This demo is implemented in demo_poisson.py. It illustrates how to:

Create a function space
Solve a linear partial differential equation

Equation and problem definition

For a domain \(\Omega \subset \mathbb{R}^n\) with boundary \(\partial \Omega = \Gamma_{D} \cup \Gamma_{N}\), the Poisson equation with particular boundary conditions reads:

\[\begin{split} - \nabla^{2} u &= f \quad {\rm in} \ \Omega, \\ u &= 0 \quad {\rm on} \ \Gamma_{D}, \\ \nabla u \cdot n &= g \quad {\rm on} \ \Gamma_{N}. \\ \end{split}\]

where \(f\) and \(g\) are input data and \(n\) denotes the outward directed boundary normal. The variational problem reads: find \(u \in V\) such that

\[ a(u, v) = L(v) \quad \forall \ v \in V, \]

where \(V\) is a suitable function space and

\[\begin{split} a(u, v) &:= \int_{\Omega} \nabla u \cdot \nabla v \, {\rm d} x, \\ L(v) &:= \int_{\Omega} f v \, {\rm d} x + \int_{\Gamma_{N}} g v \, {\rm d} s. \end{split}\]

The expression \(a(u, v)\) is the bilinear form and \(L(v)\) is the linear form. It is assumed that all functions in \(V\) satisfy the Dirichlet boundary conditions (\(u = 0 \ {\rm on} \ \Gamma_{D}\)).

In this demo we consider:

\(\Omega = [0,2] \times [0,1]\) (a rectangle)
\(\Gamma_{D} = \{(0, y) \cup (1, y) \subset \partial \Omega\}\)
\(\Gamma_{N} = \{(x, 0) \cup (x, 1) \subset \partial \Omega\}\)
\(g = \sin(5x)\)
\(f = 10\exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02)\)

Implementation

The modules that will be used are imported:

from mpi4py import MPI
from petsc4py.PETSc import ScalarType  # type: ignore

import numpy as np

import ufl
from dolfinx import fem, io, mesh, plot
from dolfinx.fem.petsc import LinearProblem
from ufl import ds, dx, grad, inner

Note that it is important to first from mpi4py import MPI to ensure that MPI is correctly initialised.

We create a rectangular Mesh using create_rectangle, and create a finite element function space \(V\) on the mesh.

msh = mesh.create_rectangle(comm=MPI.COMM_WORLD,
                            points=((0.0, 0.0), (2.0, 1.0)), n=(32, 16),
                            cell_type=mesh.CellType.triangle)
V = fem.functionspace(msh, ("Lagrange", 1))

The second argument to functionspace is a tuple (family, degree), where family is the finite element family, and degree specifies the polynomial degree. In this case V is a space of continuous Lagrange finite elements of degree 1.

To apply the Dirichlet boundary conditions, we find the mesh facets (entities of topological co-dimension 1) that lie on the boundary \(\Gamma_D\) using locate_entities_boundary. The function is provided with a ‘marker’ function that returns True for points x on the boundary and False otherwise.

facets = mesh.locate_entities_boundary(msh, dim=(msh.topology.dim - 1),
                                       marker=lambda x: np.logical_or(np.isclose(x[0], 0.0),
                                                                      np.isclose(x[0], 2.0)))

We now find the degrees-of-freedom that are associated with the boundary facets using locate_dofs_topological:

dofs = fem.locate_dofs_topological(V=V, entity_dim=1, entities=facets)

and use dirichletbc to create a DirichletBC class that represents the boundary condition:

bc = fem.dirichletbc(value=ScalarType(0), dofs=dofs, V=V)

Next, the variational problem is defined:

u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
x = ufl.SpatialCoordinate(msh)
f = 10 * ufl.exp(-((x[0] - 0.5) ** 2 + (x[1] - 0.5) ** 2) / 0.02)
g = ufl.sin(5 * x[0])
a = inner(grad(u), grad(v)) * dx
L = inner(f, v) * dx + inner(g, v) * ds

A LinearProblem object is created that brings together the variational problem, the Dirichlet boundary condition, and which specifies the linear solver. In this case an LU solver us sued. The solve computes the solution.

problem = LinearProblem(a, L, bcs=[bc], petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
uh = problem.solve()

The solution can be written to a XDMFFile file visualization with ParaView or VisIt:

with io.XDMFFile(msh.comm, "out_poisson/poisson.xdmf", "w") as file:
    file.write_mesh(msh)
    file.write_function(uh)

and displayed using pyvista.

try:
    import pyvista
    cells, types, x = plot.vtk_mesh(V)
    grid = pyvista.UnstructuredGrid(cells, types, x)
    grid.point_data["u"] = uh.x.array.real
    grid.set_active_scalars("u")
    plotter = pyvista.Plotter()
    plotter.add_mesh(grid, show_edges=True)
    warped = grid.warp_by_scalar()
    plotter.add_mesh(warped)
    if pyvista.OFF_SCREEN:
        pyvista.start_xvfb(wait=0.1)
        plotter.screenshot("uh_poisson.png")
    else:
        plotter.show()
except ModuleNotFoundError:
    print("'pyvista' is required to visualise the solution")
    print("Install 'pyvista' with pip: 'python3 -m pip install pyvista'")