Poisson equation

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

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:

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

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

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

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:

import numpy as np

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

from mpi4py import MPI
from petsc4py.PETSc import ScalarType

We create a rectangular Mesh using create_rectangle, and create a finite element FunctionSpace \(V\) 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 DirichletBCMetaClass 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 = fem.petsc.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.create_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'")