Poisson equation¶

This demo is implemented in a single Python file, demo_poisson.py, which contains both the variational forms and the solver.

This demo illustrates how to:

Solve a linear partial differential equation
Create and apply Dirichlet boundary conditions
Define a FunctionSpace

The solution for \(u\) in this demo will look as follows:

Equation and problem definition¶

The Poisson equation is the canonical elliptic partial differential equation. 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:

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

Here, \(f\) and \(g\) are input data and \(n\) denotes the outward directed boundary normal. The most standard variational form of Poisson equation 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 shall consider the following definitions of the input functions, the domain, and the boundaries:

\(\Omega = [0,1] \times [0,1]\) (a unit square)
\(\Gamma_{D} = \{(0, y) \cup (1, y) \subset \partial \Omega\}\) (Dirichlet boundary)
\(\Gamma_{N} = \{(x, 0) \cup (x, 1) \subset \partial \Omega\}\) (Neumann boundary)
\(g = \sin(5x)\) (normal derivative)
\(f = 10\exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02)\) (source term)

Implementation¶

This description goes through the implementation (in demo_poisson.py) of a solver for the above described Poisson equation step-by-step.

First, the dolfinx module is imported:

import dolfinx
import numpy as np
import ufl
from dolfinx import (DirichletBC, Function, FunctionSpace, RectangleMesh, fem,
                     plot)
from dolfinx.cpp.mesh import CellType
from dolfinx.fem import locate_dofs_topological
from dolfinx.io import XDMFFile
from dolfinx.mesh import locate_entities_boundary
from mpi4py import MPI
from petsc4py import PETSc
from ufl import ds, dx, grad, inner

We begin by defining a mesh of the domain and a finite element function space \(V\) relative to this mesh. As the unit square is a very standard domain, we can use a built-in mesh provided by the class UnitSquareMesh. In order to create a mesh consisting of 32 x 32 squares with each square divided into two triangles, we do as follows

# Create mesh and define function space
mesh = RectangleMesh(
    MPI.COMM_WORLD,
    [np.array([0, 0, 0]), np.array([1, 1, 0])], [32, 32],
    CellType.triangle, dolfinx.cpp.mesh.GhostMode.none)

V = FunctionSpace(mesh, ("Lagrange", 1))

The second argument to FunctionSpace is the finite element family, while the third argument specifies the polynomial degree. Thus, in this case, our space V consists of first-order, continuous Lagrange finite element functions (or in order words, continuous piecewise linear polynomials).

Next, we want to consider the Dirichlet boundary condition. A simple Python function, returning a boolean, can be used to define the boundary for the Dirichlet boundary condition (\(\Gamma_D\)). The function should return True for those points inside the boundary and False for the points outside. In our case, we want to say that the points \((x, y)\) such that \(x = 0\) or \(x = 1\) are inside on the inside of \(\Gamma_D\). (Note that because of rounding-off errors, it is often wise to instead specify \(x < \epsilon\) or \(x > 1 - \epsilon\) where \(\epsilon\) is a small number (such as machine precision).)

Now, the Dirichlet boundary condition can be created using the class DirichletBC. A DirichletBC takes two arguments: the value of the boundary condition and the part of the boundary on which the condition applies. This boundary part is identified with degrees of freedom in the function space to which we apply the boundary conditions. A method locate_dofs_geometrical is provided to extract the boundary degrees of freedom using a geometrical criterium. In our example, the function space is V, the value of the boundary condition (0.0) can represented using a Function and the Dirichlet boundary is defined immediately above. The definition of the Dirichlet boundary condition then looks as follows:

# Define boundary condition on x = 0 or x = 1
u0 = Function(V)
with u0.vector.localForm() as u0_loc:
    u0_loc.set(0)
facets = locate_entities_boundary(mesh, 1,
                                  lambda x: np.logical_or(np.isclose(x[0], 0.0),
                                                          np.isclose(x[0], 1.0)))
bc = DirichletBC(u0, locate_dofs_topological(V, 1, facets))

Next, we want to express the variational problem. First, we need to specify the trial function \(u\) and the test function \(v\), both living in the function space \(V\). We do this by defining a TrialFunction and a TestFunction on the previously defined FunctionSpace V.

Further, the source \(f\) and the boundary normal derivative \(g\) are involved in the variational forms, and hence we must specify these.

With these ingredients, we can write down the bilinear form a and the linear form L (using UFL operators). In summary, this reads

# Define variational problem
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
x = ufl.SpatialCoordinate(mesh)
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

# Now, we have specified the variational forms and can consider the
# solution of the variational problem. First, we need to define a
# :py:class:`Function <dolfinx.functions.fem.Function>` ``u`` to
# represent the solution. (Upon initialization, it is simply set to the
# zero function.) A :py:class:`Function
# <dolfinx.functions.fem.Function>` represents a function living in a
# finite element function space. Next, we initialize a solver using the
# :py:class:`LinearProblem <dolfinx.fem.linearproblem.LinearProblem>`.
# This class is initialized with the arguments ``a``, ``L``, and ``bc``
# as follows: :: In this problem, we use a direct LU solver, which is
# defined through the dictionary ``petsc_options``.
problem = fem.LinearProblem(a, L, bcs=[bc], petsc_options={"ksp_type": "preonly", "pc_type": "lu"})

# When we want to compute the solution to the problem, we can specify
# what kind of solver we want to use.
uh = problem.solve()

The function u will be modified during the call to solve. The default settings for solving a variational problem have been used. However, the solution process can be controlled in much more detail if desired.

A Function can be manipulated in various ways, in particular, it can be plotted and saved to file. Here, we output the solution to an XDMF file for later visualization and also plot it using the plot command:

# Save solution in XDMF format
with XDMFFile(MPI.COMM_WORLD, "poisson.xdmf", "w") as file:
    file.write_mesh(mesh)
    file.write_function(uh)


# Update ghost entries and plot
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
try:
    import pyvista

    topology, cell_types = plot.create_vtk_topology(mesh, mesh.topology.dim)
    grid = pyvista.UnstructuredGrid(topology, cell_types, mesh.geometry.x)
    grid.point_arrays["u"] = uh.compute_point_values().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 environment variable is set to off-screen (static) plotting save png
    if pyvista.OFF_SCREEN:
        pyvista.start_xvfb(wait=0.1)
        plotter.screenshot("uh.png")
    else:
        plotter.show()
except ModuleNotFoundError:
    print("pyvista is required to visualise the solution")

Poisson equation¶

Equation and problem definition¶

Implementation¶

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