Biharmonic equation

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

This demo illustrates how to:

  • Solve a linear partial differential equation

  • Use a discontinuous Galerkin method

  • Solve a fourth-order differential equation

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


Equation and problem definition

The biharmonic equation is a fourth-order elliptic equation. On the domain \(\Omega \subset \mathbb{R}^{d}\), \(1 \le d \le 3\), it reads

\[\nabla^{4} u = f \quad {\rm in} \ \Omega,\]

where \(\nabla^{4} \equiv \nabla^{2} \nabla^{2}\) is the biharmonic operator and \(f\) is a prescribed source term. To formulate a complete boundary value problem, the biharmonic equation must be complemented by suitable boundary conditions.

Multiplying the biharmonic equation by a test function and integrating by parts twice leads to a problem second-order derivatives, which would requires \(H^{2}\) conforming (roughly \(C^{1}\) continuous) basis functions. To solve the biharmonic equation using Lagrange finite element basis functions, the biharmonic equation can be split into two second-order equations (see the Mixed Poisson demo for a mixed method for the Poisson equation), or a variational formulation can be constructed that imposes weak continuity of normal derivatives between finite element cells. The demo uses a discontinuous Galerkin approach to impose continuity of the normal derivative weakly.

Consider a triangulation \(\mathcal{T}\) of the domain \(\Omega\), where the set of interior facets is denoted by \(\mathcal{E}_h^{\rm int}\). Functions evaluated on opposite sides of a facet are indicated by the subscripts ‘\(+\)’ and ‘\(-\)‘. Using the standard continuous Lagrange finite element space

\[V = \left\{v \in H^{1}_{0}(\Omega)\,:\, v \in P_{k}(K) \ \forall \ K \in \mathcal{T} \right\}\]

and considering the boundary conditions

\[\begin{split}u &= 0 \quad {\rm on} \ \partial\Omega \\ \nabla^{2} u &= 0 \quad {\rm on} \ \partial\Omega\end{split}\]

a weak formulation of the biharmonic problem reads: find \(u \in V\) such that

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

where the bilinear form is

\[a(u, v) = \sum_{K \in \mathcal{T}} \int_{K} \nabla^{2} u \nabla^{2} v \, {\rm d}x \ +\sum_{E \in \mathcal{E}_h^{\rm int}}\left(\int_{E} \frac{\alpha}{h_E} [\!\![ \nabla u ]\!\!] [\!\![ \nabla v ]\!\!] \, {\rm d}s - \int_{E} \left<\nabla^{2} u \right>[\!\![ \nabla v ]\!\!] \, {\rm d}s - \int_{E} [\!\![ \nabla u ]\!\!] \left<\nabla^{2} v \right> \, {\rm d}s\right)\]

and the linear form is

\[L(v) = \int_{\Omega} fv \, {\rm d}x\]

Furthermore, \(\left< u \right> = \frac{1}{2} (u_{+} + u_{-})\), \([\!\![ w ]\!\!] = w_{+} \cdot n_{+} + w_{-} \cdot n_{-}\), \(\alpha \ge 0\) is a penalty parameter and \(h_E\) is a measure of the cell size.

The input parameters for this demo are defined as follows:

  • \(\Omega = [0,1] \times [0,1]\) (a unit square)

  • \(\alpha = 8.0\) (penalty parameter)

  • \(f = 4.0 \pi^4\sin(\pi x)\sin(\pi y)\) (source term)


This demo is implemented in the file.

First, the necessary modules are imported:

import matplotlib.pyplot as plt
from dolfin import *

Next, some parameters for the form compiler are set:

# Optimization options for the form compiler
parameters["form_compiler"]["cpp_optimize"] = True
parameters["form_compiler"]["optimize"] = True

A mesh is created, and a quadratic finite element function space:

# Make mesh ghosted for evaluation of DG terms
parameters["ghost_mode"] = "shared_facet"

# Create mesh and define function space
mesh = UnitSquareMesh(32, 32)
V = FunctionSpace(mesh, "CG", 2)

A subclass of SubDomain, DirichletBoundary is created for later defining the boundary of the domain:

# Define Dirichlet boundary
class DirichletBoundary(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary

A subclass of Expression, Source is created for the source term \(f\):

class Source(UserExpression):
    def eval(self, values, x):
        values[0] = 4.0*pi**4*sin(pi*x[0])*sin(pi*x[1])

The Dirichlet boundary condition is created:

# Define boundary condition
u0 = Constant(0.0)
bc = DirichletBC(V, u0, DirichletBoundary())

On the finite element space V, trial and test functions are created:

# Define trial and test functions
u = TrialFunction(V)
v = TestFunction(V)

A function for the cell size \(h\) is created, as is a function for the average size of cells that share a facet (h_avg). The UFL syntax ('+') and ('-') restricts a function to the ('+') and ('-') sides of a facet, respectively. The unit outward normal to cell boundaries (n) is created, as is the source term f and the penalty parameter alpha. The penalty parameters is made a Constant so that it can be changed without needing to regenerate code.

# Define normal component, mesh size and right-hand side
h = CellDiameter(mesh)
h_avg = (h('+') + h('-'))/2.0
n = FacetNormal(mesh)
f = Source(degree=2)

# Penalty parameter
alpha = Constant(8.0)

The bilinear and linear forms are defined:

# Define bilinear form
a = inner(div(grad(u)), div(grad(v)))*dx \
  - inner(avg(div(grad(u))), jump(grad(v), n))*dS \
  - inner(jump(grad(u), n), avg(div(grad(v))))*dS \
  + alpha/h_avg*inner(jump(grad(u),n), jump(grad(v),n))*dS

# Define linear form
L = f*v*dx

A Function is created to store the solution and the variational problem is solved:

# Solve variational problem
u = Function(V)
solve(a == L, u, bc)

The computed solution is written to a file in VTK format and plotted to the screen.

# Save solution to file
file = File("biharmonic.pvd")
file << u

# Plot solution