Biharmonic equation¶

This demo is implemented in a single Python file, demo_biharmonic.py, 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)

Implementation¶

This demo is implemented in the demo_biharmonic.py 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

# 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
plot(u)
plt.show()