# 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

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

and considering the boundary conditions

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

where the bilinear form is

and the linear form is

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
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
plot(u)
plt.show()
```