UFL input for the Biharmonic equation

The finite element space is defined:

element = FiniteElement("Lagrange", triangle, 2)

On the space element, trial and test functions, and the source term are defined:

# Trial and test functions
u = TrialFunction(element)
v = TestFunction(element)
f = Coefficient(element)

Next, the outward unit normal to cell boundaries and a measure of the cell size are defined. The average size of cells sharing a facet will be used (h_avg). The UFL syntax ('+') and ('-') restricts a function to the ('+') and ('-') sides of a facet, respectively. The penalty parameter alpha is made a Constant so that it can be changed in the program without regenerating the code.

# Normal component, mesh size and right-hand side
n  = FacetNormal(triangle)
h = CellDiameter(triangle)
h_avg = (h('+') + h('-'))/2

# Parameters
alpha = Constant(triangle)

Finally the bilinear and linear forms are defined. Integrals over internal facets are indicated by *dS.

# 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

# Linear form
L = f*v*dx

Before the form file can be used in the C++ program, it must be compiled using FFC by running (on the command-line):

ffc -l dolfin Poisson.ufl