UFL input for the Poisson equation

The first step is to define the variational problem at hand. We define the variational problem in UFL terms in a separate form file Poisson.ufl. We begin by defining the finite element:

element = FiniteElement("Lagrange", triangle, 1)

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

Next, we use this element to initialize the trial and test functions (\(u\) and \(v\)) and the coefficient functions (\(f\) and \(g\)):

u = TrialFunction(element)
v = TestFunction(element)
f = Coefficient(element)
g = Coefficient(element)

Finally, we define the bilinear and linear forms according to the variational formulation of the equations:

a = inner(grad(u), grad(v))*dx
L = f*v*dx + g*v*ds

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