UFL input for mixed formulation of Poisson Equation

First we define the variational problem in UFL which we save in the file called MixedPoisson.ufl.

We begin by defining the finite element spaces. We define two finite element spaces \(\Sigma_h = BDM\) and \(V_h = DG\) separately, before combining these into a mixed finite element space:

BDM = FiniteElement("BDM", triangle, 1)
DG  = FiniteElement("DG", triangle, 0)
W = BDM * DG

The first argument to FiniteElement specifies the type of finite element family, while the third argument specifies the polynomial degree. The UFL user manual contains a list of all available finite element families and more details. The * operator creates a mixed (product) space W from the two separate spaces BDM and DG. Hence,

\[W = \{ (\tau, v) \ \text{such that} \ \tau \in BDM, v \in DG \}.\]

Next, we need to specify the trial functions (the unknowns) and the test functions on this space. This can be done as follows

(sigma, u) = TrialFunctions(W)
(tau, v) = TestFunctions(W)

Further, we need to specify the source \(f\) (a coefficient) that will be used in the linear form of the variational problem. This coefficient needs be defined on a finite element space, but none of the above defined elements are quite appropriate. We therefore define a separate finite element space for this coefficient.

CG = FiniteElement("CG", triangle, 1)
f = Coefficient(CG)

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

a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx
L = - f*v*dx