UFL Specification and User Manual 0.3 - FEniCS Project

UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Logg(v, q) = TestFunctions(TH)(u, p) = TrialFunctions(TH)f = Function(P2)a = (inner(grad(v), grad(u)) - div(v)*p + q*div(u))*dxL = dot(v, f)*dxThis example is implemented in the file stokes.ufl in the collection ofdemonstration forms included with the UFL source distribution.3.8 Mixed formulation of PoissonWe next consider the following formulation of Poisson’s equation as a pair offirst order equations for σ ∈ H(div) and u ∈ L 2 :σ +∇u = 0, (3.17)∇·σ = f. (3.18)We multiply the two equations by a pair of test functions τ and w andintegrate by parts to obtain the following variational problem: Find (σ,u) ∈V = H(div)×L 2 such thata((τ,w),(σ,u)) = L((τ,w)) ∀(τ,w) ∈ V, (3.19)wherea((τ,w),(σ,u)) =L((τ,w);f) =∫∫ΩΩτ ·σ −∇·τ u+w∇·σdx, (3.20)w ·f dx. (3.21)We may implement the corresponding forms in our form language using firstorder BDM H(div)-conforming elements for σ and piecewise constant L 2 -conforming elements for u as follows:68