Part II Implementation - FEniCS Project

The Finite Element Method3.3 Finite Element Abstract Formalism3.3.1 Linear problemsWe saw above that the finite element solution of Poisson’s equation (3.1) or (3.2)can be obtained by restricting an infinite dimensional variational problem to afinite dimensional variational problem and solving a linear system.To formalize this, we consider a general linear variational problem written inthe following canonical form: Find u ∈ V such thata(v, u) = L(v) ∀v ∈ ˆV , (3.6)where ˆV is the test space and V is the trial space. We may thus express thevariational problem in terms of a bilinear form a and linear form (functional) L,a : ˆV × V → R,L : ˆV → R.As above, we discretize the variational problem (3.6) by restricting to a pair ofdiscrete test and trial spaces: Find u h ∈ V h ⊂ V such thata(v, u h ) = L(v) ∀v ∈ ˆV h ⊂ ˆV . (3.7)To solve the discrete variational problem (3.7), we make an ansatz of the formu h =N∑U j φ j , (3.8)j=1and take v = ˆφ i , i = 1, 2, . . ., N, where {ˆφ i } N i=1 is a basis for the discrete testspace ˆV h and {φ j } N j=1 is a basis for the discrete trial space V h. It follows thatN∑U j a(ˆφ i , φ j ) = L(ˆφ i ), i = 1, 2, . . ., N.j=1We thus obtain the degrees of freedom U of the finite element solution u h bysolving a linear system AU = b, whereA ij = a(ˆφ i , φ j ), i, j = 1, 2, . . ., N,b i = L(ˆφ i ).(3.9)28