DOLFIN User Manual - FEniCS Project

DOLFIN User ManualHoffman, Jansson, Logg, Wells7.2 Variational formulationA variational formulation of (7.1) takes the form: find u ∈ V such thata(v, u) = L(v) ∀v ∈ ˆV , (7.2)where a(·, ·) : ˆV × V → R is a bilinear form acting on ˆV × V , with ˆV and Vthe test space and trial space respectively, defined by∫∫∂v ∂ua(v, u) = ∇v · ∇u dx =dx, (7.3)∂x i ∂x iΩwhere we employ tensor notation so that the double index i means summationfrom i = 1, ..., d, and L(·) : ˆV → R is a linear form acting on the test spaceˆV , defined by∫L(v) = vf dx. (7.4)For this problem we typically use V = ˆV = H 1 0(Ω), with H 1 0(Ω) the standardSobolev space of square integrable functions with also their first derivativessquare integrable (in the Lebesgue sense), with the functions being zero onthe boundary (in the sense of traces).The FEM method for (7.2) is now: find U ∈ V h such thatΩΩa(v, U) = L(v) ∀v ∈ ˆV h , (7.5)where V h ⊂ V and ˆV h ⊂ ˆV are finite dimensional subspaces of dimensionM. The finite element spaces V h , ˆV h are characterized by their sets of basisfunctions {ϕ i } M i=1 , { ˆϕ i} M i=1 . The FEM method (7.5) is thus specified by thevariational form and the basis functions of V h and ˆV h .7.3 Finite elements and FIATFinite element basis functions in DOLFIN are defined using FIAT, whichsupports the generation of arbitrary order Lagrange finite elements on lines,triangles, and tetrahedra. Upcoming versions of FIAT will also support Hermiteand nonconforming elements as well as H(div) and H(curl) elementssuch as Raviart-Thomas and Nedelec.46