DOLFIN User Manual - FEniCS Project

DOLFIN User ManualHoffman, Jansson, Logg, Wells# ./configure# make# make installMake sure that you download the latest release (which is not 0.1.0).Note that you may need to be root on your system to perform the laststep. Since DOLFIN depends on a number of other packages, including thelinear algebra package PETSc and the form compiler FFC, you may also needdownload and install these packages before you can compile DOLFIN. (SeeAppendix B for detailed instructions.)2.2 Solving Poisson’s equation with DOLFINLet’s say that we want to solve Poisson’s equation on the unit square Ω =(0, 1)×(0, 1) with homogeneous Dirichlet boundary conditions on the boundaryΓ 0 = {(x, y) ∈ ∂Ω : x = 1}, homogeneous Neumann boundary conditionson the remaining part of the boundary and right-hand side given byf = x sin(y),− ∆u(x, y) = x sin(y), x ∈ Ω = (0, 1) × (0, 1), (2.1)u(x) = 0, x ∈ Γ 0 = {(x, y) ∈ ∂Ω : x = 1}, (2.2)∂ n u(x) = 0, x ∈ ∂Ω \ Γ 0 . (2.3)To solve a partial differential equation such as Poisson with DOLFIN, itmust first be rewritten in variational form. The discrete variational (finiteelement) formulation for Poisson’s equation reads: Find u ∈ V h such thata(v, U) = L(v) ∀v ∈ ˆV h , (2.4)with ( ˆV h , V h ) a pair of suitable function spaces (the test and trial spaces).The bilinear form a : ˆV h × V h → R is given by∫a(v, U) = ∇v · ∇U dx (2.5)Ω16