Masters Thesis - TU Delft

Chapter 2Finite Element Method2.1 IntroductionWhen dealing with partial differential equations, we need to have a tool which will allow us tosolve them or at least to get an approximation of the solution. In this chapter we will presenta famous method called Finite Element Method, which let us create an approximation of thePDE we are dealing with, which preserves complex geometries and is quite easy to understand.FEM method is used all over the world in thousands of application.We will illustrate the Finite Element method with the solution of a Poisson equation withDirichlet boundary condition, where Ω is a bounded open domain in R 2 and Γ is its boundary.−∆u = f (2.1)2.2 Variational EquationTo solve this problem approximately, we will need to extract a system of algebraic equationswhich will yield the solution. To do that, we will use a common approach, namely the weakformulation of the problem. Denote:∫a(u,u) = < ∇u|∇u > dx∫Ω(f,v) = fv dxIt is easy to show, that a is bilinear. Now, from Green’s formula we get:Ωa(u,v) = −(∆u,v) = (∇u,∇v)Hence, now we can reformulate the problem into following oneFind u ∈ V such that ∀v ∈ V : a(u,v) = (f,v) (2.2)where V ⊂ L 2 is the subspace of all functions whose derivatives up to first order are in L 2 andwhich have zeros on Γ. The resulting space is called H0 1 (Ω). The above condition is called aVariational Equation.9