Mathematical Physics - Volume II - Numerical Methods

Chapter 3. Comparison of finite element method and finite difference method

68

δΩ : 2 k pu + = γ δ u δ n

δΩ 2 h

δΩ

δΩ :

1 h

1 u u = ^

k 2

y

y

k 1

x

x

(a)

(b)

Figure 3.13: Two-dimensional domain (a) and its discretization (b).

The weight function can now be observed within each individual element: v h ( ξ , η ) = v i φ i ( ξ , η ) (3.85) where v i is the weight function value in the i -th node. In a general case, Dirichlet function ˆ u , given on ∂ Ω 1 is approximated by the interpolation ˆ u h ( s ) = ˆ u i φ i ( s ) „ wherein the summing is performed along all nodes of the approximation ∂ Ω 1 h (of ∂ Ω 1 ). The approximate solution for function u h in H h is: u h ( ξ , η ) = u i φ i ( ξ , η ) (3.86) hence, u i = ˆ u i in nodes at ∂ Ω 1 h , and the following holds: Z Ω h k ∇ u · ∇ v d x d y + Z ∂ Ω 2 h ˆ σ v h d s − Z Ω h f v h d x d y = 0 (3.87) for ∀ v h ∈ H h , such that v h = 0 at ∂ Ω 1 h „ i.e. for the approximation ∂ Ω 1 . By replacing (3.85) and (3.86) into (3.87), we obtain: K i j u j = F i i , j = 1 , 2 , · · · , N (3.88) where K i j are stiffness matrix terms: K i j = Z Ω h k ∇ φ i · ∇ φ j d x d y (3.89) while F i are the components of force vector: F i = Z Ω h f φ i d x d y − Z ∂ Ω 2 h ˆ σφ i d s . (3.90) Now we need to modify the equation (3.87) in order to include Dirichlet’s conditions and solve the resulting system of unknown nodal values, u j , thus determining the finite element approximation of the equation, (3.84) solution, u . It is necessary to point out the similarities between this approach and the on-dimensional problem: