Diploma thesis as pdf file - Johannes Kepler University, Linz

3.2 Finite element discretization 27
We use standard Galerkin method. Let T h be regular triangulation of ¯Ω :
¯Ω = ∪ T ∈Th T , T - triangle of the mesh;
for any two distinct triangles T 1 and T 2 : intT 1 ∩ intT 2 = ∅;
any non-empty intersections of two distinct triangles equals one common edge
E ∈ E (T h ) or a node x ∈ N (T h ).
{
∣ ∣ }
∣∣T
Let V h = v h ∈ C( ¯Ω)| v h ∈ P 1 ∣∣∂Ω
(T ) ∀T ∈ T h , v h = 0 ⊂ V = H0 1(Ω).
Let {φ j }, j = 1,N be piecewise linear basis functions in V h with compact support:
∀x i ∈ N (T h )
φ j (x i ) = δ i j ; N < ∞ number of nodes in the mesh, dimension of
V h . Then we can use the representation
u h = ∑u j φ j ,
j
δu h = ∑δu j φ j .
j
Substituting this in (3.2) we obtain the linear system of equations :
Aδu h = b,
where A ∈ R N×N : V h → R N has components:
∫
A i j =
b ∈ R N with components
Ω
∫
φ i φ j
∇φ i ∇φ j dω + κ
Ω (u h − ψ) 2 dω,
∫
∫
∫
b i = −∑u j ∇φ i ∇φ j dω + f φ i dω + κ
j Ω
Ω
Ω
and δu h ∈ R N with components δu j .
Assembling the stiffness matrix
φ i
u h − ψ dω,
For a triangular element T ∈ T h let (x 1 ,y 1 ),(x 2 ,y 2 ),(x 3 ,y x ) be the vertices and φ 1 ,
φ 2 , φ 3 be the corresponding basis functions in V h . We denote by |T | the area of the
triangle. Then
⎛ ⎞
|T | = 1 1 1 1
⎝x 1 x 2 x 3
⎠
2
y 1 y 2 y 3