Page 431 - Applied Numerical Methods Using MATLAB

420 PARTIAL DIFFERENTIAL EQUATIONS
9.4 FINITE ELEMENT METHOD (FEM) FOR SOLVING PDE

The FEM method is another procedure used in finding approximate numerical
solutions to BVPs/PDEs. It can handle irregular boundaries in the same way as
regular boundaries [R-1, S-2, Z-1]. It consists of the following steps to solve the
elliptic PDE:

2
2
∂ u(x, y) ∂ u(x, y)
+ + g(x, y)u(x, y) = f (x, y) (9.4.1)
∂x 2 ∂y 2
for the domain D enclosed by the boundary B on which the boundary condition
is given as
u(x, y) = b(x, y) on the boundary B (9.4.2)

1. Discretize the (two-dimensional) domain D into, say, N s subregions
{S 1 ,S 2 ,. ..,S Ns } such as triangular elements, neither necessarily of the
same size nor necessarily covering the entire domain completely and
exactly.
2. Specify the positions of N n nodes and number them starting from the
boundary nodes, say, n = 1,... ,N b , and then the interior nodes, say, n =
N b + 1,..., N n .
3. Define the basis/shape/interpolation functions

φ n (x, y) ={φ n,s , for s = 1,...,N s }∀ (x, y) ∈ D (9.4.3a)
φ n,s (x, y) = p n,s (1) + p n,s (2)x + p n,s (3)y

for each subregion S s (9.4.3b)

collectively for all subregions s = 1: N s and for each node n = 1: N n ,so
that φ n is 1 only at node n, and 0 at all other nodes. Then, the approxi-
mate solution of the PDE is a linear combination of basis functions
φ n (x, y) as

N n N b N n

T T T
u(x, y) = c ϕ(x, y) = c n φ n (x, y) = c n φ n + c n φ n = c ϕ 1 + c ϕ 2
1
2
n=1 n=1 n=N b +1
(9.4.4)
where
T
ϕ 1 = [ φ 1 φ 2 · φ N b ] , c 1 = [ c 1 c 2 · c N b ] T (9.4.5a)
T
ϕ 2 = [ φ N b +1 φ N b +2 · φ N n ] , c 2 = [ c N b +1 c N b +2 · c N n ] T
(9.4.5b)