Page 432 - Applied Numerical Methods Using MATLAB

FINITE ELEMENT METHOD (FEM) FOR SOLVING PDE 421
For each subregion s = 1,... ,N s , this solution can be written as

N n N n

φ s (x, y) = c n φ n,s (x, y) = c n (p n,s (1) + p n,s (2)x + p n,s (3)y)
n=1 n=1
(9.4.6)
4. Set the values of the boundary node coefficients in c 1 to the boundary
values according to the boundary condition.
5. Determine the values of the interior node coefficients in c 2 by solving the
system of equations
A 2 c 2 = d (9.4.7)

where

N s T T
∂ ∂ ∂ ∂
T
A 1 = ϕ 2,s ϕ 1,s + ϕ 2,s ϕ 1,s − g(x s ,y s )ϕ 2,s ϕ 1,s S s
∂x ∂x ∂y ∂y
s=1
(9.4.8)
ϕ 1,s = [ φ 1,s φ 2,s · φ N b ,s ] T
∂
ϕ 1,s = [ p 1,s (2)p 2,s (2) · p N b ,s (2) ] T
∂x
∂
ϕ 1,s = [ p 1,s (3)p 2,s (3) · p Nb,s (3) ] T
∂y

N s T T
∂ ∂ ∂ ∂
T
A 2 = ϕ 2,s ϕ 2,s + ϕ 2,s ϕ 2,s − g(x s ,y s )ϕ 2,s ϕ 2,s S s
∂x ∂x ∂y ∂y
s=1
(9.4.9)
ϕ 2,s = [ φ Nb+1,s φ Nb+2,s · φ Nn,s ] T
∂
ϕ 2,s = [ p Nb+1,s (2)φ Nb+2,s (2) · φ Nn,s (2) ] T
∂x
∂ T
ϕ 2,s = [ p Nb+1,s (3)φ Nb+2,s (3) · φ Nn,s (3) ]
∂y
N s

d =−A 1 c 1 − f(x s ,y s )ϕ 2,s S (9.4.10)
s=1
(x s ,y s ): the centroid (gravity center) of the sth subregion S s
The FEM is based on the variational principle that a solution to Eq. (9.4.1)
can be obtained by minimizing the functional

2 2

∂ ∂
I = u(x, y) + u(x, y)
R ∂x ∂y

2
− g(x, y)u (x, y) + 2f (x, y)u(x, y) dx dy (9.4.11)