Introduction to Computational Linear Algebra

Sparse Systems to Solve Poisson Differential Equations 207

Recall from (7.59) that the finite element system of linear equations is given

by:

n

X

j=2

[A(ψ

j

, ψ

i

)]U

j

= F (ψ

i

) − αA(ψ

1

, ψ

i

), 2 ≤ i ≤ n.

where

A(ψ

i

, ψ

j

) =

1

Z

0

h

a(x)

dψ

i

dx

dψ

j

dx

+ b(x)ψ

i

ψ

j

i

dx.

{ψ

i

} is a set of functions with compact support. supp(ψ

i

) = [x

i−1

, x

i+1

].

Definition 7.4 Given a node i from the set of points that partition Ω we

define:

E(i) = {E ∈ Π |i is a boundary point to E}.

Example 7.1 In the figure above, E(3) = {E

2

, E

3

}, E(1) = {E

1

}, E(5) =

{E

4

}.

Lemma 7.1

A(ψ

i

, ψ

j

) =

Z

E(i)∩E(j)

h

a(x)

dψ

i

dx

dψ

j

dx

+ b(x)ψ

i

ψ

j

i

dx

=

(

0 if i and j do not border the same element

6= 0 if i and j border the same element

Corollary ...