Introduction to Computational Linear Algebra

Sparse Systems to Solve Poisson Differential Equations 187

Proof. From Theorem 7.1-1, the matrix A is spd and therefore the system

(7.15) has a unique solution. Furthermore, since:

U

T

AU = U

T

F,

then using part 4 of Theorem 7.1, one has:

||U||

2

2

≤

L

2

γ

0

U

T

AU =

L

2

γ

0

U

T

F ≤

L

2

γ

0

||U||

2

||F ||

2

≤

L

2

γ

0

||U||

2

||F ||

2

.

Simplifying by ||U ||

2

leads to the result of this theorem.

Consequently, one obtains convergence of the discrete solution of (7.15) to the

exact solution of (7.1) in addition to error estimates. Specifically if the solution

u to (7.1) is such that, u ∈ C

k

(Ω) ∩ C(Ω), k ≥ 3, then the approximation

U

h

= U = {U

i

} to u

h

= {u

i

= u(x

i

)} verifies the estimates:

||u

h

− U

h