Uniqueness

We shall first show that the difference system obtained above has a unique solution.

Theorem: Let the element of A satisfy

Then A is non singular and the quantities and are bounded by

Proof: If for the factorization is valid. Then

so that A is nonsingular

From the hypothesis, .

For an inductive proof of (a), assume that for . But we know that

and thus , so part a) follows. Now we use in and take absolute values to conclude part (b).

Cor. Let and satisfy the inequalities

and the mesh spacing satisfy

. Then the finite difference system has a unique solution.