Combining this estimate with Lemma 4.2.2, we find that when the solution to
(3.7) is substituted in (4.5), we obtain
O(At)' + O(h2),
where m = 3 if p = 1/2, and m = 2 if p # 1/2. The finite difference scheme
(4.5) yields a matrix-vector relation of the form
Ax"p+ AA 2" = h [(1 j)BqIn + Bn+l] 1 e -I, (4.6)
7 e
where In corresponds to the term V J of (4.5) evaluated at time level n + 5.
The next theorem gives us the max-norm error estimate when the solution to
(3.7) is substituted into Equation (4.6). Define the max-norm for a vector x as
|ix|oo = max |xil,
1