where S2 is the matrix
I+2A1-1
-A1-1
I+ 2A,
-A-1 I+ 2Af-1 -A,-1
-A,-1 I+ 2A-1
Thus,
h|IA2-1B21. -< hi AH-A BIjlIjS2|oo.
By Lemma 5.3.1, we have IAi-1B1Io = O(At/h). Hence,
hIA2- B21loo < O(At)||S2||oo.
Note that S2 is difficult to analyze using the max-norm. Hence, S2 is evaluated on
a computer to get a numerical value for J|S21oo. For example, if A1 is 10 x 10 and
m = 10, so that S2 is 100 x 100, then ||S21|oo 1.6687. If A1 is 20 x 20 and m = 20,
so that S2 is 400 x 400, then J|S2|1K| 2.0794. If A1 is 40 x 40 and m = 40, so that
S2 is 1600 x 1600, then |IS2||oo ; 2.5041. Therefore, we have
hIIA21-B2[[o, = O(At).
-A,-1
(5.12)