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Exercises  45


         1.13 In the assumptions of Theorem 1.9 substitute (1.32) (4) by (4)* and
        amend (6) to
           #
        (6)  Condition (1.32) (6) is valid and
             M 1
                                                                ˆ
                (A h ) rs > 0 ⇒ there exists s ∈{M 1 ,... ,M} such that (A h ) rs < 0.
             s=1
        Under these conditions prove a weak maximum principle as in Theorem 1.9.

         1.14 Assuming the existence of w h ∈ R M 1  such that A h w h ≥ 1 and
        |w h | ∞ ≤ C for some constant C independent of h, show directly (without
        Theorem 1.14) a refined order of convergence estimate on the basis of an
        order of consistency estimate in which also the shape of w h appears.
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