In section 46 we obtained a priori estimates of

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Unformatted text preview: y assumptions. Recall (cf. De nition 4.6.1), that we considered a family of nite element meshes h which became ner as h ! 0. The uniformity condition implied that all vertex angles were bounded away from 0 and and that all aspect ratios were bounded away from 0 as h ! 0. Uniformity ensured that transformations from the physical to the computational space were well behaved. Thus, with uniform meshes, we were able to show (cf. Theorem 4.6.5) that the error in interpolating a function u 2 H p+1 by a complete polynomial W of degree p satis es ku ; W ks C hp+1;skukp+1 s = 0 1: (7.2.10a) The norm on the right can be replaced by the seminorm juj2+1 = p X j j=p+1 kD uk2 0 (7.2.10b) to produce a more precise estimate, but this will not be necessary for our present application. If singularities are present so that u 2 H q+1 with q < p then, instead of (7.2.10a), we nd ku ; W k1 C hq kukq+1: (7.2.10c) 8 Analysis of the Finite Element Method With optimality (or near optimality) established and interp...
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