12 2 analysis of the finite element method 1 showing

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Unformatted text preview: of the Finite Element Method 1. showing that U is optimal in the sense that the error u ; U satis es ku ; U k = minN ku ; W k W 2SE (7.1.3) in an appropriate norm, and 2. nding an upper bound for the right-hand side of (7.1.3). The appropriate norm to use with (7.1.3) for the model problem (7.1.1) is the strain energy norm p kv kA = A(v v ): (7.1.4) The nite element solution might not satisfy (7.1.3) with other norms and/or problems. For example, nite element solutions are not optimal in any norm for non-self-adjoint problems. In these cases, (7.1.3) is replaced by the weaker statement ku ; U k C minN ku ; W k W 2S0 (7.1.5) C > 1. Thus, the solution is \nearly best" in the sense that it only di ers by a constant from the best possible solution in the space. Upper bounds of the right-hand sides of (7.1.3) or (7.1.5) are obtained by considering the error of an interpolant W of u. Using Theorems 2.6.4 and 4.6.5, for example, we could conclude that ku ; W ks C hp+1;skukp+1 s=0 1 (7.1.6) if S N consists...
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