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38a 16 analysis of the finite element method where

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Unformatted text preview: n @ ~ . (Thus, S0 is not a subspace of H01.) Figure 7.3.1: Approximation of a curved boundary by a polygon. For piecewise linear polynomial approximations on triangles they show that ku ; U k1 = O(h) and for piecewise quadratic approximations ku ; U k1 = O(h3=2). The poor accuracy with quadratic polynomials is due to large errors in a narrow \boundary layer" near @ . Large errors are con ned to the boundary layer and results are acceptable elsewhere. Wait and Mitchell 21], Chapter 6, quote other results which prove that ku ; U k1 = O(hp) for pth degree piecewise polynomial approximations when the distance between @ and @ ~ is O(hp+1). Such is the case when @ is approximated by p th degree piecewise-polynomial interpolation. 7.4 A Posteriori Error Estimation In previous sections of this chapter, we considered a priori error estimates. Thus, we can, without computation, infer that nite element solutions converge at a certain rate depending on the exact solution's smoothness. Error bounds are expressed in terms of unknown constants which are di cult, i...
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