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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