This preview shows page 1. Sign up to view the full content.
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...
View Full Document
This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14