0 with such uniform meshes we can combine theorems 462

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Unformatted text preview: (4.6.11) 4.4. Three-Dimensional Shape Functions 35 Proof. Use the bounds on and with (4.6.9) and (4.6.10) to rede ne the constant C and obtain (4.6.11). Theorems 4.6.2 - 4.6.5 only apply when u 2 H p+1. If u has a singularity and belongs to H q+1, q < p, then the convergence rate is reduced to ju ; U js C hq+1;sjujq+1 s = 0 1: (4.6.12) Thus, there appears to be little bene t to using p th-degree piecewise-polynomial interpolants in this case. However, in some cases, highly graded nonuniform meshes can be created to restore a higher convergence rate. 36 Finite Element Approximation Bibliography 1] S. Adjerid, M. Ai a, and J.E. Flaherty. Hierarchical nite element bases for triangular and tetrahedral elements. Computer Methods in Applied Mechanics and Engineering, 2000. to appear. 2] O. Axelsson and V.A. Barker. Finite Element Solution of Boundary Value Problems. Academic Press, Orlando, 1984. 3] S.C. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 1994. 4] P. Carnevali, R.V. Morric, Y.Tsuji, and B. Taylor. New basis functions and computational procedures for p-version nite element analysis. International Journal of Numerical Methods in Enginneering, 36:3759{3779, 1993. 5] S. Dey, M.S. Shephard, and J.E. Flaherty. Geometry-based issues associated with p-version nite element computations. Computer Methods in Applied Mechanics and Engineering, 150:39 { 50, 1997. 6] M.S. Shephard, S. Dey, and J.E. Flaherty. A straightforward structure to construct shape functions for variable p-order meshes. Computer Methods in Applied Mechanics and Engineering, 147:209{233, 1997. 7] B. Szabo and I. Babuska. Finite Element Analysis. John Wiley and Sons, New York, 1991. 8] O.C. Zienkiewicz. The Finite Element Method. McGraw-Hill, New York, third edition, 1977. 37...
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