HP FEM for reaction-diffusion equations I: Robust exponential convergence

SIAM Journal on Numerical Analysis

Volumn 35, Issue 4, 1998, Pages 1520-1557

  Melenk, J M ^a   Schwab, C ^a  

^a ETH ZURICH   (Switzerland)

Author keywords

Boundary layer; Hp finite element method; Robust method; Singularly perturbed problem; Spectral element method

Indexed keywords

EID: 0344307755     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142997317602     Document Type: Article

References (25)

1. K. W. MORTON, Numerical solution of convection-diffusion problems, Appl. Math. Math. Comput., 12 (1996).
2. H.-G. ROOS, M. STYNES, AND L. TOBISKA, Nmerical Methods for Singularly Perturbed Differential Equations, Springer Ser. Comput. Math. 24, 1996.
3. J. J. H. MILLER, E. O'RIORDAN, AND G. I. SHISHKIN, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Rivers Edge, NJ, 1996.
4. I. BABUŠKA AND M. SURI, The p and h-p versions of the finite element method, basic principles and properties, SIAM Rev., 36 (1994), pp. 578-632.
5. C. XENOPHONTOS, The hp Finite Element Method for Singularly Perturbed Problems, Ph.D. thesis, University of Maryland Baltimore County, Catonsville, MD, 1996.
6. C. SCHWAB, M. SURI, AND C. XENOPHONTOS, The hp Finite Element Method for Problems in Mechanics with Boundary Layers, Tech. report 96-20, Seminar für Angewandte Mathematik, ETH Zürich, CH-8092 Zürich, Switzerland, 1996.
7. R. B. KELLOGG, Boundary layers and corner singularities for a self-adjoint problem, in Boundary Value Problems and Integral Equations in Non-Smooth Domains, Lecture Notes in Pure and Applied Mathematics 167, M. Costabel, M. Dauge, and S. Nicaise, eds., Marcel Dekker, New York, 1995, pp. 121-149..
8. J. M. MELENK, Hp FEM for Reaction-Diffusion Equations III: Non-Smooth Domains, in preparation.
9. C. SCHWAB AND M. SURI, The p and hp versions of the finite element method for problems with boundary layers, Math. Comp., 65 (1996), pp. 1403-1429.
10. 3AS, 8 (1998), pp. 299-326.
11. Y. MADAY AND E. M. RØNQUIST, Optimal error analysis of spectral methods with emphasis on nonconstant coefficients and deformed geometries, Comput. Methods Appl. Mech. Engrg., 80 (1990), pp. 91-115.
12. U. BANERJEE AND M. SURI, The effect of numerical quadrature in the p-version of the finite element method, Math. Comp., 59 (1992), pp. 1-20.
13. J. M. MELENK AND C. SCHWAB, Hp FEM for Reaction-Diffusion Equations II: Regularity, Research Report 97-04, Seminar für Angewandte Mathematik, ETH Zürich, CH-8092 Zürich, 1997.
14. J. M. MELENK, On robust exponential convergence of hp finite element methods for problems with boundary layers, IMA J. Numer. Anal., 17 (1997), pp. 577-601.
15. C. B. MORREY, Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966.
16. I. BABUS̃KA AND B. Guo, The hp version of the finite element method for domains with curved boundaries, SIAM J. Numer. Anal., 25 (1988), pp. 837-861.
17. I. BABUS̃KA AND A. K. Aziz, On the angle condition in the finite element method, SIAM J. Numer. Anal., 13 (1976), pp. 214-226.
18. P. J. DAVIS, Interpolation and Approximation, Dover Publications, New York, 1974.
19. B. SÜNDERMANN, Lebesgue constants in Lagrangian interpolation at the Fekete points, Ergebnisberichte der Lehrstühle Mathematik III und VIII (Angewandte Mathematik) 44, Universität Dortmund, Germany, 1980.
20. C. BERNARDI AND Y. MADAY, Approximations spectrales de problèmes aux limites elliptiques, Mathématiques &: Applications, Springer-Verlag, Paris, 1992.
21. I. BABUS̃KA AND M. SURI, The optimal convergence rate of the p-version of the finite element method, SIAM J. Numer. Anal., 24 (1987), pp. 750-776.
22. I. BABUS̃KA, B. A. SZABO, AND I. N. KATZ, The p version of the finite element method, SIAM J. Numer. Anal., 18 (1981), pp. 515-545.
23. G. STRANG, Variational crimes in the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972, pp. 689-710.
24. G. STRANG AND G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, New York, 1973.
25. A. BERGER, R. SCOTT, AND G. STRANG, Approximate boundary conditions in the finite element method, Sympos. Math., 10 (1972), pp. 295-313.