On n-Widths for Elliptic Problems

Journal of Mathematical Analysis and Applications

Volumn 247, Issue 1, 2000, Pages 272-289

  Melenk, J M ^a  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

N width; singularly perturbed problem; rough coefficients

Indexed keywords

EID: 0034215872     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.2000.6862     Document Type: Article

References (21)

1. Kolmogorov A. N. Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse. Ann. Math. 37:1936;107-111.
2. Jerome J. W. On n-widths in Sobolev spaces and applications to elliptic boundary value problems. J. Math. Anal. Appl. 29:1970;201-215.
3. Kellogg R. B., Stynes M. Optimal approximability of solutions of singularly perturbed differential equations. SIAM J. Numer. Anal. 34:1997;1808-1816.
4. Kellogg R. B., Stynes M. n-widths and singularly perturbed boundary value problems. SIAM J. Numer. Anal. 36:1999;1604-1620.
5. Triebel H. Interpolation Theory, Function Spaces, and Differential Operators. 1995;Johann Ambrosius Barth, Heidelberg.
6. Thomée V. Galerkin Finite Element Methods for Parabolic Problems. 1987;Springer-Verlag, New York/Berlin.
7. Lorentz G. G. Approximation of Functions. 1986;Chelsea, New York.
8. Pinkus A. n-Widths in Approximation Theory. 1984;Springer-Verlag, New York/Berlin.
9. Levendorskǐ S. Asymptotic Distribution of Eigenvalues of Differential Operators. 1990;Kluwer Academic, Dordrecht.
10. Stein E. M. Singular Integrals and Differentiability Properties of Functions. 1970;Princeton Univ. Press, Princeton.
11. Aziz A. K., Babuška I. M. Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. 1972;Academic Press, New York.
12. Melenk J. M., Schwab C. hp FEM for reaction-diffusion equations. I. Robust exponentional convergence. SIAM J. Numer. Anal. 35:1998;1520-1557.
13. Lax P. D., Milgram A. N. Parabolic Equations. Annals of Mathematics Studies. 33:1954;Princeton Univ. Press, Princeton. p. 167-190.
14. Raugel G. Résolution numérique par une méthode d'élements finis du probleme de Dirichlet pour le Laplacien dans un polygone. C. R. Acad. Sci. Paris Ser. A. 286:1978;791-794.
15. Babuška I., Kellogg R. B., Pitkäranta J. Direct and inverse estimates for finite elements with mesh refinement. Numer. Math. 33:1979;447-471.
16. Babuška I., Caloz G., Osborn J. Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31:1994;945-981.
17. Melenk J. M., Babuška I. The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139:1996;289-314.
18. Miller J. J. H., O'Riordan E., Shishkin G. I. Fitted Numerical Methods for Singular Perturbation Problems. 1996;Word Scientific, Singapore.
19. Sun G., Stynes M. Finite-element methods for singularly perturbed high-order elliptic boundary value problems. I. Reaction-diffusion-type problems. IMA J. Numer. Anal. 15:1995;117-139.
20. Dobrowolski M., Roos H.-G. A priori estimates for the solution convection-diffusion problems and interpolation on Shishkin meshes. Z. Anal. Anwendungen. 16:1997;1001-1012.
21. Melenk J. M. On the robust exponential convergence of hp finite element methods for problems with boundary layers. IMA J. Numer. Anal. 17:1997;577-601.