hp-interpolation of nonsmooth functions and an application to hp-A posteriori error estimation

SIAM Journal on Numerical Analysis

Volumn 43, Issue 1, 2005, Pages 127-155

  Melenk, J M ^a,b  

^a MPI für Mathematik in Den Naturwissenschaften ^*  (Germany)

^b UNIVERSITY OF READING   (United Kingdom)

Author keywords

A posteriori error estimation; Cl ment interpolant; hp FEM; Polynomial inverse estimates; Quasi interpolation

Indexed keywords

FINITE ELEMENT METHOD; INVERSE PROBLEMS; MATHEMATICAL OPERATORS; OPTIMIZATION; POLYNOMIALS;

A POSTERIORI ERROR ESTIMATION; CLÉMENT INTERPOLANT; POLYNOMIAL INVERSE ESTIMATES; QUASI INTERPOLATION;

FUNCTIONS;

EID: 33644524981     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903432930     Document Type: Article

References (40)

1. R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. M. AINSWORTH AND D. KAY, The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs, Numer. Math., 82 (1999), pp. 351-388.
3. M. AINSWORTH AND D. KAY, Approximation theory for the hp-version finite element method and application to the non-linear Laplacian, Appl. Numer. Math., 34 (2000), pp. 329-344.
4. M. AINSWORTH AND T.J. ODEN, A Posteriori Error Estimation in Finite Element Analysis, Wiley, New York, 2000.
5. T. APEL, Anisotropic Finite Elements: Local Estimates and Applications, Adv. Numer. Math., Teubner, Leipzig, 1999.
6. I. BABUŠKA, A. CRAIG, J. MANDEL, AND J. PITKÄRANTA, Efficient preconditioning for the p version finite element method in two dimensions, SIAM J. Numer. Anal., 28 (1991), pp. 624-661.
7. I. BABUŠKA AND T. STROUBOULIS, The Finite Element Method and its Reliability, Oxford University Press, Oxford, UK, 2001.
8. I. BABUŠKA 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.
9. IVO BABUSKA AND BENQI GUO, Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions, Numer. Math., 85 (2000), pp. 219-255.
10. IVO BABUSKA AND BENQI GUO, Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces. I: Approximability of functions in the weighted Besov spaces, SIAM J. Numer. Anal., 39 (2001), pp. 1512-1538.
11. F. BEN BELGACEM, Relèvement polynômiaux sur un cube, Technical report HI-72/7780, Electricité de France, Clamart, France, 1992.
12. F. BEN BELGACEM, Polynomial extensions of compatible polynomial traces in three dimensions, Comput. Meth. Appl. Mech. Engrg., 116 (1994), pp. 235-241.
13. C. BERNARDI, Optimal finite element interpolation, SIAM J. Numer. Anal, 26 (1989), pp. 430-454.
14. C. BERNARDI, M. DAUGE, AND Y. MADAY, Trace liftings which preserve polynomials, C. R. Acad. Sci. Paris Sér. I, 315 (1992), pp. 333-338.
15. C. BERNARDI, M. DAUGE, AND Y. MADAY, Interpolation of nullspaces for polynomial approximation of divergence-free functions in a cube, in Boundary value Problems and Integral Equations in Nonsmooth Domains, Lectures Notes in Pure and Appl. Math., 167, M. Costabel, M. Dauge, and S. Nicaise, eds., Dekker, New York, 1995, pp. 27-46.
16. C. BERNARDI AND V. GIRAULT, A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal., 35 (1998), pp. 1893-1916.
17. C. BERNARDI AND Y. MADAY, Spectral methods, in Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, eds., Vol. 5, North-Holland, Amsterdam, 1997, pp. 209-485.
18. S. BÖRM, M. LÖHNDORF, AND J. M. MELENK, Approximation of integral operators by variable-order interpolation, Numer. Math., 99 (2005), pp. 605-643.
19. S. C. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, 1994.
20. C. CARSTENSEN, Quasi interpolation and a posteriori error analysis in finite element methods, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 1187-1202.
21. C. CARSTENSEN AND R. VERFÜRTH, Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal., 36 (1999), pp. 1571-1587.
22. PH. CLÉMENT, Approximation by finite element functions using local regularization, RAIRO Anal. Numérique, 9 (1975), pp. 77-84.
23. L. DEMKOWICZ, T. J. ODEN, W. RACHOWICZ, AND O. HARDY, Towards a universal hp finite element strategy. Part 1. Constrained approximation and data structure, Comput. Meth. Appl. Mech. Engrg., 77 (1989), pp. 79-112.
24. R. A. DEVORE AND G. G. LORENTZ, Constructive Approximation, Springer-Verlag, Berlin, 1993.
25. T. EIBNER AND J. M. MELENK, Local error analysis of the boundary concentrated fem, IMA J. Numer. Anal., to appear.
26. E. GAGLIARDO, Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova, 27 (1957), pp. 284-305.
27. V. GIRAULT AND L. R. SCOTT, Hermite interpolation of nonsmooth functions preserving boundary conditions, Math. Comput., 73 (2002), pp. 1043-1074.
28. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.
29. G. HARDY, J. E. LITTLEWOOD, AND G. PÓLYA, Inequalities, in Cambridge Mathematical Library, Cambridge University Press, Cambridge 1991.
30. J. L. LIONS, Théorèmes de trace et d'interpolation (IV), Math. Annalen, 152 (1963), pp. 42-56.
31. Y. MADAY, Relèvement de traces polyômiales et interpolations hilbertiennes entres espaces de polynômes, C. R. Acad. Sci. Paris, Série I, 309 (1989), pp. 463-468.
32. J. M. MELENK, hp-interpolation of nonsmooth functions, Technical report NI03050, Isaac New-ton Institute for Mathematics Sciences, Cambridge, UK, 2003.
33. J. M. MELENK AND I. BABUŠKA, The partition of unity finite element method: basic theory and applications, Comput. Meth. Appl. Mech. Engrg., 139 (1996), pp. 289-314.
34. J. M. MELENK AND B. WOHLMUTH, On residual-based a posteriori error estimation in hp-FEM, Adv. Comp. Math., 15 (2001), pp. 311-331.
35. R. MUÑOZ-SOLA, Polynomial liftings on a tetrahedron and applications to the hp-version of the finite element method in three dimensions, SIAM J. Numer. Anal., 34 (1997), pp. 282-314.
36. N. NEUSS, Homogenisierung und Mehrgitter, Ph.D. thesis, University of Heidelberg, 1996.
37. C. SCHWAB, p- and hp-Finite Element Methods, Oxford University Press, Oxford, 1998.
38. L. R. SCOTT AND S. ZHANG, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput., 54 (1990), pp. 483-493.
39. E. M. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.
40. R. VERFÜRTH, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Teubner-Wiley, Leipzig, 1996.