Improving the rate of convergence of 'high order finite elements' on polygons and domains with cusps

Numerische Mathematik

Volumn 100, Issue 2, 2005, Pages 165-184

  Bǎcuţǎ, Constantin ^a   Nistor, Victor ^b   Zikatanov, Ludmil T ^b  

^a UNIVERSITY OF DELAWARE   (United States)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17444419688     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-005-0588-3     Document Type: Article

References (49)

1. Ammann, B., Lauter, R., Nistor, V.: Pseudodifferential operators on manifolds with a Lie structure at infinity. Math.DG/0304044
2. Arnold, D.N., Boffi, D., Falk, R.S., Gastaldi, L.: Finite element approximation on quadrilateral meshes. Communications in Numerical Methods in Engineering 17, 805-812 (2001)
3. Atiyah, M., Bott, R.: The index problem for anifolds with boundary. Differential Analysis (papers presented at the Bombay Colloquium 1964). Oxford University Press, 175-186
4. Babuska, I., Aziz, A. K.:Survey lectures on the mathematical foundations of the finite element method. In: A.K. Aziz (Ed.), The mathematical foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, London, 1972, pp 3-359
5. Babuška, I., Aziz, A. K.:On the angle condition in the finite element method. SIAM J. Numer. Anal. 13, 214-226 (1976)
6. Babuška, I., Elman, H.C.: Performance of the h-p version of the finite element method with various elements. Pnt. J. Num. Methods Engrg. 36, 2503-1459 (1993)
7. Babuška, I., Elman, H.C., Markle, K.: Parallel implementation of the h-p version of the finite element method on shared memory architecture. SIAM J. Numer. Anal. 13, 1433-1459 (1993)
8. Babuška, I. : Finite Element Method for Domains with Corners. Computing 6, 264-273 (1970)
9. Babuška, I., Kellog, R.B., Pitkäranta, J.: Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33, 441-411 (1979)
10. Babuška, I., Suri, M.: The P and H-P Versions of the finite element method, Basic principles and properties. SIAM Review 36, 578-632 (1994)
11. Bǎcuţǎ, C., Bramble, J. H., Pasciak, J.: New interpolation results and applications to finite element methods for elliptic boundary value problems. Numerical Linear Algebra with Applications, Volume 10 (1-2), 33-64 (2003)
12. Bǎcuţǎ, C., Bramble, J. H., Xu, J.: Regularity estimates for elliptic boundary value problems in Besov spaces. Mathematics of Computation, 72, 1577-1595 (2003)
13. Bǎcuţǎ, C., Nistor, V., Zikatanov, L.: Regularity and well posedness for the Laplace operator on polyhedral domains. Math.AP/0410207
14. Blum, H.: Treatment of corner singularities. In: E. Stein, W.L. Wendland, (eds.), Finie Element and Boundary Element Techniques from the Mathematical and Engineering Point of View, CISM Courses and Lectures, vol. 301, Springer, Wien, 1988, pp. 172-212
15. Ciarlet, P.: Basic error estimates for elliptic problems, chapter in Handbook of Numerical Analysis (P. G. Ciarlet and J. L. Lions, eds.), Part 1, North Holland, (1991), 17-352. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996
16. Ciarlet, P.: Basic error estimates for elliptic problems, chapter in Handbook of Numerical Analysis (P. G. Ciarlet and J. L. Lions, eds.), Part 1, North Holland, (1991), 17-352. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996
17. Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics 1341, Springer-Verlag, Berlin, 1988
18. Dauge, M., Nicaise, S., Bourland, M., Lubuma, M.S.: Coefficients des singularitiés pour des problèmes aux limities elliptiques sur un domaine à points coniques. I: Résultats généraux pour le problème de Dirichlet. RAIRO Model. Math. Anal. Numér. 24, 27-52 (1990)
19. Dauge, M., Nicaise, S., Bourland, M., Lubuma, M.S.: Coefficients of the singularities for elliptic boundary value problems on domains with conical points III: Finite Element Methods on Polygonal Domains. SIAM J. Numer. Anal. 29, 136-155 (1992)
20. DeVore, R., Lorentz, G.G.: Constructive Approximation. Grundlehren, Springer Verlag, New York, 1993
21. DeVore, R., Dahlke, S.: Besov Regularity for elliptic boundary value problems. Comm. PDE 22, 1-16 (1997)
22. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985
23. Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris, 1992
24. Grubb, G.: Functional calculus of pseudodifferential boundary value problems. Second edition. Progress in Mathematics 65, Birkhäuser, Boston, 1996
25. Kellogg, R. B.: Interpolation between subspaces of a Hubert space. Technical note BN-719. Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, 1971
26. Kondratiev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Transl. Moscow Math. Soc. 16, 227-313 (1967)
27. Kozlov, V., Mazya, V., Rossmann, J.: Elliptic boundary value problems in domains with point singularities. Amer. Math. Soc., Providence, RI: 1997
28. Kozlov, V., Mazya, V., Rossmann, J.: Spectral problems associated with corner singularities of solutions to elliptic equations. Amer. Math. Soc., Providence, RI: 2001
29. Laubin, P.: High order convergence for collocation of second kind boundary integral equations on polygons. Numer. Math. 79, 107-140 (1998)
30. Lewis, J., Parenti, C.: Pseudodifferential operators of Melin type. Comm. Part. Diff. Eq. 8, 477-544 (1983)
31. Lubuma, J., Nicaise, S.: Dirichlet problems in polyhedral domains H: Approximation by FEM and BEM. J. Comp. and Applied Math. 61, 13-27 (1995)
32. Lubuma, J., Nicaise, S.: Finite element method for elliptic problems with edge singularities. J. Comp. and Applied Math. 106, 145-168 (1999)
33. p estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains. ZAMM Z. Angew. Math. Mech. 83, 435-467 (2003)
34. Melrose, R.B.: The Atiyah-Patodi-Singer index theorem. Research Notes in Mathematics, 4, A.K. Peters, Ltd., MA 1993, xiv+377
35. Melrose, R.B.: Geometric scattering theory. Stanford Lectures, Cambridge University Press, Cambridge, 1995
36. Mitrea, M., Nistor, V: Boundary layer potentials on manifolds with cylindrical ends. ESI preprint 2002
37. Nečas, J.: Les Methodes Directes en Theorie des Equations Elliptiques. Academia, Prague, 1967
38. Nazarov, S.A., Plamenevsky, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Expositions in Mathematics 13, de Gruyter, New York, 1994
39. Raugel, G.: Résolution numérique par une méthode d'éléments finis du problème de Dirichlet pour le laplacien dans un polygone. C.R. Acad. Sci. Paris 286, 791-794 (1978)
40. Raugel, G. : Résolution numérique de problèmes elliptiques dans domaines avec coins. Thesis, University of Rennes, 1978
41. Schatz, A. H., Wahlbin, L. B.: On the Quasi-Optimality in L of the H -Projection into Finite Element Spaces. Math. Comp. 38, 1-22 (1982)
42. Schatz, A. H., Sloan, I., Wahlbin, L. B.: Superconvergence in finite element methods and meshes which are locally symmetric with respect to a point. SLAM Journal of Numerical Analysis, to appear
43. Schrohe, E.: Fréchet algebra technique for boundary value problems on noncompact manifolds: Fredholm criteria and functional calculus via spectral invariance. Math. Nach. 1999, 145-185
44. Taylor, M. E.: Partial differential equations, Applied Mathematical Sciences, vol. I, Springer-Verlag, New York, (1995-1997)
45. Taylor, M. E.: Partial differential equations, Applied Mathematical Sciences, vol. II, Springer-Verlag, New York, (1995-1997)
46. Verfurth, V: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996 (150 pp)
47. Wahlbin, L. B.: Local Behavior in Finite Element Methods, chapter in Handbook of Numerical Analysis (P. G. Ciarlet and J. L. Lions, eds.), Part 1, North Holland, (1991), 353-522. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996 (150 pp)
48. Wahlbin, L. B.: Local Behavior in Finite Element Methods, chapter in Handbook of Numerical Analysis (P. G. Ciarlet and J. L. Lions, eds.), Part 1, North Holland, (1991), 353-522. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996 (150 pp)
49. Wahlbin, L. B.: Superconvergence in Galerkin Finite Element Methods. Springer Lecture Notes in Mathematics 1605, Springer-Verlag New York, 1995