Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements

IMA Journal of Numerical Analysis

Volumn 28, Issue 1, 2008, Pages 1-24

  Schöberl, Joachim ^a   Melenk, Jens M ^b   Pechstein, Clemens ^c   Zaglmayr, Sabine ^d  

^a RWTH AACHEN UNIVERSITY   (Germany)

^b VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^c JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

^d JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

Author keywords

High order finite element method; Preconditioning

Indexed keywords

NUMBER THEORY;

ADDITIVE SCHWARZ PRECONDITIONING; ELASTICITY PROBLEMS; FINITE ELEMENT SPACE; HIGH-ORDER FINITE ELEMENT METHOD; NUMERICAL EXPERIMENTS; P-VERSION FINITE ELEMENT METHOD; PRECONDITIONING; TETRAHEDRAL FINITE ELEMENTS;

FINITE ELEMENT METHOD;

EID: 37849008769     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/drl046     Document Type: Article

References (36)

1. AINSWORTH, M. (1996a) A preconditioner based on domain decomposition for h-p finite element approximation on quasi-uniform meshes. SIAM J. Numer. Anal., 33, 1358-1376.
2. AINSWORTH, M. (1996b) Hierarchical domain decomposition preconditioner for h-p finite element approximation on locally refined meshes. SIAM J. Sci. Comput, 17, 1395-1413.
3. BABUŠKA, I., CRAIG, A. W., MANDEL, J. & PITKÄRANTA, J. (1991) Efficient preconditioning for the p version of the finite element method in two dimensions. SIAM J. Numer. Anal., 28, 624-661.
4. BABUŠKA, I. & SURI, M. (1994) The p and hp versions of the finite element method, basic principles and properties. SIAM Rev., 36, 578-632.
5. BEĆIROVÍ, A., PAULE, P., PILLWEIN, V., RIESE, A., SCHNEIDERS, C. & SCHÖBERL, J. (2006) Hypergeometric summation methods for high order finite elements. Computing, 78, 235-249. Available at http://www.risc.uni-linz.ac.at/publications/download/risc_2949/FEM.pdf. Preprint.
6. BERNARDI, C. & MADAY, Y. (1992) Approximations spectrales de problèmes aux limites elliptiques. France, Paris: Springer.
7. BERNARDI, C. & MADAY, Y. (1997) Spectral methods. Handbook of Numerical Analysis (P. G. Ciarlet & J. L. Lions eds), vol. V. Amsterdam, The Netherlands: North-Holland, pp. 209-485.
8. BEUCHLER, S., SCHNEIDER, R. & SCHWAB, C. (2004) Multiresolution weighted norm equivalences and applications. Numer. Math., 98, 67-97.
9. BEUCHLER, S. & SCHÖBERL, J. (2005) Extension operators on tensor product structures. Appl. Numer. Math., 54, 391-405.
10. BICA, I. (1997) Iterative substructuring algorithms for the p-version finite element method for elliptic problems. Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University. New York, NY, USA.
11. CANUTO, C., HUSSAINI, M. Y., QUARTERONI, A., & ZHANG, T. A. (1988) Spectral Methods in Fluid Dynamics. New York: Springer.
12. CASARIN, M. A. (1997) Quasi-optimal Schwarz methods for the conforming spectral element discretization. SIAM J. Numer. Anal., 34, 2482-2502.
13. CLÉMENT, P. (1975) Approximation by finite element functions using local regularization. RAIRO Anal. Numer., 9, 77-84.
14. DEVILLE, M. O., FISCHER, P. F., & MUND, E. H. (2002) High Order Methods for Incompressible Fluid Flow. Cambridge: Cambridge University Press.
15. DRYJA, M. & WIDLUND, O. B. (1990) Towards a unified theory of domain decomposition algorithms for elliptic problems. Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) (T. F. Chan, R. Glowinski, J. Périaux & O. B. Widlund eds). Philadelphia, PA: SIAM, pp. 3-21.
16. EIBNER, T. & MELENK, J. M. (2006) A local error analysis of the boundary concentrated FEM. IMA J. Numer. Anal., 26, 752-778.
17. GOTTLIEB, D. & ORSZAG, S. A. (1977) Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia, PA: SIAM-CMBS.
18. GRIEBEL, M. & OSWALD, P. (1995) On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math., 70, 163-180.
19. GUO, B. & CAO, W. (1998) An additive Schwarz method for the h-p version of the finite element method in three dimensions. SIAM J. Numer. Anal., 35, 632-654.
20. KARNIADAKIS, G. & SHERWIN, S. (1999) Spectral/hp Element Methods for CFD. Oxford: Oxford University Press.
21. KORNEEV, V. & JENSEN, S. (1999) Domain decomposition preconditioning in the hierarchical p-version of the finite element method. Appl. Numer. Math., 29, 479-518.
22. MANDEL, J. (1990) Iterative solvers by substructuring for the p-version finite element method. Comput. Methods Appl. Mech Eng., 80, 117-128.
23. MELENK, J. M. (2002) On condition numbers in hp-FEM with Gauss-Lobatto based shape functions. J. Comput. Appl. Math., 139, 21-48.
24. MUÑOZ-SOLA, R. (1997) Polynomial liftings on a tetrahedron and applications to the hp-version of the finite element method in three dimensions. SIAM J. Numer. Anal., 34, 282-314.
25. ORSZAG, S. A. (1980) Spectral methods for problems in complex geometries. J. Comput. Phys., 37, 70-92.
26. PAVARINO, L. & WARBURTON, T. (2000) Overlapping Schwarz methods for unstructured spectral elements. J. Comput. Phys., 160, 298-317.
27. PAVARINO, L. F. (1994) Additive Schwarz methods for the p-version finite element method. Numer. Math., 66, 493-515.
28. QUARTERONI, A. & VALLI, A. (1999) Domain Decomposition Methods for Partial Differential Equations. Oxford: Oxford University Press.
29. SCHÖBERL, J., MELENK, J. M., PECHSTEIN, C. G. A. & ZAGLMAYR, S. C. (2007) Schwarz preconditioning for high order simplicial finite elements. Domain Decomposition Methods in Science and Engineering XVI (D. Keyes & O. Widlund eds). Lecture Notes in Computational Science and Engineering, vol. 55. New York: Springer. Available at http://cims.nyu.edu/dd16/proceedings.html.
30. SCHWAB, C. (1998) p- and hp-Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics. Oxford: Oxford Science Publications.
31. SHERWIN, S. J. & CASARIN, M. (2001) Low energy basis preconditioning for elliptic substructured solvers based on unstructured spectral/hp element discretisations. J. Comput. Phys., 171, 394-417.
32. SMITH, B. F., BJØRSTAD, P. E. & GROPP, W. D. (1996) Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge: Cambridge University Press.
33. ŠOLIN, P., SEGETH, K. & DOLEŽ EL, I. (2003) Higher-Order Finite Element Methods. Boca Raton, FL: Chapman & Hall.
34. SZABÓ, B. & BABUŠKA, I. (1991) Finite Element Analysis. New York: John Wiley.
35. SZABÓ, B., DÜSTER, A. & RANK, E. (2004) The p-version of the finite element method. Encyclopedia of Computational Mechanics (E. Stein, R. de Borst & T. J. R. Hughes eds). Chichester: John Wiley.
36. TOSELLI, A. & WIDLUND, O. (2005) Domain Decomposition Methods - Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Berlin, Germany: Springer.