Two-level non-overlapping schwarz preconditioners for a discontinuous Galerkin approximation of the biharmonic equation

Journal of Scientific Computing

Volumn 22-23, Issue , 2005, Pages 289-314

  Feng, Xiaobing ^a   Karakashian, Ohannes A ^a  

^a UNIVERSITY OF TENNESSEE   (United States)

Author keywords

Biharmonic equation; Discontinuous Galerkin methods; Domain decomposition; Schwarz methods

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL COMPLEXITY; HARMONIC ANALYSIS; INTERPOLATION; PROBLEM SOLVING; THEOREM PROVING;

BIHARMONIC EQUATION; DISCONTINUOUS GALERKIN METHOD; DOMAIN DECOMPOSITION; SCHWARZ METHODS;

GALERKIN METHODS;

EID: 17444401742     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-004-4141-9     Document Type: Article

References (22)

1. Arnold, D. N. (1982). An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742-760.
2. Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, L. D. (2002). Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749-1779.
3. Baker, G. A. Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31, 1977, pp.45-59.
4. Baker, G. A., Jureidini, W. N., and Karakashian, O. A. (1990). Piecewise solenoidal vector fields and the Stokes problems. SIAM J. Numer. Anal. 27, 1466-1485.
5. Bramble, J. H., Pasciak, J. E., Wang, J., and Xu, J. (1991). Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comp. 57, 1-21.
6. 1 functions. SIAM J. Numer. Anal. 41, 306-324.
7. Brenner, S. (1996). Two-level additive Schwarz preconditioners for nonconfroming plate elements. Numer. Math. 72, 419-447.
8. Ciarlet, P. G. (1978). The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam.
9. Douglas, Jr. J., and Dupont, T. (1976). Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods, Lecture Notes In Physics Vol. 58, Springer-Verlag, Berlin.
10. Dryja, M., and Widlund, O. B. (1990). Towards a unified theory of domain decomposition algorithms for elliptic problems. In T. Chan (ed.), Proceedings of Third International Symposium on Domain Decomposition Methods for Partial Differential Equations SIAM, Philadelphia, 3-21.
11. Feng, X., and Karakashian, O. A. (2001). Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 39, 1343-1365.
12. Feng, X., and Karakashian, O. A. Two-level overlapping Schwarz methods for a discontinuous Galerkin approximation of fourth order elliptic problems, preprint.
13. Feng, X., and Rahman, M. T. (2002). An additive average Schwarz method for the plate bending problem. J. Numer. Math. 10, 109-125.
14. Griebel, M., and Oswald, P. (1995). On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. 70, 163-180.
15. Grisvard, P. Singularities in Boundary Value Problems. Research Notes in Applied Mathematics, Vol. 22, MASSON and Springer-Verlag, Berlin.
16. Houston, P., Süli, E., and Schwab, C. (2002). Discontinuous hp-finite element methods for advection-diffusion problems. Tech. Report # 2000-07, Seminar für Angewandte Mathenatik, ETH, Zürich, SIAM J. Numer. Anal. 39, 2133-2163.
17. Karakashian, O. A., and Jureidini, W. N. (1998). A nonconforming unite element method for the stationary Navier-Stokes Equations. SIAM J. Numer. Anal. 35, 93-120.
18. Oden, J. T., and Baumann, C. E. (2000). A conservative DGM for convection-diffusion and Navier-Stokes problems. In Cockburn, B. karniadakis, G. E. Shu C.-W. (eds.),. Proceedings of the International Symposium on the Discontinuous Galerkin Method Springer lecture notes in Computational Science and Eng. Vol. 11, 179-196.
19. Quarteroni, A. and Valli, A. (1999). Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, New York.
20. Smith, B. E., Bjørstad, P. E., and Gropp, W. D. (1996). Domain Decomposition, Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, New York.
21. Wheeler, M. (1978). An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal 15, 152-161
22. Xu, J. (1992). Iterative methods by space decomposition and subspace correction. SIAM Rev. 34, 581-613.