Overlapping additive Schwarz preconditioners for boundary element methods

Journal of Integral Equations and Applications

Volumn 12, Issue 2, 2000, Pages 177-208

  Tran, Thanh ^a  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

Additive Schwarz; Galerkin boundary element method; H version; Overlapping; P version; Preconditioned conjugate gradient

Indexed keywords

EID: 0004303257     PISSN: 08973962     EISSN: None     Source Type: Journal    
DOI: 10.1216/jiea/1020282169     Document Type: Article

References (41)

1. M. Ainsworth, W. McLean and T. Tran, The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling, SIAM J. Numer. Anal. 36 (1999), 1901-1932.
2. 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), 624-661.
3. J.H. Bramble, Z. Leyk and J.E. Pasciak, The analysis of multigrid algorithms for pseudodifferential operators of order minus one, Math. Comp. 63 (1994), 461-478.
4. J.H. Bramble, J.E. Pasciak and A.H. Schatz, The construction of preconditioners for elliptic problems by substructuring, I, Math. Comp. 47 (1986), 103-134.
5. X.-C. Cai and O.B. Widlund, Domain decomposition algorithms for indefinite elliptic problems, SIAM J. Sci. Stat. Comput. 13 (1992), 243-258.
6. T.F. Chan and T.P. Mathew, Domain decomposition algorithms, Acta Numerica (1994), 61-143.
7. M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 19 (1988), 613-626.
8. W. Dahmen, S. Prössdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution, Adv. Comput. Math. 1 (1993), 259-335.
9. W. Dahmen, S. Prössdorf and R. Schneider, Multiscale methods for pseudodifferential equations, in Wavelet analysis and its applications, Recent advances in wavelet analysis (L.L. Schumaker and G. Webb, eds.), Academic Press, San Diego, 1994.
10. W. Dahmen, S. Prössdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations I: Stability and convergence, Math. Z. 215 (1994), 583-620.
11. M. Dryja and O.B. Widlund, Towards a unified theory of domain decomposition algorithms for elliptic problems, in Third international symposium on domain decomposition methods for partial differential equations (T.F. Chan, R. Glowinski, J. Périaux and O.B. Widlund, eds.), SIAM, Philadelphia, 1990.
12. M. Dryja and O.B. Widlund, Additive Schwarz methods for elliptic finite element problems in three dimensions, in Fifth international symposium on domain decomposition methods for partial differential equations (D.E. Keyes, T.F. Chan, G.A. Meurant, J.S. Scroggs and R.G. Voigt, eds.), SIAM, Philadelphia, 1992.
13. M. Dryja and O.B. Widlund, Domain decomposition algorithms with small overlap, SIAM J. Sci. Comput. 15 (1994), 604-620.
14. P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, Boston, 1985.
15. 3 the p version, in Boundary elements: Implementation and analysis of advanced algorithms (W. Hackbusch and G. Wittum, eds.), Vieweg-Verlag, Braunschweig, 1996.
16. G.C. Hsiao and W.L. Wendland, Domain decomposition in boundary element methods, in Proc. of the 4th international symposium on domain decomposition methods for partial differential equations (R. Glowinski, Y.A. Kuznetsov, G.A. Meurant, J. Périaux and O.B. Widlund, eds.), SIAM, Philadelphia, 1991.
17. B.N. Khoromskij and S. Prössdorf, Multilevel preconditioning on the refined interface and optimal boundary solvers for the Laplace equation, Adv. Comput. Math. 4 (1995), 331-355.
18. B.N. Khoromskij and W.L. Wendland, Spectrally equivalent preconditioners for boundary equations in substructuring techniques, East-West J. Numer. Math. 1 (1992), 1-26.
19. J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications I, Springer-Verlag, New York, 1972.
20. P.L. Lions, On the Schwarz alternating method, I, in Proc. of the 1st international symposium on domain decomposition methods for partial differential equations (R. Glowinski, G.H. Golub, G.A. Meurant and J. Périaux, eds.), SIAM, Philadelphia, 1988.
21. P.L. Lions, On the Schwarz alternating method, II, in Proc. of the 2nd international symposium on domain decomposition methods for partial differential equations (T.F. Chan, R. Glowinski, J. Périaux and O.B. Widlund, eds.), SIAM, Philadelphia, 1989.
22. M. Maischak, E.P. Stephan and T. Tran, Multiplicative Schwarz algorithms for the Galerkin boundary element method, Institute für Angewandte Mathematik, Universität Hannover, 1997.
23. W. McLean and T. Tran, A preconditioning strategy for boundary element Galerkin methods, Numer. Methods Partial Differential Equations 13 (1997), 283-301.
24. S.V. Nepomnyaschikh, Domain decomposition and Schwarz methods in a subspace for approximate solution of elliptic boundary value problems, Ph.D.Thesis, Computing Center of the Siberian Branch of the USSR Academy of Sciences, Novosibirsk, USSR, 1986.
25. L.F. Pavarino, Additive Schwarz methods for the p version finite element method, Numer. Math. 66 (1994), 493-515.
26. B. Smith, P. Bjørstad and W. Gropp, Domain decomposition Parallel multilevel methods for elliptic partial differential equations, Cambridge University Press, Cambridge, 1996.
27. B.F. Smith, An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems, SIAMJ. Sci. Stat.Comput. 13 (1992), 364-378.
28. O. Steinbach and W.L. Wendland, The construction of some efficient preconditioners in the boundary element method, Adv. Comput. Math. 9 (1998), 191-216.
29. O. Steinbach and W.L. Wendland, Hierarchical boundary element preconditioners in domain decomposition method, in Proceedings of the 9th internat. conf. on domain decomposition methods (P. Bjørstad, M. Espedal and D. Keyes, eds.), http://www.ddm.org/DD9/Steinbach.pdf.
30. T. Tran, Additive Schwarz algorithms and the Galerkin boundary element method, in Computational techniques and applications: CTAC97 (J. Noye, M. Teubner and A. Gill, eds.), World Scientific, Singapore, 1998.
31. T. Tran and E.P. Stephan, Additive Schwarz methods for the h version boundary element method, Appl. Anal. 60 (1996), 63-84.
32. T. Tran and E.P. Stephan, Additive Schwarz algorithms for the p version of the Galerkin boundary element method, Numer. Math. 85 (2000), 433-468.
33. T. Tran, E.P. Stephan, and P. Mund, Hierarchical basis preconditioners for first kind integral equations, Appl. Anal. 65 (1997), 353-372.
34. T. Tran, E.P. Stephan and S. Zaprianov, Wavelet-based preconditioners for boundary integral equations, Adv. Comput. Math. 9 (1998), 233-249.
35. T. von Petersdorff, Randwertprobleme der Elastizitätstheorie für Polyeder Singularitäten und Approximation mit Randelementmethoden, Ph.D. Thesis, Technische Hochschule Darmstadt, Darmstadt, 1989.
36. T. von Petersdorff and C. Schwab, Wavelet approximations for first kind boundary integral equations on polygons, Numer. Math. 74 (1996), 479-516.
37. T. von Petersdorff and E.P. Stephan, On the convergence of the multigrid method for a hypersingular integral equation of the first kind, Numer. Math. 57 (1990), 379-391.
38. T. von Petersdorff and E.P. Stephan, Multigrid solvers and preconditioners for first kind integral equations, Numer. Methods Partial Differential Equations 8 (1992), 443-450.
39. W.L. Wendland and E.P. Stephan, A hypersingular boundary integral method for two-dimensional screen and crack problems, Arch. Rational Mech. Anal. 112 (1990), 363-390.
40. O.B. Widlund, Optimal iterative refinement methods, in Proc. of the 2nd international symposium on domain decomposition methods for partial differential equations (T.F. Chan, R. Glowinski, J. Périaux and O.B. Widlund, eds.), SIAM, Philadelphia, 1989.
41. X. Zhang, Multilevel Schwarz methods, Numer. Math. 63 (1992), 521-539.