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A convergence theory for an overlapping Schwarz algorithm using discontinuous iterates. (English) Zbl 1067.65112

The author proposes a new type of overlapping Schwarz methods, using discontinuous iterates. The proposed algorithm allows for discontinuous iterates across the artificial interfaces. A new theory using Lagrange multipliers is developed and conditions are found for the existence of an almost uniform convergence factor for the dual variables, which implies rapid convergence factor of the primal variables, in the two overlapping subdomanin case. Numerical examples are also presented.

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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[1] Benzi, M., Frommer, A., Nabben, R., Szyld, D.B.: Algebraic theory of multiplicative Schwarz methods. Numerische Mathematik. 89, 605-639 (2001) · Zbl 0991.65037
[2] Bjørstad, P.E., Widlund, O.B.: Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 23(6), 1093-1120 (1986) · Zbl 0615.65113
[3] Cai, X.-C., Casarin, M.A., Elliott, F.W., Widlund, O.B.: Overlapping Schwarz algorithms for solving Helmholtz?s equation. In: Jan Mandel, Charbel Farhat, and Xiao-Chuan Cai, (eds.), Tenth International Conference of Domain Decomposition Methods. volume 218 of Contemporary Mathematics. AMS, 391-399 (1998) · Zbl 0909.65104
[4] Casarin, M.A., Widlund, O.B.: Overlapping Schwarz algorithms for solving Helmholtz?s equation. In: Choi-Hong Lai, Petter E. Bjørstad, Mark Cross, and Olof B. Widlund, (eds.), Eleventh International Conference of Domain Decomposition Methods. DDM.org, 1999
[5] Després, B.: Méthodes de Décomposition de Domaine pour les Problémes de Propagation d?Ondes en Régime Harmonique. PhD thesis, Paris IX Dauphine, October 1991
[6] Dryja, M., Smith, B.F., Widlund, O.B.: Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. SIAM J. Numer. Anal. 31(6), 1662-1694, December 1994 · Zbl 0818.65114 · doi:10.1137/0731086
[7] Dryja, M., Widlund, O.B.: Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15(3), 604-620, May 1994 · Zbl 0802.65119 · doi:10.1137/0915040
[8] Eijkhout, V., Vassilevski, P.: The role of the strengthened Cauchy-Buniakowskii-Schwarz inequality in multilevel methods. SIAM Review. 33(3), 405-419 (1991) · Zbl 0737.65026 · doi:10.1137/1033098
[9] Farhat, C., Roux, F.-X.: An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems. SIAM J. Sci. Stat. Comput. 13(1), 379-396 (1992) · Zbl 0746.65086
[10] Farhat, C., Roux, F.-X.: Implicit parallel processing in structural mechanics. In: J. Tinsley Oden, (ed.), Computational Mechanics Advances. volume 2(1), North-Holland, 1-124 (1994) · Zbl 0805.73062
[11] Gander, M.J., Halpern, L., Nataf, F.: Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation. In: C-H. Lai, P. Bjørstad, M. Cross, and O. Widlund, (eds.), Eleventh international Conference of Domain Decomposition Methods. ddm.org, 1999
[12] Kimn, J.-H.: Overlapping Schwarz Algorithms using Discontinuous Iterates for Poisson?s Equation. PhD thesis, Courant Institute, New York University, May 2001. Tech. Rep. 817, Department of Computer Science, Courant Institute
[13] Lions, P.L.: On the Schwarz alternating method. III: a variant for nonoverlapping subdomains. In: Tony F. Chan, Roland Glowinski, Jacques Périaux, and Olof Widlund, (eds.), Third International Symposium on Domain Decomposition Methods for Partial Differential Equations , held in Houston, Texas, March 20-22, 1989. Philadelphia, PA, 1990. SIAM
[14] Mandel, J.: On block diagonal and Schur complement preconditioning. Numer. Math. 58, 79-93 (1990) · Zbl 0699.65025 · doi:10.1007/BF01385611
[15] Nataf, F.: Absorbing boundary conditions in block Gauss-Seidel methods for the convection problems. Math. Models Methods Appl Sci. 6(4), 481-502 (1996) · Zbl 0857.65032 · doi:10.1142/S0218202596000183
[16] Ne?as, J. Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967
[17] Quarteroni, A., Valli, A. Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, 1999 · Zbl 0931.65118
[18] Schwarz, H.A.: Über einen Grenzübergang durch alternierendes Verfahren. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich. 15, 272-286, May 1870
[19] Smith, B.F.: An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems. SIAM J. Sci. Stat. Comput. 13(1), 364-378, January 1992 · Zbl 0751.73059
[20] Smith, B.F., Bjørstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996 · Zbl 0857.65126
[21] Toselli, A., Widlund, O.: Domain Decomposition Methods: Algorithms and Theory. volume 34 of Springer Series in Computational Mathematics. Springer-Verlag, 2004 · Zbl 1069.65138
[22] Widlund, O.B.: An extension theorem for finite element spaces with three applications. In: Wolfgang Hackbusch and Kristian Witsch, (eds.), Numerical Techniques in Continuum Mechanics. pages 110-122, Braunschweig/Wiesbaden, 1987. Notes on Numerical Fluid Mechanics, v. 16, Friedr. Vieweg und Sohn. Proceedings of the Second GAMM-Seminar, Kiel, January, 1986
[23] Widlund, O.B.: Iterative substructuring methods: Algorithms and theory for elliptic problems in the plane. In: Roland Glowinski, Gene H. Golub, Gérard A. Meurant, and Jacques Périaux, (eds.), First International Symposium on Domain Decomposition Methods for Partial Differential Equations. Philadelphia, PA, 1988. SIAM · Zbl 0662.65097
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