Total overlapping Schwarz' preconditioners for elliptic problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 1, p. 91-113

A variant of the Total Overlapping Schwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1 Math. 336 (2003) 277-282] as an iterative algorithm to approximate the absorbing boundary condition, in unbounded domains. That same method turns to be an efficient tool to make numerical zooms in regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture and the reliability to obtain the behavior of the solution at infinity, when handling exterior problems. The main aim of the paper is to use this modified Schwarz procedure as a preconditioner to Krylov subspaces methods so to accelerate the calculations. A detailed study concludes to a super-linear convergence of GMRES and enables us to state accurate estimates on the convergence speed. Afterward, some implementation hints are discussed. Analytical and numerical examples are also provided and commented that demonstrate the reliability of the TOS-preconditioner.

DOI : https://doi.org/10.1051/m2an/2010032
Classification:  65F08,  65N12,  65N80
Keywords: total overlapping Schwarz method, minimum residual Krylov methods, numerical zooms
@article{M2AN_2011__45_1_91_0,
author = {Ben Belgacem, Faker and Gmati, Nabil and Jelassi, Faten},
title = {Total overlapping Schwarz' preconditioners for elliptic problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {45},
number = {1},
year = {2011},
pages = {91-113},
doi = {10.1051/m2an/2010032},
zbl = {1270.65073},
mrnumber = {2781132},
language = {en},
url = {http://www.numdam.org/item/M2AN_2011__45_1_91_0}
}

Ben Belgacem, Faker; Gmati, Nabil; Jelassi, Faten. Total overlapping Schwarz' preconditioners for elliptic problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 1, pp. 91-113. doi : 10.1051/m2an/2010032. http://www.numdam.org/item/M2AN_2011__45_1_91_0/

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR 450957 | Zbl 1098.46001

[2] J.B. Apoung-Kamga and O. Pironneau, Numerical zoom for multiscale problems with an application to nuclear waste disposal. J. Comput. Phys. 224 (2007) 403-413. | MR 2322278 | Zbl 1261.76014

[3] F. Ben Belgacem, M. Fournié, N. Gmati, F. Jelassi, Handling boundary conditions at infinity for some exterior problems by the alternating Schwarz method. C. R. Acad. Sci., Sér. 1 Math. 336 (2003) 277-282. | MR 1968273 | Zbl 1027.65161

[4] F. Ben Belgacem, M. Fournié, N. Gmati and F. Jelassi, On the Schwarz algorithms for the elliptic exterior boundary value problems. ESAIM: M2AN 39 (2005) 693-714. | Numdam | MR 2165675 | Zbl 1089.65126

[5] C. Bernardi, Y. Maday and A.T. Patera, A New Non Conforming Approach to Domain Decomposition: The Mortar Element Method, in Non-linear Partial Differential Equations and their Applications 11, H. Brezis and J.-L. Lions Eds., Pitman/Wiley, London/New York (1994) 13-51. | MR 1268898 | Zbl 0797.65094

[6] S. Bertoluzza, M. Ismaïl and B. Maury, The Fat Boundary Method: Semi-Discrete Scheme and Some Numerical experiments, in Domain decomposition methods in science and engineering, Lect. Notes Comput. Sci. Eng. 40, Springer, Berlin (2005) 513-520. | MR 2235779 | Zbl 1067.65121

[7] F. Brezzi, J.L. Lions and O. Pironneau, On the chimera method. C. R. Acad. Sci., Sér. 1 Math. 332 (2001) 655-660. | MR 1842464 | Zbl 0988.65117

[8] H.D. Bui, Fracture Mechanics: Inverse Problems and Solutions, Solid Mechanics and Its Applications 139. Springer (2006). | Zbl 1108.74002

[9] P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications 4. North Holland (1978). | MR 520174 | Zbl 0383.65058

[10] D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences 93. Springer (1992). | MR 1183732 | Zbl 0760.35053

[11] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Second edition, Masson, Paris (1988). | Zbl 0642.35001

[12] G. Dolzmann and S. Müller, Estimates for Green's matrices of elliptic systems by Lp theory. Manuscripta Math. 88 (1995) 261-273. | MR 1354111 | Zbl 0846.35040

[13] V. Frayssé, L. Giraud, G. Gratton and J. Langou, A Set of GMRES Routines for Real and Complex Arithmeticcs on High Performance Computers. CERFACS Technical Report TR/PA/03/3 (2003). | Zbl 1070.65527

[14] R. Glowinski, J. He, J. Rappaz and J. Wagner, Approximation of multi-scale elliptic problems using patches of finite elements. C. R. Acad. Sci., Sér. 1 Math. 337 (2003) 679-684. | MR 2030111 | Zbl 1036.65090

[15] R. Glowinski, J. He, J. Rappaz and J. Wagner, A multi-domain method for solving numerically multi-scale elliptic problems. C.R., Math. 338 (2004) 741-746. | MR 2065385 | Zbl 1049.65145

[16] N. Gmati and B. Philippe, Comments on the GMRES convergence for preconditioned systems, in 6th International Conference on Large-Scale Scientific Computations, June 5-9, 2007, I. Lirkov, S. Margenov and J. Waśniewski Eds., Lect. Notes Comput. Sci. 4818, Springer-Verlag (2008) 40-51. | MR 2518410 | Zbl 1229.65063

[17] P. Grisvard, Boundary value problems in non-smooth domains, Monographs and Studies in Mathematics 24. Pitman, London (1985). | Zbl 0695.35060

[18] M. Grüter and K.-O Widman, The Green function for uniformly elliptic equations. Manuscripta Math. 37 (1982) 303-342. | MR 657523 | Zbl 0485.35031

[19] J. He, A. Lozinski and J. Rappaz, Accelerating the method of finite element patches using approximately harmonic functions. C. R. Acad. Sci., Sér. 1 Math. 345 (2007) 107-112. | MR 2343562 | Zbl 1120.65126

[20] F. Hecht, EMC2, Éditeur de Maillage et de Contours en 2 Dimensions. http://www-rocq1.inria.fr/gamma/cdrom/www/emc2.

[21] F. Hecht, A. Lozinski and O. Pironneau, Numerical Zoom and the Schwarz Algorithm, in Domain Decomposition Methods in Science and Engineering XVIII, Lecture Notes in Computational Science and Engineering 70, M. Bercovier, M.J. Gander, R. Kornhuber and O. Widlund Eds., Springer (2008). | MR 2230692 | Zbl 1183.65162

[22] M. Ismaïl, The Fat Boundary Method for the Numerical Resolution of Elliptic Problems in Perforated Domains. Application to 3D Fluid Flows. Ph.D. thesis, Université UPMC, Paris VI, France (2004).

[23] F. Jelassi, Sur les méthodes de Schwarz pour les problèmes extérieurs. Application au calcul des courants de Foucault en électrotechnique. Ph.D. Thesis, Université Paul Sabatier, Toulouse III, France (2006).

[24] P.-L. Lions, On the alternating Schwarz method I., in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Gowinski, G.H. Golub, G.A. Meurant and J. Périaux Eds., SIAM, Philadelphia (1988) 1-42. | MR 972510 | Zbl 0658.65090

[25] J. Liu and J.M. Jin, A novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation problems. IEEE Trans. Antennas Propag. 49 (2001) 1794-1806. | Zbl 1001.78021

[26] B. Lucquin and O. Pironneau, Introduction to Scientific Computing. John Wiley & Sons Ltd., Inc., New York (1998). | MR 1627818 | Zbl 0899.65062

[27] D. Martin, MELINA, Guide de l'utilisateur. I.R.M.A.R., Université de Rennes I/E.N.S.T.A. Paris, France (2000). http://perso.univ-rennes1.fr/daniel.martin/melina.

[28] B. Maury, A fat boundary method for the Poisson equation in a domain with holes. J. Sci. Comp. 16 (2001) 319-339. | MR 1873286 | Zbl 0995.65115

[29] I. Moret, A note on the superlinear convergence of GMRES. SIAM J. Numer. Anal. 34 (1997) 513-516. | MR 1442925 | Zbl 0873.65054

[30] J.C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems. Springer (2000). | MR 1822275 | Zbl 0981.35002

[31] O. Pironneau, Numerical Zoom for Localized Multi-Scale Problems. Invited conference, MAFELAP, Brunel University, London (2009).

[32] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999). | MR 1857663 | Zbl 0931.65118

[33] A. Quarteroni, A Veneziani and P. Zunino, A domain decomposition method for advection-diffusion processes with application to blood solutes. SIAM J. Sci. Comput. 23 (2002) 1959-1980. | MR 1923721 | Zbl 1032.76036

[34] Y. Saad, Iterative methods for sparse linear systems. Second edition, SIAM (2003). | MR 1990645 | Zbl 1031.65046

[35] R. Schinzinger and P.A.A. Laura, Conformal Mapping: Methods and Applications. Amsterdam: Elsevier Science Publishers (1991). | MR 1159967 | Zbl 0863.30010

[36] A. Toselli and O.B. Widlund, Domain decomposition methods-algorithms and theory, Springer Series in Computational Mathematics 34. Springer-Verlag, Berlin (2005). | Zbl 1069.65138

[37] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71 (1912) 441-479. | JFM 43.0436.01 | MR 1511670