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Total overlapping Schwarz'preconditioners for elliptic problems

Published online by Cambridge University Press:  15 April 2010

Faker Ben Belgacem ,
Nabil Gmati  and
Faten Jelassi
Faker Ben Belgacem
Affiliation:
LMAC, Université de Technologie de Compiègne, BP 20529, 60205 Compiègne cedex, France.
Nabil Gmati
Affiliation:
LAMSIN, École Nationale d'Ingénieurs de Tunis, B.P. 37, 1002 Le Belvédère, Tunisia. [email protected]
Faten Jelassi
Affiliation:
LMAC, Université de Technologie de Compiègne, EA 2222, BP 20529, 60205 Compiègne cedex, France. LAMSIN, Faculté des Sciences de Bizerte, Jarzouna, 7021 Bizerte, Tunisia.
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Abstract

A variant of the Total OverlappingSchwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1Math.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 toolto make numerical zoomsin regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture andthe reliability to obtain the behavior of the solution at infinity, when handling exterior problems. The main aim of the paper isto use this modified Schwarz procedure as a preconditioner to Krylovsubspaces methods so to accelerate the calculations. A detailed study concludes toa super-linear convergence ofGMRES and enables us to state accurate estimates on the convergence speed. Afterward, some implementation hints are discussed. Analytical and numericalexamples are also provided and commented that demonstrate the reliability of theTOS-preconditioner.

Keywords

Total Overlapping Schwarz methodminimum residual Krylov methodsnumerical zooms
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 1 , January 2011 , pp. 91 - 113
Copyright
© EDP Sciences, SMAI, 2010

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References

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
Apoung-Kamga, J.B. and Pironneau, O., Numerical zoom for multiscale problems with an application to nuclear waste disposal. J. Comput. Phys. 224 (2007) 403–413. CrossRef
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.
Ben Belgacem, F., Fournié, M., Gmati, N. and Jelassi, F., On the Schwarz algorithms for the elliptic exterior boundary value problems. ESAIM: M2AN 39 (2005) 693–714. CrossRef
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.
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.
F. Brezzi, J.L. Lions and O. Pironneau, On the chimera method. C. R. Acad. Sci., Sér. 1 Math. 332 (2001) 655–660.
H.D. Bui, Fracture Mechanics: Inverse Problems and Solutions, Solid Mechanics and Its Applications 139. Springer (2006).
P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications 4. North Holland (1978).
D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences 93. Springer (1992).
R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Second edition, Masson, Paris (1988).
Dolzmann, G. and Müller, S., Estimates for Green's matrices of elliptic systems by Lp theory. Manuscripta Math. 88 (1995) 261–273. CrossRef
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).
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.
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.
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.
P. Grisvard, Boundary value problems in non-smooth domains, Monographs and Studies in Mathematics 24. Pitman, London (1985).
Grüter, M. and Widman, K.-O, The Green function for uniformly elliptic equations. Manuscripta Math. 37 (1982) 303–342. CrossRef
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.
F. Hecht, EMC2, Éditeur de Maillage et de Contours en 2 Dimensions. http://www-rocq1.inria.fr/gamma/cdrom/www/emc2.
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).
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).
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).
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.
Liu, J. and Jin, J.M., 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.
B. Lucquin and O. Pironneau, Introduction to Scientific Computing. John Wiley & Sons Ltd., Inc., New York (1998).
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.
Maury, B., A fat boundary method for the Poisson equation in a domain with holes. J. Sci. Comp. 16 (2001) 319–339. CrossRef
Moret, I., A note on the superlinear convergence of GMRES. SIAM J. Numer. Anal. 34 (1997) 513–516. CrossRef
J.C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems. Springer (2000).
O. Pironneau, Numerical Zoom for Localized Multi-Scale Problems. Invited conference, MAFELAP, Brunel University, London (2009).
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999).
Quarteroni, A., Veneziani, A and Zunino, P., A domain decomposition method for advection-diffusion processes with application to blood solutes. SIAM J. Sci. Comput. 23 (2002) 1959–1980. CrossRef
Y. Saad, Iterative methods for sparse linear systems. Second edition, SIAM (2003).
R. Schinzinger and P.A.A. Laura, Conformal Mapping: Methods and Applications. Amsterdam: Elsevier Science Publishers (1991).
A. Toselli and O.B. Widlund, Domain decomposition methods–algorithms and theory, Springer Series in Computational Mathematics 34. Springer-Verlag, Berlin (2005).
Weyl, H., Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71 (1912) 441–479. CrossRef