# Total overlapping Schwarz' preconditioners for elliptic problems

Faker Ben Belgacem; Nabil Gmati; Faten Jelassi

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 45, Issue: 1, page 91-113
- ISSN: 0764-583X

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topBen Belgacem, Faker, Gmati, Nabil, and Jelassi, Faten. "Total overlapping Schwarz' preconditioners for elliptic problems." ESAIM: Mathematical Modelling and Numerical Analysis 45.1 (2011): 91-113. <http://eudml.org/doc/197464>.

@article{BenBelgacem2011,

abstract = {
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.
},

author = {Ben Belgacem, Faker, Gmati, Nabil, Jelassi, Faten},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Total Overlapping Schwarz method;
minimum residual Krylov methods; numerical zooms; cracked or perforated domains; Laplace boundary value problems; total overlapping Schwarz preconditioner; minimum residual Krylov algorithms; convergence; Laplace equation; generalized minimal residual (GMRES) method},

language = {eng},

month = {1},

number = {1},

pages = {91-113},

publisher = {EDP Sciences},

title = {Total overlapping Schwarz' preconditioners for elliptic problems},

url = {http://eudml.org/doc/197464},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Ben Belgacem, Faker

AU - Gmati, Nabil

AU - Jelassi, Faten

TI - Total overlapping Schwarz' preconditioners for elliptic problems

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2011/1//

PB - EDP Sciences

VL - 45

IS - 1

SP - 91

EP - 113

AB -
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.

LA - eng

KW - Total Overlapping Schwarz method;
minimum residual Krylov methods; numerical zooms; cracked or perforated domains; Laplace boundary value problems; total overlapping Schwarz preconditioner; minimum residual Krylov algorithms; convergence; Laplace equation; generalized minimal residual (GMRES) method

UR - http://eudml.org/doc/197464

ER -

## References

top- R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
- 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.
- 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.
- F. Ben Belgacem, M. Fournié, N. Gmati and F. Jelassi, On the Schwarz algorithms for the elliptic exterior boundary value problems. ESAIM: M2AN39 (2005) 693–714.
- 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 Applications11, 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 Applications139. Springer (2006).
- P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications4. North Holland (1978).
- D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences93. Springer (1992).
- R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Second edition, Masson, Paris (1988).
- G. Dolzmann and S. Müller, Estimates for Green's matrices of elliptic systems by Lp theory. Manuscripta Math.88 (1995) 261–273.
- 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 Mathematics24. Pitman, London (1985).
- M. Grüter and K.-O Widman, The Green function for uniformly elliptic equations. Manuscripta Math.37 (1982) 303–342.
- 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 EngineeringXVIII, Lecture Notes in Computational Science and Engineering70, 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.
- 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.
- 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). . URIhttp://perso.univ-rennes1.fr/daniel.martin/melina
- B. Maury, A fat boundary method for the Poisson equation in a domain with holes. J. Sci. Comp.16 (2001) 319–339.
- I. Moret, A note on the superlinear convergence of GMRES. SIAM J. Numer. Anal.34 (1997) 513–516.
- 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).
- 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.
- 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 Mathematics34. Springer-Verlag, Berlin (2005).
- H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann.71 (1912) 441–479.

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