Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-13T02:51:38.397Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Schwarz algorithms for the Elliptic Exterior Boundary Value Problems

Published online by Cambridge University Press:  15 August 2005

Faker Ben Belgacem ,
Michel Fournié ,
Nabil Gmati  and
Faten Jelassi
Faker Ben Belgacem
Affiliation:
MIP (UMR CNRS 5640), UPS, 118 route de Narbonne, 31062 Toulouse, France. belgacem@mip.ups-tlse.fr; fournie@mip.ups-tlse.fr
Michel Fournié
Affiliation:
MIP (UMR CNRS 5640), UPS, 118 route de Narbonne, 31062 Toulouse, France. belgacem@mip.ups-tlse.fr; fournie@mip.ups-tlse.fr
Nabil Gmati
Affiliation:
LAMSIN, IPEIN, Campus Universitaire, Route Mrazka, 8000 Nabeul, Tunisia. nabil.gmati@ipein.rnu.tn
Faten Jelassi
Affiliation:
LAMSIN, FSB, Jarzouna, 7021 Bizerte, Tunisia. faten.jelassi@lamsin.rnu.tn 

Get access

Abstract

Tuning the alternating Schwarz method to the exterior problems is the subject of this paper. We present the original algorithm and we propose a modification of it, so that the solution of the subproblem involving the condition at infinity has an explicit integral representation formulas while the solution of the other subproblem, set in a bounded domain, is approximated by classical variational methods. We investigate many of the advantages of the new Schwarz approach: a geometrical convergence rate, an easy implementation, a substantial economy in computational costs and a satisfactory accuracy in the numerical results as well as their agreement with the theoretical statements.

Keywords

Boundary integral equationsboundary element methodsfinite element methodscoupling methodsdomain decomposition techniquesSchwarz algorithm.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 4 , July 2005 , pp. 693 - 714
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams, Sobolev Spaces. Academic Press (1975).
Albuquerque, C. and Cottet, G.-H., Coupling finite difference methods and integral formulas for elliptic problems arising in fluid mechanics. Numer. Methods Partial Differential Equations 20 (2003) 199–229. CrossRef
F. Ben Belgacem, M. Fournié, N. Gmati and F. Jelassi, Sur le traitement des conditions aux limites à l'infini pour quelques problèmes extérieurs par la méthode de Schwarz alternée. (French) [Handling boundary conditions at infinity for some exterior problems by the alternating Schwarz method] C. R. Math. Acad. Sci. Paris 336 (2003) 277–282.
P.E. Bjørstad, Multiplicative and additive Schwarz methods: Convergence in the two domain case, in Second International Domain Decomposition Methods for Partial Differential Equations, T.F. Chan, R. Glowinski, J. Périaux and O.B. Widlund Eds., SIAM, Philadelphia (1989) 147–159.
M. Bonnet, Equations intégrales et éléments de frontière. CNRS, Éditions Eyrolles, Paris (1995).
P. Chassaing, Mécanique des fluides. Eléments d'un premier parcours. Collection Polytech, Cépaduès-Editions, 2e édition (2000).
Christiansen, S. and Nédélec, J.C., A preconditioner for the Electric Field Integral Equation based on Calderon formula. SIAM J. Numer. Anal. 40 (2002) 1100–1135. CrossRef
D.L. Colton and R. Kress, Integral equation methods in scattering theory. Pure Appl. Math., Wiley-Interscience, John Wiley & Sons, New York (1983).
G.-H. Cottet, Particle grid domain decomposition methods for the Navier-Stokes equations in exterior domains. C. Anderson and C. Greengard Eds., Vortex dynamics and vortex methods, American Mathematical Society, Rhode Island (1991) 103–117.
R. Dautray and J.-L.Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Collaboration avec M. Artola, P. Bénilan, M. Bernadou, M. Cessenat, J.-C. Nédélec et J. Planchard. Réimprimé à partir de l'édition de 1984. INSTN: Collection Enseignement, Masson, Paris (1988).
M. Dryja and O.B. Widlund, Some domain decomposition algorithms for elliptic problems, in iterative Methods for Large Linear Systems. L. Hayes and D. Kincaid Eds., Academic Press, San Diego, CA (1989).
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, O.B. Widlund, Eds., SIAM, Philadelphia (1990) 273–291.
D. Euvrard, Résolution numérique des équations aux dérivées partielles. Masson, Paris (1990).
M. Feistauer and C. Schwab, Coupled Problems for Viscous Incompressible Flow in Exterior Domains. A. Sequeira et al. Eds., Kluwer/Plenum Publ. Appl. Nonlinear Anal. (1999) 97–116.
P. Germain and P. Muller, Introduction à la mécanique des milieux continus. 2e édition, Masson, Paris (1990).
J. Giroire, Études de quelques problèmes aux limites extérieurs et résolution par équations intégrales. Thèse de Doctorat d'État, Université Pierre et Marie Curie, Paris VI (1987).
Giroire, J. and Nédélec, J.-C., Numerical solution of an exterior Neumann problem using a double-layer potential. Math. Comp. 32 (1978) 973–990. CrossRef
Greengard, L. and Rokhlin, V., A new version of the fast multipole method for the Laplace equation in 3 dimensions. Cambridge University Press, Cambridge, UK. Acta Numerica 6 (1997) 226–269. CrossRef
J.D. Jackson, Classical electrodynamics. Wiley, New York, third edition (1999).
Jami, A., Résolution numérique des problèmes de Helmholtz extérieurs par couplage entre éléments finis et représentation intégrale. C. R. Acad. Sci. Paris Sér. A-B 287 (1978) A799–A801.
Jami, A. and Lenoir, M., Formulation variationnelle pour le couplage entre une méthode d'éléments finis et une représentation intégrale. C. R. Acad. Sci. Paris Sér. A-B 285 (1977) A269–A272.
Jami, A. and Lenoir, M., A new numerical method for solving exterior linear elliptic problems. Sixth International Conference on Numerical Methods in Fluid Dynamics (Proc. Conf., Tbilisi, 1978), Springer, Berlin-New York. Lect. Notes Phys. 90 (1979) 292–298. CrossRef
M.A. Jawson and G.T. Symm, Integral Equations Methods in Potential Theory and Elastostatics. Academic Press, New York (1977).
F. Jelassi, Ph.D. Thesis of the University Paul Sabatier Toulouse (France) and the École Nationale d'Ingénieurs de Tunis (Tunisia).
Johnson, C. and Nédélec, J.-C., On the coupling of boundary integral and finite element methods. Math. Comput. 35 (1980) 1063–1079. CrossRef
C. Hazard and M. Lenoir, Modélisation et résolution des problèmes de diffraction. Cours de L'ENSTA et de DEA de Mécanique, Paris VI, ENSTA SMP, Centre de l'Yvette, Palaiseau (1995).
M.-N. Le Roux, Résolution numérique du problème du potentiel dans le plan par une méthode variationnelle d'éléments finis. Thèse de Doctorat de 3e cycle, Université de Rennes (1974).
Le Roux, M.-N., Méthode d'éléments finis Résolution pour la résolution des problèmes extérieurs en dimension deux. RAIRO Anal. Numér. 11 (1977) 27–60. CrossRef
M. Lenoir, Equations intégrales et problèmes de diffraction. Cours de L'ENSTA, Paris VI, ENSTA SMP, Centre de l'Yvette, Palaiseau (2003).
P.-L. Lions, On the alternating Schwarz method I, in First 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) 1–42.
P.-L. Lions, On the alternating Schwarz method II, in Second International Domain Decomposition Methods for Partial Differential Equations. T.F. Chan, R. Glowinski, J. Périaux and O.B. Widlund, Eds., SIAM, Philadelphia, (1989) 47–70.
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 Propagation 49 (2001) 1794–1806.
Liu, J. and Jin, J.-M., Highly, A effective preconditioner for solving the finite element-boundary integral matrix equation of 3-D scattering. IEEE Trans. Antennas and Propagation 50 (2002) 1212–1221.
D. Martin, MELINA, Guide de l'utilisateur. IRMAR, Université de Rennes I et ENSTA Paris (2000). http://perso.univ-rennes1.fr/daniel.martin/melina/
Matsokin, A.M. and Nepomnyaschikh, S.V., Schwarz, A alternating method in a subspace. Soviet Math. 29 (1985) 78–84.
J.-C. Nédélec, Approximation des équations intégrales en mécanique et en physique. Cours de DEA, Centre de mathématiques appliquées-école polytechnique (1977).
J.-C. Nédélec, Acoustic and Electromagnetic equations. Integral Representations for Harmonic Problems. Springer-Verlag, New-York, Appl. Math. Sci. 144 (2001).
A. Quarteroni and R. Valli, Domain Decomposition Methods for Partial Differential Equations, Numerical mathematics and Scientific computation. Oxford Science Publications (1999).
H.A. Schwarz, Gesammelte Mathematische Abhandlungen, Volume 2. Springer, Berlin (1890). First published in Vierteljahrsschrift Naturforsch. Ges. Zurich (1870).
A. Sequeira, Couplage de la méthode des éléments finis et des équations intégrales – Application au problème de Stokes stationnaire dans le plan. Thèse de Doctorat de 3e cycle, Université Paris 6 (1981).
Sequeira, A., The Coupling of boundary integral and finite element methods for the bi-dimensional exterior steady Stokes problem. Math. Methods Appl. Sci. 5 (1983) 356–375. CrossRef
Silvester, P. and Hsieh, M.S., Finite element solution for two-dimensional exterior field problem. Proc. Inst. Electr. Eng. 118 (1971) 1743–1746. CrossRef
B. Smith, P. Bjørstad and W. Gropp, Domain Decomposition Method Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge university press, Cambridge (1996).
I. Stakgold, Green functions and boundary value problems. Pure Appl. Math., Wiley-Interscience, John Wiley & Sons, New York, Second edition (1998).
Steinbach, O. and Wendland, W.L., The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9 (1998) 191–216. CrossRef
W.L. Wendland, Boundary element methods for elliptic problems, in Mathematical Theory of Finite Element and Boundary Element Methods. A.H. Schatz, V. Thomée, W.L. Wendland Eds., Birkhäuser Verlag, Bazsel (1990).
Zienkiewicz, O., D.W. kelly and P. Bettess, The coupling of the finite element method and boundary solution procedures. Internat. J. Numer. Methods Engrg. 11 (1977) 355–375. CrossRef