A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 3, pp. 487-506.

A new Schwarz method for nonlinear systems is presented, constituting the multiplicative variant of a straightforward additive scheme. Local convergence can be guaranteed under suitable assumptions. The scheme is applied to nonlinear acoustic-structure interaction problems. Numerical examples validate the theoretical results. Further improvements are discussed by means of introducing overlapping subdomains and employing an inexact strategy for the local solvers.

DOI : 10.1051/m2an/2009010
Classification : 74F10, 65B99, 65M12
Mots-clés : Schwarz method, fluid-structure interaction, coupled problems, nonlinear elasticity, nonlinear acoustics, elasto-acoustic 
@article{M2AN_2009__43_3_487_0,
     author = {Ernst, Roland and Flemisch, Bernd and Wohlmuth, Barbara},
     title = {A multiplicative {Schwarz} method and its application to nonlinear acoustic-structure interaction},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {487--506},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {3},
     year = {2009},
     doi = {10.1051/m2an/2009010},
     mrnumber = {2536246},
     zbl = {1165.74017},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2009010/}
}
TY  - JOUR
AU  - Ernst, Roland
AU  - Flemisch, Bernd
AU  - Wohlmuth, Barbara
TI  - A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2009
SP  - 487
EP  - 506
VL  - 43
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2009010/
DO  - 10.1051/m2an/2009010
LA  - en
ID  - M2AN_2009__43_3_487_0
ER  - 
%0 Journal Article
%A Ernst, Roland
%A Flemisch, Bernd
%A Wohlmuth, Barbara
%T A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2009
%P 487-506
%V 43
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2009010/
%R 10.1051/m2an/2009010
%G en
%F M2AN_2009__43_3_487_0
Ernst, Roland; Flemisch, Bernd; Wohlmuth, Barbara. A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 3, pp. 487-506. doi : 10.1051/m2an/2009010. http://www.numdam.org/articles/10.1051/m2an/2009010/

[1] H.-B. An, On convergence of the additive Schwarz preconditioned inexact Newton method. SIAM J. Numer. Anal. 43 (2005) 1850-1871. | MR | Zbl

[2] A. Bermúdez, R. Rodríguez and D. Santamarina, Finite element approximation of a displacement formulation for time-domain elastoacoustic vibrations. J. Comput. Appl. Math. 152 (2003) 17-34. | MR | Zbl

[3] C. Bernardi, Y. Maday and A.T. Patera, Domain decomposition by the mortar element method, in Asymptotic and numerical methods for partial differential equations with critical parameters (Beaune, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 384, Kluwer Acad. Publ., Dordrecht (1993) 269-286. | MR | Zbl

[4] C. Bernardi, Y. Maday and A.T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, in Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. XI (Paris, 1989-1991), Pitman Res. Notes Math. Ser. 299, Longman Sci. Tech., Harlow (1994) 13-51. | MR | Zbl

[5] X.-C. Cai and D.E. Keyes, Nonlinearly preconditioned inexact Newton algorithms. SIAM J. Sci. Comput. 24 (2002) 183-200. | MR | Zbl

[6] P.G. Ciarlet, Mathematical elasticity, Vol. I: Three-dimensional elasticity, Studies in Mathematics and its Applications 20. North-Holland Publishing Co., Amsterdam (1988). | MR | Zbl

[7] M. Dryja and W. Hackbusch, On the nonlinear domain decomposition method. BIT 37 (1997) 296-311. | MR | Zbl

[8] B. Flemisch, M. Kaltenbacher and B.I. Wohlmuth, Elasto-acoustic and acoustic-acoustic coupling on non-matching grids. Int. J. Numer. Meth. Engng. 67 (2006) 1791-1810. | MR | Zbl

[9] M.F. Hammilton and D.T. Blackstock, Nonlinear Acoustics. Academic Press (1998).

[10] T. Hughes, The Finite Element Method. Prentice-Hall, New Jersey (1987). | MR | Zbl

[11] M. Kaltenbacher. Numerical Simulation of Mechatronic Sensors and Actuators. Springer, Berlin-Heidelberg-New York (2007). | Zbl

[12] D. Kuhl and M.A. Crisfield, Energy-conserving and decaying algorithms in non-linear structural dynamics. Int. J. Numer. Meth. Engng. 45 (1999) 569-599. | MR | Zbl

[13] V.I. Kuznetsov, Equations of nonlinear acoustics. Soviet Phys.-Acoust. 16 (1971) 467-470.

[14] N.M. Newmark, A method of computation for structural dynamics. J. Engng. Mech. Div., Proc. ASCE 85 (EM3) (1959) 67-94.

[15] A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (1999). | MR | Zbl

[16] A.-M. Sändig, Nichtlineare Funktionalanalysis mit Anwendungen auf partielle Differentialgleichungen. Vorlesung im Sommersemester 2006, IANS preprint 2006/012, Technical report, University of Stuttgart, Germany (2006).

[17] B.F. Smith, P.E. Bjørstad and W.D. Gropp, Domain decomposition, Parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge (1996). | MR | Zbl

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

Cité par Sources :