# A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction

Roland Ernst; Bernd Flemisch; Barbara Wohlmuth

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

- Volume: 43, Issue: 3, page 487-506
- ISSN: 0764-583X

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topErnst, Roland, Flemisch, Bernd, and Wohlmuth, Barbara. "A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction." ESAIM: Mathematical Modelling and Numerical Analysis 43.3 (2009): 487-506. <http://eudml.org/doc/250602>.

@article{Ernst2009,

abstract = {
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.
},

author = {Ernst, Roland, Flemisch, Bernd, Wohlmuth, Barbara},

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

keywords = {Schwarz method; fluid-structure interaction; coupled problems; nonlinear elasticity; nonlinear acoustics; elasto-acoustic.; straightforward additive scheme; convergence; overlapping subdomains},

language = {eng},

month = {4},

number = {3},

pages = {487-506},

publisher = {EDP Sciences},

title = {A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction},

url = {http://eudml.org/doc/250602},

volume = {43},

year = {2009},

}

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: Mathematical Modelling and Numerical Analysis

DA - 2009/4//

PB - EDP Sciences

VL - 43

IS - 3

SP - 487

EP - 506

AB -
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.

LA - eng

KW - Schwarz method; fluid-structure interaction; coupled problems; nonlinear elasticity; nonlinear acoustics; elasto-acoustic.; straightforward additive scheme; convergence; overlapping subdomains

UR - http://eudml.org/doc/250602

ER -

## References

top- H.-B. An, On convergence of the additive Schwarz preconditioned inexact Newton method. SIAM J. Numer. Anal.43 (2005) 1850–1871. Zbl1111.65046
- 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. Zbl1107.76345
- 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.
- 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. Zbl0797.65094
- X.-C. Cai and D.E. Keyes, Nonlinearly preconditioned inexact Newton algorithms. SIAM J. Sci. Comput.24 (2002) 183–200. Zbl1015.65058
- P.G. Ciarlet, Mathematical elasticity, Vol. I: Three-dimensional elasticity, Studies in Mathematics and its Applications20. North-Holland Publishing Co., Amsterdam (1988). Zbl0648.73014
- M. Dryja and W. Hackbusch, On the nonlinear domain decomposition method. BIT37 (1997) 296–311. Zbl0891.65126
- 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. Zbl1127.74042
- M.F. Hammilton and D.T. Blackstock, Nonlinear Acoustics. Academic Press (1998).
- T. Hughes, The Finite Element Method. Prentice-Hall, New Jersey (1987). Zbl0634.73056
- M. Kaltenbacher. Numerical Simulation of Mechatronic Sensors and Actuators. Springer, Berlin-Heidelberg-New York (2007). Zbl1072.78001
- 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. Zbl0946.74078
- V.I. Kuznetsov, Equations of nonlinear acoustics. Soviet Phys.-Acoust.16 (1971) 467–470.
- N.M. Newmark, A method of computation for structural dynamics. J. Engng. Mech. Div., Proc. ASCE85 (EM3) (1959) 67–94.
- A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (1999). Zbl0931.65118
- 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).
- 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). Zbl0857.65126
- A. Toselli and O. Widlund, Domain decomposition methods – algorithms and theory, Springer Series in Computational Mathematics34. Springer-Verlag, Berlin (2005). Zbl1069.65138

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