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Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumannboundary condition

Published online by Cambridge University Press:  05 December 2008

Eliane Bécache ,
Jeronimo Rodríguez  and
Chrysoula Tsogka
Eliane Bécache
Affiliation:
POEMS, INRIA-Rocquencourt, BP 105, 78153 Le Chesnay Cédex, France. [email protected]
Jeronimo Rodríguez
Affiliation:
POEMS, ENSTA, 32 boulevard Victor, 75739 Paris Cedex 15, France. [email protected]
Chrysoula Tsogka
Affiliation:
Dept. of Applied Mathematics, University of Crete & IACM/FORTH, Crete, Greece. [email protected]
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Abstract

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly taken into account when the first family of mixed finite elements is used. We, therefore, introduce the second family of mixed finite elements for which a theoretical convergence analysis is presented and error estimates are obtained. A numerical study of the convergence is also considered for a particular object geometry which shows that our theoretical error estimates are optimal.

Keywords

Mixed finite elementsfictitious domain methoddomain embedding methodacoustic wavesconvergence analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 2 , March 2009 , pp. 377 - 398
Copyright
© EDP Sciences, SMAI, 2009

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References

Babuška, I., The finite element method with lagrangian multipliers. Numer. Math. 20 (1973) 179–192. CrossRef
Bécache, E., Joly, P. and Tsogka, C., Éléments finis mixtes et condensation de masse en élastodynamique linéaire, (i) Construction. C. R. Acad. Sci. Paris Sér. I 325 (1997) 545–550. CrossRef
Bécache, E., Joly, P. and Tsogka, C., An analysis of new mixed finite elements for the approximation of wave propagation problems. SINUM 37 (2000) 1053–1084. CrossRef
Bécache, E., Joly, P. and Tsogka, C., Fictitious domains, mixed finite elements and perfectly matched layers for 2d elastic wave propagation. J. Comp. Acoust. 9 (2001) 1175–1203. CrossRef
Bécache, E., Joly, P. and Tsogka, C., A new family of mixed finite elements for the linear elastodynamic problem. SINUM 39 (2002) 2109–2132. CrossRef
Bécache, E., Chaigne, A., Derveaux, G. and Joly, P., Time-domain simulation of a guitar: Model and method. J. Acoust. Soc. Am. 6 (2003) 3368–3383.
E. Bécache, J. Rodríguez and C. Tsogka, On the convergence of the fictitious domain method for wave equation problems. Technical Report RR-5802, INRIA (2006).
Bécache, E., Rodríguez, J. and Tsogka, C., A fictitious domain method with mixed finite elements for elastodynamics. SIAM J. Sci. Comput. 29 (2007) 1244–1267. CrossRef
Bérenger, J.P., A perfectly matched layer for the absorption of electromagnetic waves. J. Comp. Phys. 114 (1994) 185–200. CrossRef
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
Collino, F. and Tsogka, C., Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media. Geophysics 66 (2001) 294–307. CrossRef
Collino, F., Joly, P. and Millot, F., Fictitious domain method for unsteady problems: Application to electromagnetic scattering. J. Comput. Phys. 138 (1997) 907–938. CrossRef
S. Garcès, Application des méthodes de domaines fictifs à la modélisation des structures rayonnantes tridimensionnelles. Ph.D. Thesis, SupAero, France (1998).
Girault, V. and Glowinski, R., Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan J. Indust. Appl. Math. 12 (1995) 487–514. CrossRef
V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations - Theory and algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag, Berlin (1986).
R. Glowinski, Numerical methods for fluids, Part 3, Chapter 8, in Handbook of Numerical Analysis IX, P.G. Ciarlet and J.L. Lions Eds., North-Holland, Amsterdam (2003) x+1176.
Glowinski, R. and Kuznetsov, Y., On the solution of the Dirichlet problem for linear elliptic operators by a distributed Lagrange multiplier method. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 693–698. CrossRef
Glowinski, R., Pan, T.W. and Periaux, J., A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Engrg. 111 (1994) 283–303. CrossRef
P. Grisvard, Singularities in boundary value problems. Springer-Verlag, Masson (1992).
Heikkola, E., Kuznetsov, Y.A., Neittaanmäki, P. and Toivanen, J., Fictitious domain methods for the numerical solution of two-dimensional scattering problems. J. Comput. Phys. 145 (1998) 89–109. CrossRef
Heikkola, E., Rossi, T. and Toivanen, J., A domain embedding method for scattering problems with an absorbing boundary or a perfectly matched layer. J. Comput. Acoust. 11 (2003) 159–174. CrossRef
E. Hille and R.S. Phillips, Functional analysis and semigroups, Colloquium Publications 31. Rev. edn., Providence, R.I., American Mathematical Society (1957).
Joly, P. and Rhaouti, L.. Analyse numérique - Domaines fictifs, éléments finis H(div) et condition de Neumann : le problème de la condition inf-sup. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1225–1230. CrossRef
Yu.A. Kuznetsov, Fictitious component and domain decomposition methods for the solution of eigenvalue problems, in Computing methods in applied sciences and engineering VII (Versailles, 1985), North-Holland, Amsterdam (1986) 155–172.
Nédélec, J.C., A new family of mixed finite elements in . Numer. Math. 50 (1986) 57–81. CrossRef
L. Rhaouti, Domaines fictifs pour la modélisation d'un probème d'interaction fluide-structure : simulation de la timbale. Ph.D. Thesis, Paris IX, France (1999).