# Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition

Eliane Bécache; Jeronimo Rodríguez; Chrysoula Tsogka

- Volume: 43, Issue: 2, page 377-398
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBécache, Eliane, Rodríguez, Jeronimo, and Tsogka, Chrysoula. "Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 43.2 (2009): 377-398. <http://eudml.org/doc/244953>.

@article{Bécache2009,

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.},

author = {Bécache, Eliane, Rodríguez, Jeronimo, Tsogka, Chrysoula},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {mixed finite elements; fictitious domain method; domain embedding method; acoustic waves; convergence analysis; convergence; numerical results; error estimates},

language = {eng},

number = {2},

pages = {377-398},

publisher = {EDP-Sciences},

title = {Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition},

url = {http://eudml.org/doc/244953},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Bécache, Eliane

AU - Rodríguez, Jeronimo

AU - Tsogka, Chrysoula

TI - Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2009

PB - EDP-Sciences

VL - 43

IS - 2

SP - 377

EP - 398

AB - 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.

LA - eng

KW - mixed finite elements; fictitious domain method; domain embedding method; acoustic waves; convergence analysis; convergence; numerical results; error estimates

UR - http://eudml.org/doc/244953

ER -

## References

top- [1] I. Babuška, The finite element method with lagrangian multipliers. Numer. Math. 20 (1973) 179–192. Zbl0258.65108MR359352
- [2] E. Bécache, P. Joly and C. Tsogka, Éléments finis mixtes et condensation de masse en élastodynamique linéaire, (i) Construction. C. R. Acad. Sci. Paris Sér. I 325 (1997) 545–550. Zbl0895.73064MR1692323
- [3] E. Bécache, P. Joly and C. Tsogka, An analysis of new mixed finite elements for the approximation of wave propagation problems. SINUM 37 (2000) 1053–1084. Zbl0958.65102MR1756415
- [4] E. Bécache, P. Joly and C. Tsogka, Fictitious domains, mixed finite elements and perfectly matched layers for 2d elastic wave propagation. J. Comp. Acoust. 9 (2001) 1175–1203. MR1871063
- [5] E. Bécache, P. Joly and C. Tsogka, A new family of mixed finite elements for the linear elastodynamic problem. SINUM 39 (2002) 2109–2132. Zbl1032.74049MR1897952
- [6] E. Bécache, A. Chaigne, G. Derveaux and P. Joly, Time-domain simulation of a guitar: Model and method. J. Acoust. Soc. Am. 6 (2003) 3368–3383. Zbl1047.76115
- [7] 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). Zbl1161.65071
- [8] E. Bécache, J. Rodríguez and C. Tsogka, A fictitious domain method with mixed finite elements for elastodynamics. SIAM J. Sci. Comput. 29 (2007) 1244–1267. Zbl05288533MR2318705
- [9] J.P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comp. Phys. 114 (1994) 185–200. Zbl0814.65129MR1294924
- [10] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). Zbl0788.73002MR1115205
- [11] F. Collino and C. Tsogka, Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media. Geophysics 66 (2001) 294–307.
- [12] F. Collino, P. Joly and F. Millot, Fictitious domain method for unsteady problems: Application to electromagnetic scattering. J. Comput. Phys. 138 (1997) 907–938. Zbl1126.78311MR1607498
- [13] S. Garcès, Application des méthodes de domaines fictifs à la modélisation des structures rayonnantes tridimensionnelles. Ph.D. Thesis, SupAero, France (1998).
- [14] V. Girault and R. Glowinski, Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan J. Indust. Appl. Math. 12 (1995) 487–514. Zbl0843.65076MR1356667
- [15] 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). Zbl0585.65077MR851383
- [16] 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. Zbl1020.00003MR2009814
- [17] R. Glowinski and Y. Kuznetsov, 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. Zbl1005.65127MR1652674
- [18] R. Glowinski, T.W. Pan and J. Periaux, A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Engrg. 111 (1994) 283–303. Zbl0845.73078MR1259864
- [19] P. Grisvard, Singularities in boundary value problems. Springer-Verlag, Masson (1992). Zbl0766.35001MR1173209
- [20] E. Heikkola, Y.A. Kuznetsov, P. Neittaanmäki and J. Toivanen, Fictitious domain methods for the numerical solution of two-dimensional scattering problems. J. Comput. Phys. 145 (1998) 89–109. Zbl0909.65119MR1640154
- [21] E. Heikkola, T. Rossi and J. Toivanen, A domain embedding method for scattering problems with an absorbing boundary or a perfectly matched layer. J. Comput. Acoust. 11 (2003) 159–174. Zbl1035.65127MR2013686
- [22] E. Hille and R.S. Phillips, Functional analysis and semigroups, Colloquium Publications 31. Rev. edn., Providence, R.I., American Mathematical Society (1957). Zbl0078.10004MR89373
- [23] P. Joly and L. Rhaouti. 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. Zbl0931.65112MR1701390
- [24] 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. Zbl0677.65033
- [25] J.C. Nédélec, A new family of mixed finite elements in ${\mathbb{R}}^{3}$. Numer. Math. 50 (1986) 57–81. Zbl0625.65107
- [26] 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).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.