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

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2009)

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

Abstract

top
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.

How to cite

top

Bécache, 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. [1] I. Babuška, The finite element method with lagrangian multipliers. Numer. Math. 20 (1973) 179–192. Zbl0258.65108MR359352
  2. [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. [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. [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. [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. [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. [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. [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. [9] J.P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comp. Phys. 114 (1994) 185–200. Zbl0814.65129MR1294924
  10. [10] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). Zbl0788.73002MR1115205
  11. [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. [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. [13] S. Garcès, Application des méthodes de domaines fictifs à la modélisation des structures rayonnantes tridimensionnelles. Ph.D. Thesis, SupAero, France (1998). 
  14. [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. [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. [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. [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. [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. [19] P. Grisvard, Singularities in boundary value problems. Springer-Verlag, Masson (1992). Zbl0766.35001MR1173209
  20. [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. [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. [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. [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. [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. [25] J.C. Nédélec, A new family of mixed finite elements in ℝ 3 . Numer. Math. 50 (1986) 57–81. Zbl0625.65107
  26. [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 ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.