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.
Mots clés : mixed finite elements, fictitious domain method, domain embedding method, acoustic waves, convergence analysis
@article{M2AN_2009__43_2_377_0, author = {B\'ecache, Eliane and Rodr{\'\i}guez, Jeronimo and Tsogka, Chrysoula}, title = {Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a {Neumann} boundary condition}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {377--398}, publisher = {EDP-Sciences}, volume = {43}, number = {2}, year = {2009}, doi = {10.1051/m2an:2008047}, mrnumber = {2512501}, zbl = {1161.65071}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2008047/} }
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: Modélisation mathématique et analyse numérique PY - 2009 SP - 377 EP - 398 VL - 43 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008047/ DO - 10.1051/m2an:2008047 LA - en ID - M2AN_2009__43_2_377_0 ER -
%0 Journal Article %A Bécache, Eliane %A Rodríguez, Jeronimo %A Tsogka, Chrysoula %T Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 377-398 %V 43 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008047/ %R 10.1051/m2an:2008047 %G en %F M2AN_2009__43_2_377_0
Bécache, Eliane; Rodríguez, Jeronimo; Tsogka, Chrysoula. Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 2, pp. 377-398. doi : 10.1051/m2an:2008047. http://www.numdam.org/articles/10.1051/m2an:2008047/
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