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 , Tome 43 (2009) no. 2, pp. 377-398.

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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.

DOI : 10.1051/m2an:2008047
Classification : 65M60, 65M12, 65M15, 65C20, 74S05
Mots-clés : mixed finite elements, fictitious domain method, domain embedding method, acoustic waves, convergence analysis 
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     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: Mathematical Modelling and Numerical Analysis },
     pages = {377--398},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {2},
     year = {2009},
     doi = {10.1051/m2an:2008047},
     zbl = {1161.65071},
     mrnumber = {2512501},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2008047/}
}
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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: Mathematical Modelling and Numerical Analysis , Tome 43 (2009) no. 2, pp. 377-398. doi : 10.1051/m2an:2008047. https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2008047/

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