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

¹ POEMS, INRIA-Rocquencourt, BP 105, 78153 Le Chesnay Cédex, France. eliane.becache@inria.fr
² POEMS, ENSTA, 32 boulevard Victor, 75739 Paris Cedex 15, France. jeronimo.rodriguez@ensta.fr
³ Dept. of Applied Mathematics, University of Crete & IACM/FORTH, Crete, Greece. tsogka@tem.uoc.gr

Received: 22 February 2008
Revised: 22 September 2008

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.