A Finite Element Method Whose Shape Functions are Constructed Thanks to a Boundary Element Method,

International audienceThe numerical solution of wave propagation problems on very large domains (with respect to the wave length) is a very challenging problem. This is particularly true in the context of three dimensional problems where the large number of degrees of freedom can rapidly exceeds the capacity in storage and in computing even in the framework of massively parallel computer platforms. In the context of time harmonic wave equation, classical volume methods, like finite differences, finite volumes or finite elements suffer from the so called numerical pollution effect. To obtain a given precision, the number of degrees of freedom is not proportional to the volume of the computational domain. Roughly speaking, it is necessary to augment the density of nodes to maintain a given level of accuracy, when increasing the size of the computational domain, see [Ihlenburg-Babuska:95]. This is mainly due to a bad approximation of the speed of propagation of the wave. However, these methods are well adapted to heterogeneous media.Numerical methods based on integral equation formulation, like boundary finite element method, see for example \cite{Sauter-Schwab:11}, or collocation method, see [Bruno:03], do not suffer from the pollution effect since these technique are based on the evaluation of the Green function which contains exact information about the phase. In contrary, these techniques are hard to implement for heterogeneous media.During this talk we would like to present a numerical method which benefits from the advantages of both methods: low pollution effect and ability to handle heterogeneous media. This Trefftz method, see for example [Pluymers-et-al:07], can either be interpreted as a finite element method whose shape functions are constructed thanks to a boundary element method or as a boundary element method where the coupling between sub-domains is achieved thanks to a Galerkin formulation