A Coupled Mixed Finite Element Method for the Interaction Problem between an Electromagnetic Field and an Elastic Body

Abstract

Abstract. This paper deals with the coupled problem arising from the interaction of a timeharmonic electromagnetic field with a three-dimensional elastic body. More precisely, we consider a suitable transmission problem holding between the solid and a sufficiently large annular region surrounding it, and aim to compute both the magnetic component of the scattered wave and the stresses that take place in the obstacle. To this end, we assume Voigt's model, which allows interaction only through the boundary of the body, and employ a dual-mixed variational formulation in the solid medium. As a consequence, one of the two transmission conditions becomes essential, whence it is enforced weakly through the introduction of a Lagrange multiplier. An abstract framework developed recently, which is based on regular decompositions of the spaces involved, is applied next to show that our coupled variational formulation is well-posed. In addition, we define the corresponding Galerkin scheme by using PEERS in the solid and using the edge finite elements of Nédélec in the electromagnetic region. Then, we prove that the resulting coupled mixed finite element scheme is uniquely solvable and convergent. Moreover, optimal a priori error estimates are derived in the usual way. Finally, some numerical results illustrating the analysis and the good performance of the method are also reported. Key words. Maxwell equations, edge finite elements, elastodynamics equations, PEERS AMS subject classifications. 65N30, 65N12, 65N15, 74F10, 74B05, 35J05DOI. 10.1137/090754212 [4] to analyze, at the continuous and discrete levels, a class of variational formulations defined by noncoercive bilinear (or sesquilinear) forms. More precisely, though the analysis in [4] was originally motivated by the study of Maxwell equations, the author succeeded in setting up the corresponding technique in a quite general framework. In fact, the key issue is the utilization of a Helmholtz-type decomposition of the main unknown, which allows us to reveal hidden compactness properties of the formulation, and hence the classical results connecting Fredholm alternative and projection methods (see, e.g., [18], [21]) can be applied straightforwardly. Introduction. A successful strategy has been developed inThe method from [4] was applied recently in [12] and [14] to deal with a timeharmonic fluid-solid interaction problem posed in the plane. The model consists of an elastic body occupying a region Ω s , which is subject to a given incident acoustic wave that travels in the fluid surrounding it. In [12] the fluid is supposed to occupy an annular region Ω f , and a Robin boundary condition imitating the behavior of the

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