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Pre-Publicación 2019-22

Gabriel N. Gatica, Salim Meddahi:

Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems

Abstract:

This paper extends the applicability of the combined use of the virtual element method (VEM) and the boundary element method (BEM), recently introduced to solve the coupling of linear elliptic equations in divergence form with the Laplace equation, to the case of acoustic scattering problems in 2D and 3D. As a model we consider a bounded obstacle with piecewise constant refractive index, and a time harmonic incident wave, so that the scattered field, and hence the total wave as well, satisfies the homogeneous Helmholtz equation in the unbounded exterior region. The resulting coupled problem is complemented with suitable transmission conditions and the Sommerfeld radiation condition at infinity. The usual primal formulation and the corresponding VEM approach are then employed in the obstacle, which is combined, by means of either the Costabel and Han approach or a modification of it, with the boundary integral equation method in the exterior domain, thus yielding two possible VEM/BEM schemes. The first one of them, which is valid only in 2D, considers the main variable and its normal derivative as unknowns, whereas the second one, valid in 2D and 3D, adds the trace of the original unknown. In both procedures, the above mentioned boundary unknowns are non-virtual, and hence they are approximated by usual finite element subspaces. In addition, the discrete setting certainly requires virtual element subspaces for the main unknowns, and suitable projection and interpolation operators that are employed to define the corresponding discrete bilinear forms. The well-posedness of the continuous and discrete formulations are established, and the key aspects of the associated analyses include the fact that the boundary integral operators of the Helmholtz equation are compact perturbations of those for the Laplacian, the use of the Fredholm alternative, and the introduction of Galerkin projection-type operators. Finally, Cea-type estimates and consequent rates of convergence for the solutions are also derived.

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