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Preprint 2018-33

Gabriel N. Gatica, Salim Meddahi:

On the coupling of VEM and BEM in two and three dimensions

Abstract:

This paper introduces and analyzes the combined use of the virtual element method (VEM) and the boundary element method (BEM) to numerically solve linear transmission problems in 2D and 3D. As a model we consider an elliptic equation in divergence form holding in an annular domain coupled with the Laplace equation in the corresponding unbounded exterior region, together with transmission conditions on the interface and a suitable radiation condition at infinity. We employ the usual primal formulation in the bounded region, and combine it, by means of the Costabel and Han approach, with the boundary integral equation method in the exterior domain. As a consequence, and besides the original unknown of the model, its normal derivative in 2D, and both its normal derivative and its trace in the 3D case, are introduced as auxiliary non-virtual unknowns. Moreover, for the latter case, a new and more suitable variational formulation for the coupling is introduced. In turn, the main ingredients required by the discrete analyses include the virtual element subspaces for the domain unknowns, explicit polynomial subspaces for the boundary unknowns, and suitable projection and interpolation operators that allow to define the corresponding discrete bilinear forms. In particular, two VEM/BEM schemes are proposed in the three-dimensional case, one of them mimicking the non-symmetric interior penalty discontinuous Galerkin method. Then, as for the continuous formulations, the classical Lax-Milgram lemma is employed to derive the well-posedness of our coupled VEM-BEM scheme. Finally, a priori error estimates in the energy and weaker norms, and corresponding rates of convergence for the solution as well as for a fully computable projection of the virtual component of it, are provided.

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