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 two and three dimensions. 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 in addition to the original unknown of the model, its normal derivative in two dimensions, and both its normal derivative and its trace in the three-dimensional case, are introduced as auxiliary nonvirtual 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 us to define the corresponding discrete bilinear forms. Then, as for the continuous formulations, the classical Lax--Milgram lemma is employed to derive the well-posedness of our coupled VEM/BEM scheme. 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 its virtual component, are provided. Finally, a couple of numerical examples in two dimensions illustrating the performance of the derived VEM/BEM schemes are reported.