A Virtual Marriage à la Mode: Some Recent Results on the Coupling of VEM and BEM

Abstract

AbstractThe aim of this chapter is to present in a unified way some recent results on 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 models we consider an elliptic equation in divergence form holding in an annular domain coupled with the Laplace equation in the corresponding unbounded exterior region, and an acoustic scattering problem determined by a bounded obstacle and a time harmonic incident wave, so that the scattered field, and hence the total wave as well, satisfies the homogeneous Helmholtz equation. Both sets of corresponding equations are complemented with proper transmission conditions at the respective interfaces, and suitable radiation conditions at infinity. We employ the usual primal formulation and the associated VEM approach in the respective bounded regions, and combine it, by means of either the Costabel & Han approach or a recent modification of it, with the boundary integral equation method in the exterior domain, thus yielding two possible VEM/BEM schemes. The first method is valid only in 2D and considers the main variable and its normal derivative as unknowns, whereas the second one, which includes additionally the trace of the former as a third unknown, is applicable in both dimensions. The well-posedness of the continuous and discrete formulations is established and a priori error estimates together with corresponding rates of convergence are derived. Finally, several numerical examples in 2D illustrating the performance of the proposed discrete schemes are reported.