Bend 3d Mixed Virtual Element Method for Elliptic Problems

Abstract: In this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty, here proposed, is that curved elements are naturally handled without any degradation of the solution accuracy. In fact, in presence of curved boundaries, or internal interfaces, the geometrical error introduced by planar approximations may dominate the convergence rate limiting the benefit of high-order approximations. We consider the Darcy problem in its mixed form to directly obtain, with our numerical scheme, accurate and mass conservative fluxes without any post-processing. An important step to derive this new scheme is the actual computation of polynomials over curved polyhedrons, here presented and discussed. Finally, we show the theoretical analysis of the scheme as well as several numerical examples to support our findings