Bend 3d mixed virtual element method for Darcy problems

In this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty is that curved elements are naturally handled without any degradation of the solution accuracy. Indeed, 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 accurate and mass conservative fluxes without any post-processing. An important step to derive the proposed scheme is the integration 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.