Discontinuous Galerkin Approximation of Flows in Fractured Porous Media on Polytopic Grids

Abstract

We present a numerical approximation of Darcy's flow through a fractured porous medium which employs discontinuous Galerkin methods on polytopic grids. For simplicity, we analyze the case of a single fracture represented by a $(d-1)$-dimensional interface between two $d$-dimensional subdomains, $d=2,3$. We propose a discontinuous Galerkin finite element approximation for the flow in the porous matrix which is coupled with a conforming finite element scheme for the flow in the fracture. Suitable (physically consistent) coupling conditions complete the model. We theoretically analyze the resulting formulation, prove its well-posedness, and derive optimal a priori error estimates in a suitable (mesh-dependent) energy norm. Two-dimensional numerical experiments are reported to assess the theoretical results.