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.

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We present a comprehensive review of the current development of PolyDG methods for geophysical applications, addressing as paradigmatic applications the numerical modeling of seismic wave propagation and fracture reservoir simulations. We first recall the theoretical background of the analysis of PolyDG methods and discuss the issue of its efficient implementation on polytopic meshes. We address in detail the issue of numerical quadrature and recall the new quadrature free algorithm for the numerical evaluation of the integrals required to assemble the mass and stiffness matrices introduced in [22]. Then we present PolyDG methods for the approximate solution of the elastodynamics equations on computational meshes consisting of polytopic elements. We review the well-posedness of the numerical formulation and hp-version a priori stability and error estimates for the semi-discrete scheme, following [10]. The computational performance of the fully-discrete approximation obtained based on employing the PolyDG method for the space discretization coupled with the leap-frog time marching scheme are demonstrated through numerical experiments. Next, we address the problem of modeling the flow in a fractured porous medium and we review the unified construction and analysis of PolyDG methods following [16]. We show, in a unified setting, the well-posedness of the numerical formulations and hp-version a priori error bounds, that are then validated through numerical tests. We also briefly discuss the extendability of our approach to handle networks of partially immersed fractures and networks of intersecting fractures, recently proposed in [15].

We propose a formulation based on discontinuous Galerkin methods on polygonal/polyhedral grids for the simulation of flows in fractured porous media. We adopt a model for single-phase flows where the fracture is modelled as a (d − 1)-dimensional interface in a d-dimensional bulk domain and the flow is governed by the Darcy's law in both the bulk and the fracture. The two problems are then coupled through physically consistent conditions. We focus on the numerical approximation of the coupled bulk-fracture problem, discretizing the bulk problem in mixed form and the fracture problem in primal form. We present an priori h-and p-version error estimate in a suitable (mesh-dependent) energy norm and numerical tests assessing it.

Galerkin methods on polygonal and polyhedral grids Flow through a porous medium Networks of fracturesWe present a numerical approximation of Darcy's flow through a porous medium that incorporates networks of fractures with non empty intersection. Our scheme employs PolyDG methods, i.e. discontinuous Galerkin methods on general polygonal and polyhedral (polytopic, for short) grids, featuring elements with edges/faces that may be in arbitrary number (potentially unlimited) and whose measure may be arbitrarily small. Our approach is then very well suited to tame the geometrical complexity featured by most of applications in the computational geoscience field. From the modelling point of view, we adopt a reduction strategy that treats fractures as manifolds of codimension one and we employ the primal version of Darcy's law to describe the flow in both the bulk and in the fracture network. In addition, some physically consistent conditions couple the two problems, allowing for jump of pressure at their interface, and they as well prescribe the behaviour of the fluid along the intersections, imposing pressure continuity and flux conservation. Both the bulk and fracture discretizations are obtained employing the Symmetric Interior Penalty DG method extended to the polytopic setting. The key instrument to obtain a polyDG approximation of the problem in the fracture network is the generalization of the concepts of jump and average at the intersection, so that the contribution from all the fractures is taken into account. We prove the well-posedness of the discrete formulation and perform an error analysis obtaining a priori ℎ𝑝-error estimates. All our theoretical results are validated performing preliminary numerical tests with known analytical solution.

We present a numerical approximation of Darcy's flow through a porous medium that incorporates networks of fractures with non empty intersection. Our scheme employs PolyDG methods, i.e. discontinuous Galerkin methods on general polygonal and polyhedral (polytopic, for short) grids, featuring elements with edges/faces that may be in arbitrary number (potentially unlimited) and whose measure may be arbitrarily small.Our approach is then very well suited to tame the geometrical complexity featured by most of applications in the computational geoscience field. From the modelling point of view, we adopt a reduction strategy that treats fractures as manifolds of codimension one and we employ the primal version of Darcy's law to describe the flow in both the bulk and in the fracture network. In addition, some physically consistent conditions couple the two problems, allowing for jump of pressure at their interface, and they as well prescribe the behaviour of the fluid along the intersections, imposing pressure continuity and flux conservation. Both the bulk and fracture discretizations are obtained employing the Symmetric Interior Penalty DG method extended to the polytopic setting. The key * Paola F. Antonietti and Chiara Facciolà have been supported by SIR Project n. RBSI14VT0S "PolyPDEs: Non-conforming polyhedral finite element methods for the approximation of partial differential equations" funded by MIUR. Marco Verani has been partially supported by the Italian research grant Prin 2012 2012HBLYE4 "Metodologie innovative nella modellistica differenziale numerica" and by INdAM-GNCS.