High–order Discontinuous Galerkin Methods on Polyhedral Grids for Geophysical Applications: Seismic Wave Propagation and Fractured Reservoir Simulations

We present a comprehensive review of the current development of discontinuous Galerkin methods on polytopic grids (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].