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

Abstract

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].

Stability analysis of polytopic Discontinuous Galerkin approximations of the Stokes problem with applications to fluid-structure interaction problems

“…where U ∈ R Nu and P ∈ R Np are the vectors collecting the expansion coefficients {u j } j and {p j } j , respectively, whereas A h , B h , S h , and f h denote the matrix representations of the discrete bilinear forms in (4b)-(4d) and right-hand side in (5).…”

“…In this section, we prove the well-posedness of problem (5) in the stationary case making use of the Banach-Nečas-Babuška theorem.…”

“…In this work, we consider the Discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) that extends the standard DG method to polytopic meshes; see, e.g., [11,6,62,27,4,25,7,5]. In this framework, we study the discrete stability and well-posedness for the Stokes problem, by presenting an analysis that covers at once the two-and three-dimensional cases.…”

Machine Learning based refinement strategies for polyhedral grids with applications to Virtual Element and polyhedral Discontinuous Galerkin methods

“…To overcome this limitation, in the last years there has been a great interest in developing FEMs that can employ general polygons and polyhedra as grid elements for the numerical discretizations of partial differential equations. We mention the mimetic finite difference method [1,2,3,4], the hybridizable discontinuous Galerkin method [5,6,7,8], the Polyhedral Discontinuous Galerkin (PolyDG) method [9,10,11,12,13,14,15], the Virtual Element Method (VEM) [16,17,18,19,20,21] and the Hybrid High-Order method [22,23,24,25,26]. This calls for the need to develop effective algorithms to handle polygonal and polyhedral grids and to assess their quality (see e.g.…”

A high-order discontinuous Galerkin approach to the elasto-acoustic problem

“…Moreover, in the proposed semidiscrete dG formulation, the coupling between the acoustic and the poroelastic domains is introduced by considering (physically consistent) interface conditions, naturally incorporated in the scheme based on employing the For early results in the field of dG methods on polygonal and polyhedral grids we refer, for example, to [8,18,16,4] for second-order elliptic problems problems, to [15] for time dependent problems, to [6,7] for flows in fractured porous media, to [2] for fluid structure interaction problems, to [10] for non-liner sound waves, cf. also [17] for a comprehensive monograph and to [5] for a recent review in the context of geophysical applications.…”