V-cycle Multigrid Algorithms for Discontinuous Galerkin Methods on Non-nested Polytopic Meshes

Abstract

In this paper we analyse the convergence properties of V-cycle multigrid algorithms for the numerical solution of the linear system of equations arising from discontinuous Galerkin discretization of second-order elliptic partial differential equations on polytopal meshes. Here, the sequence of spaces that stands at the basis of the multigrid scheme is possibly non nested and is obtained based on employing agglomeration with possible edge/face coarsening. We prove that the method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. J satisfies a similar stability estimate as the one of I J J−1 , that is

High-order Discontinuous Galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes

“…Recently, DG methods have been shown to be naturally well suited to handle meshes composed by arbitrarily-shaped polygonal/polyhedral (polytopic, for short) elements, see e.g. [17,18,19,20,21,22,23,24,25,26,27,28]. The flexibility in the process of mesh design offered by polytopic elements is a great advantage whenever the differential problem at hand is posed on complicated domains featuring internal layers, microstructures, fractures or heterogeneities, as for example in geophysical applications, fluid-structure interaction or crack propagation problems.…”

An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids

“…Remark 5.8, we get that ∇z L 2 (Dj ) v h − v h,j L 2 (Dj ) . By inserting this bound into (22) we obtain (15). In order to show (16) we first select w = v h,j ∈ V j in (14); then, using the Cauchy-Schwarz inequality we obtain:…”

“…A variety of two-level and multigrid/multilevel techniques have been proposed, both in the geometric and algebraic settings, for the solution of DG discretizations; see, for example, [47,38,26,25,16]. In particular, the availability of efficient geometric multilevel solvers is strongly related to the possibility of employing general-shaped polytopic grids; indeed, if polytopic grids can be employed, then the sequence of grids which are required within a multilevel iteration can be defined by agglomeration; see [11,15] for details. Besides multigrid, a recent strand in the literature has focused on Schwarz domain decomposition methods; see, for example, [62], for a general abstract overview of these methods.…”

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

“…Such a discretization is then performed using general polygonal or polyhedral (briefly, polytopic) elements, with no restriction on the number of faces each element can possess, and possibly allowing for face degeneration in mesh refinement. The dG method has been recently proven to successfully support polytopic meshes: we refer the reader, e.g., to [7,8,9,10,11,12,13,14,15], as well as to the comprehensive research monograph by Cangiani et al [16]. In addition to the dG method, several other methods are capable to support polytopic meshes, such as the Polygonal Finite Element method [17,18,19,20], the Mimetic Finite Difference method [21,22,23,24], the Virtual Element method [25,26,27,28], the Hybridizable Discontinuous Galerkin method [29,30,31,32,33], and the Hybrid High-Order method [34,35,36,37,38].…”