ℎ𝑝-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements

Abstract

We extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for curved element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical trace and Markov-type inverse estimates to arbitrary element shapes. A further new H 1 − L 2 -type inverse estimate on essentially arbitrary element shapes enables the proof of inf-sup stability of the method in a streamline-diffusion-like norm. These inverse estimates may be of independent interest. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach.

On Mathematical and Numerical Modelling of Multiphysics Wave Propagation with Polytopal Discontinuous Galerkin Methods: a Review

“…Inserting the previous bounds into (35) and taking the maximum over t ∈ [0, T ], yields the assertion.…”

“…Ω p corresponds to the quantity defined in (32). Therefore, to conclude it only remains to bound the right-hand side of (35). To do so, we apply the Cauchy-Schwarz and Young inequalities to infer…”

“…For early results in the field of dG methods on polygonal/polyhedral grids we refer the reader to [4,5,22,[33][34][35]43] for second-order elliptic problems, to [36] for parabolic differential equations, to [6][7][8] for flows in fractured porous media, and to [9] for fluid structure interaction problems; cf. also the review paper in the context of geophysical applications [10].…”

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

“…Remark 3.2. Following the recent approach of [24], it is possible to prove the inverse estimates in Lemmata 3.1 and 3.6 using assumptions milder than Assumptions 3.1 and 3.2. Notably, the theory therein presented covers very general geometries, including C 1 -curved faces and possibly the presence of arbitrary number of faces.…”

A posteriori error estimates for discontinuous Galerkin methods on polygonal and polyhedral meshes

“…To the best of our knowledge, Assumption 2.1 allows for the most general polytopic meshes for which a posteriori error bounds are proven, for any Galerkin discretization and for any PDE problem. Nevertheless, it is, perhaps inevitably, more restrictive compared to the respective ones required for stability and a priori error analysis of dG methods; see [9], [10, Section 4] and [8], for details. The key advantage of the present setting is that it allows us to be as explicit as possible in the constants involved in the a posteriori error bounds.…”