Inverse Estimates and Polynomial Approximation on Polytopic Meshes

Abstract

This chapter develops the key mathematical tools needed to study the stability and convergence properties of hp-version discontinuous Galerkin finite element methods on polytopic meshes. A key issue in this setting is that general shape-regular polytopic meshes in $\mathbb{R}^{d}$, d > 1, may, under mesh refinement, possess elements with (d − k)-dimensional facets, k = 1, 2, …, d − 1, which degenerate as the mesh size tends to zero. Thereby, care must be taken to ensure that the resulting inverse estimates and polynomial approximation results are sensitive to this type of degeneracy. The key approach adopted here is to exploit known results for standard elements, both within an L ²- and L ^∞-setting, and to take the minimum of the resulting bounds. In this way, bounds which are optimal in both the h-version and p-version setting may be deduced, which directly account for (d − k)-dimensional facet degeneration, k = 1, 2, …, d − 1.