$hp$-Version robust interior penalty discontinuous Galerkin methods for the $p$-Laplacian on simplicial and on essentially arbitrarily-shaped element meshes

TL;DR

This paper develops and analyzes an $hp$-version interior penalty discontinuous Galerkin method for the $p$-Laplacian, proving stability and error estimates on complex meshes with numerical validation.

Contribution

It introduces novel trace-type inverse estimates and extends the method to arbitrarily-shaped elements, providing new theoretical insights and practical error bounds.

Findings

Unconditional stability of the method is proven.

Error estimates are established for $hp$-version approximations.

Numerical experiments confirm the theoretical results.

Abstract

We consider the discretization of the $p$-Laplacian equation with an interior penalty discontinuous Galerkin method. We prove novel trace-type inverse estimates, leading to unconditional stability of the method. Further, $hp$-version a priori norm and quasi-norm error estimates are established, subordinate to available polynomial approximation results. The analysis is extended to discontinuous Galerkin methods, based on meshes with essentially arbitrarily-shaped, curved polygonal/polyhedral elements. This extension requires the proof of new $hp$-version weighted inverse estimates on essentially arbitrarily-shaped elements. Numerical experiments are also presented, highlighting the relevance of the theoretical findings.