Trace inequalities for piecewise $W^{1,p}$ functions over general polytopic meshes

TL;DR

This paper establishes new trace inequalities for piecewise $W^{1,p}$ functions on general polytopic meshes, which are vital for analyzing stability and convergence in nonconforming finite element methods.

Contribution

It introduces novel trace inequalities applicable to general polytopic meshes without relying on finite dimensional arguments, extending previous results.

Findings

Valid for arbitrary polytopic meshes with small facets

Applicable for a range of Lebesgue indices

Does not depend on inverse estimates or averaging operators

Abstract

Trace inequalities are crucial tools to derive the stability of partial differential equations with inhomogeneous, natural boundary conditions. In the analysis of corresponding Galerkin methods, they are also essential to show convergence of sequences of discrete solutions to the exact one for data with minimal regularity under mesh refinements and/or degree of accuracy increase. In nonconforming discretizations, such as Crouzeix-Raviart and discontinuous Galerkin, the trial and test spaces consists of functions that are only piecewise continuous: standard trace inequalities cannot be used in this case. In this work, we prove several trace inequalities for piecewise $W^{1,p}$ functions. Compared to analogous results already available in the literature, our inequalities are established: (i) on fairly general polytopic meshes (with arbitrary number of facets and arbitrarily small facets); (ii) without the need of finite dimensional arguments (e.g., inverse estimates, approximation properties of averaging operators); (iii) for different ranges of maximal and nonmaximal Lebesgue indices.