Compact embedding in the space of piecewise H1 functions

TL;DR

This paper establishes a compact embedding theorem for piecewise H1 functions on shape-regular triangulations, generalizing classical results and enabling new inequalities relevant to elastic shell analysis.

Contribution

It introduces a generalized compact embedding theorem for piecewise H1 spaces on non-quasi-uniform meshes, extending classical functional analysis results.

Findings

Proves a generalized Rellich--Kondrachov theorem for piecewise functions.

Derives nonstandard Poincaré--Friedrichs inequalities for these spaces.

Provides tools for Korn inequalities in elastic shell modeling.

Abstract

We prove a compact embedding theorem in a class of spaces of piecewise H1 functions subordinated to a class of shape regular, but not necessarily quasi-uniform triangulations of a polygonal domain. This result generalizes the Rellich--Kondrachov theorem. It is used to prove generalizations to piecewise functions of nonstandard Poincar\'e--Friedrichs inequalities. It can be used to prove Korn inequalities for piecewise functions associated with elastic shells.