Quasi-optimal nonconforming methods for symmetric elliptic problems. I -- Abstract theory

TL;DR

This paper analyzes nonconforming methods for symmetric elliptic problems, characterizing their quasi-optimality through stability and consistency, and reducing their construction to choosing appropriate smoothing operators.

Contribution

It provides a theoretical framework for understanding quasi-optimal nonconforming methods and links their construction to the selection of specific linear smoothing operators.

Findings

Quasi-optimality constant is explicitly characterized.

Impact of nonconformity on stability is quantified.

Structure of quasi-optimal methods is identified.

Abstract

We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined and the possible impact of nonconformity on its size is quantified by means of two alternative consistency measures. Identifying the structure of quasi-optimal methods, we show that their construction reduces to the choice of suitable linear operators mapping discrete functions to conforming ones. Such smoothing operators are devised in the forthcoming parts of this work for various finite element spaces.