Legendre-Fenchel duality and a generalized constitutive relation error

TL;DR

This paper introduces a generalized constitutive relation error based on Legendre-Fenchel duality, providing a new framework for error estimation in numerical solutions of elliptic variational inequalities.

Contribution

It develops a novel error measure grounded in Fenchel-Young inequality, applicable to a broad class of problems and useful for a posteriori error estimation.

Findings

Derived strict upper bounds for global energy errors.

Applied the approach to elliptic variational inequalities.

Demonstrated wide applicability in numerical error estimation.

Abstract

A generalized constitutive relation error is proposed in an analogous form to Fenchel-Young inequality on the basis of the key idea of Legendre-Fenchel duality theory. The generalized constitutive relation error is linked with the global errors of some admissible solutions for the problem in question, and is of wide applicability, especially in a posteriori error estimations of numerical methods. A class of elliptic variational inequalities is examined using the proposed approach and a strict upper bound of global energy errors of admissible solutions is obtained.