# Error identities for variational problems with obstacles

## TL;DR

This paper develops error identities for nonlinear free boundary problems solved via variational methods, linking a nonlinear measure of solution accuracy to the duality gap and providing computable bounds.

## Contribution

It introduces specific error identities that relate a nonlinear measure of solution accuracy to the duality gap, enabling better error estimation in variational free boundary problems.

## Key findings

- Error identities relate nonlinear measures to the duality gap.
- Two-sided bounds provide efficient error estimates.
- Different minimization sequences affect error component dominance.

## Abstract

The paper is concerned with a class of nonlinear free boundary problems, which are usually solved by variational methods based on primal (or primal-dual) variational settings. We deduce and investigate special relations (error identities). They show that a certain nonlinear measure of the distance to the exact solution (specific for each problem) is equivalent to the respective duality gap, which minimization is a keystone of all variational numerical methods. Therefore, the identity defines the measure that contains maximal quantitative information on the quality of a numerical solution available through these methods. The measure has quadratic terms generated by the linear part of the differential operator and nonlinear terms associated with free boundaries. We obtain fully computable two sided bounds of this measure and show that they provide efficient estimates of the distance between the minimizer and any function from the corresponding energy space. Several examples show that for different minimization sequence the balance between different components of the overall error measure may be different and domination of nonlinear terms may indicate that coincidence sets are approximated incorrectly.