# Primal-dual gap estimators for a posteriori error analysis of nonsmooth minimization problems

## TL;DR

This paper develops primal-dual gap estimators for a posteriori error analysis in nonsmooth minimization problems, providing reliable bounds and adaptive schemes demonstrated on nonlinear Laplace and image denoising problems.

## Contribution

It introduces a new primal-dual gap error estimator with proven reliability and applies it to nonlinear problems with adaptive finite element methods.

## Key findings

- Reliable error bounds with constant one for energy error
- Effective local mesh refinement near singularities
- Moderate overestimation of the error

## Abstract

The primal-dual gap is a natural upper bound for the energy error and, for uniformly convex minimization problems, also for the error in the energy norm. This feature can be used to construct reliable primal-dual gap error estimators for which the constant in the reliability estimate equals one for the energy error and equals the uniform convexity constant for the error in the energy norm. In particular, it defines a reliable upper bound for any functions that are feasible for the primal and the associated dual problem. The abstract a posteriori error estimate based on the primal-dual gap is provided in this article, and the abstract theory is applied to the nonlinear Laplace problem and the Rudin-Osher-Fatemi image denoising problem. The discretization of the primal and dual problems with conforming, low-order finite element spaces is addressed. The primal-dual gap error estimator is used to define an adaptive finite element scheme and numerical experiments are presented, which illustrate the accurate, local mesh refinement in a neighborhood of the singularities, the reliability of the primal-dual gap error estimator and the moderate overestimation of the error.