Explicit and efficient error estimation for convex minimization problems

TL;DR

This paper introduces a systematic, constant-free a posteriori error estimation method for convex minimization problems, applicable to various variational problems, and demonstrates its reliability and efficiency through specific examples.

Contribution

It combines convex duality and a generalized Marini formula to derive a broad class of a posteriori error estimates for convex minimization problems.

Findings

Error estimates are essentially constant-free.

Estimates are reliable and efficient for the p-Dirichlet problem.

Applicable to degenerate minimization, obstacle, and image de-noising problems.

Abstract

We combine a systematic approach for deriving general a posteriori error estimates for convex minimization problems based on convex duality relations with a recently derived generalized Marini formula. The a posteriori error estimates are essentially constant-free and apply to a large class of variational problems including the $p$-Dirichlet problem, as well as degenerate minimization, obstacle and image de-noising problems. In addition, these a posteriori error estimates are based on a comparison to a given non-conforming finite element solution. For the $p$-Dirichlet problem, these a posteriori error bounds are equivalent to residual type a posteriori error bounds and, hence, reliable and efficient.