Volume 54, Number 5, September-October 2020
|Page(s)||1635 - 1660|
|Published online||28 July 2020|
Primal-dual gap estimators for a posteriori error analysis of nonsmooth minimization problems
Department of Applied Mathematics, Mathematical Institute, University of Freiburg, Hermann-Herder-Str. 9, 79104 Freiburg im Breisgau, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 16 October 2019
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.
Mathematics Subject Classification: 49M29 / 65K15 / 65N15 / 65N50
Key words: Convex minimization / primal-dual gap / adaptive mesh refinement / nonlinear Laplace / image denoising
© EDP Sciences, SMAI 2020
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