A short perspective on a posteriori error control and adaptive discretizations

TL;DR

This paper reviews the development and application of a posteriori error control and adaptive discretizations, emphasizing theoretical progress and industrial implementation since their emergence in the late 1970s.

Contribution

It provides a comprehensive overview of the evolution, achievements, and future perspectives of a posteriori error estimators and adaptive discretizations.

Findings

Significant theoretical advancements have been made since the 1970s.

Initial industrial applications have successfully integrated these methods.

The paper discusses future directions for research and application.

Abstract

Error control by means of a posteriori error estimators or indica-tors and adaptive discretizations, such as adaptive mesh refinement, have emerged in the late seventies. Since then, numerous theoretical developments and improvements have been made, as well as the first attempts to introduce them into real-life industrial applications. The present introductory chapter provides an overview of the subject, highlights some of the achievements to date and discusses possible perspectives.