# The Prager-Synge theorem in reconstruction based a posteriori error estimation

## TL;DR

This paper reviews the Prager-Synge hypercircle method and its influence on a posteriori error estimation, focusing on the Braess-Schöberl estimator for the Poisson problem, and demonstrates convergence and optimality of adaptive finite element schemes.

## Contribution

It provides a comprehensive review of the Prager-Synge theorem's application to a posteriori error estimation and proves convergence and optimality of related adaptive algorithms.

## Key findings

- The Braess-Schöberl estimator effectively estimates errors in Poisson problems.
- Adaptive finite element schemes based on these estimators converge.
- The algorithms achieve optimal error reduction.

## Abstract

In this paper we review the hypercircle method of Prager and Synge. This theory inspired several studies and induced an active research in the area of a posteriori error analysis. In particular, we review the Braess--Sch\"oberl error estimator in the context of the Poisson problem. We discuss adaptive finite element schemes based on two variants of the estimator and we prove the convergence and optimality of the resulting algorithms.