# Residual estimates for post-processors in elliptic problems

## TL;DR

This paper develops a unified framework for a posteriori error estimation in elliptic problems, enhancing the reliability and efficiency of various post-processing techniques including superconvergent methods.

## Contribution

It introduces a general post-processing operator that improves error bounds for multiple existing reconstruction techniques in elliptic boundary value problems.

## Key findings

- Optimal error control achieved for superconvergent and other reconstruction operators
- The proposed framework applies to popular methods like SIAC filter and Superconvergent Patch Recovery
- Numerical tests confirm the theoretical error estimates and improvements

## Abstract

In this work we examine a posteriori error control for post-processed approximations to elliptic boundary value problems. We introduce a class of post-processing operator that `tweaks' a wide variety of existing post-processing techniques to enable efficient and reliable a posteriori bounds to be proven. This ultimately results in optimal error control for all manner of reconstruction operators, including those that superconverge. We showcase our results by applying them to two classes of very popular reconstruction operators, the Smoothness-Increasing Accuracy-Enhancing filter and Superconvergent Patch Recovery. Extensive numerical tests are conducted that confirm our analytic findings.