# Oscillation in a posteriori error estimation

## TL;DR

This paper introduces a novel approach to a posteriori error estimation that effectively controls oscillation terms, ensuring they are dominated by the actual error and capturing unquantifiable residual parts.

## Contribution

It proposes a new method using a locally stable projection onto discretized residuals to improve error estimation in finite element analysis.

## Key findings

- Oscillation is now dominated by the error regardless of mesh or data regularity.
- The approach captures residual parts that cannot be quantified with finite information.
- The method enhances the reliability of a posteriori error estimates.

## Abstract

In a posteriori error analysis, the relationship between error and estimator is usually spoiled by so-called oscillation terms, which cannot be bounded by the error. In order to remedy, we devise a new approach where the oscillation has the following two properties. First, it is dominated by the error, irrespective of mesh fineness and the regularity of data and the exact solution. Second, it captures in terms of data the part of the residual that, in general, cannot be quantified with finite information. The new twist in our approach is a locally stable projection onto discretized residuals.