# Explicit a posteriori local error estimation for FEM solutions

## TL;DR

This paper introduces an explicit local a posteriori error estimator for FEM solutions of Poisson's equation using the Hypercircle method, effective even without $H^2$ regularity, validated through numerical experiments.

## Contribution

It proposes a novel explicit local error estimation technique for FEM solutions of Poisson's equation applicable without $H^2$ regularity assumptions.

## Key findings

- Effective local error estimates demonstrated on 2D and 3D problems.
- Method works even when solution lacks $H^2$ regularity.
- Numerical experiments confirm the efficiency of the approach.

## Abstract

For the finite element solution of Poisson's equation, a local a posteriori error estimation based on the Hypercircle method is proposed. Even for the solution of Poisson's equation without the $H^2$ regularity, this method can provide explicit local error estimation. The efficiency of the proposed method is demonstrated by numerical experiments for the boundary value problem of Poisson's equation defined on the 2D and 3D domains.