# A posteriori error estimates for the Allen-Cahn problem

## TL;DR

This paper establishes a posteriori error estimates for fully-discrete Galerkin methods solving the Allen-Cahn equation, providing bounds that depend polynomially on the inverse interface length and improving existing error bounds.

## Contribution

It introduces new a posteriori error estimates for the Allen-Cahn problem using spectral estimates and elliptic reconstruction, enhancing previous bounds in various norms.

## Key findings

- Error estimates depend polynomially on inverse interface length
- Spectral estimate for linearized operator is crucial
- Improved bounds in $L_2(H^1)$ and $L_ extinfty(L_2)$ norms

## Abstract

This work is concerned with the proof of \emph{a posteriori} error estimates for fully-discrete Galerkin approximations of the Allen-Cahn equation in two and three spatial dimensions. The numerical method comprises of the backward Euler method combined with conforming finite elements in space. For this method, we prove conditional type \emph{a posteriori} error estimates in the $L^{}_4(0,T;L^{}_4(\Omega))$-norm that depend polynomially upon the inverse of the interface length $\epsilon$. The derivation relies crucially on the availability of a spectral estimate for the linearized Allen-Cahn operator about the approximating solution in conjunction with a continuation argument and a variant of the elliptic reconstruction. The new analysis also appears to improve variants of known \emph{a posteriori} error bounds in $L_2(H^1)$, $L_\infty^{}(L_2^{})$-norms in certain regimes.