# Numerical approximations for a fully fractional Allen-Cahn equation

## TL;DR

This paper introduces and analyzes a finite element scheme for a fully fractional Allen-Cahn equation involving nonlocal operators, addressing error estimates, implementation, and asymptotic behavior as fractional parameters vanish.

## Contribution

It presents a novel numerical method for the fractional Allen-Cahn equation with comprehensive analysis and insights into asymptotic limits.

## Key findings

- Error estimates for the numerical scheme
- Implementation strategies for nonlocal operators
- Asymptotic behavior of solutions as fractional parameters tend to zero

## Abstract

A finite element scheme for an entirely fractional Allen-Cahn equation with non-smooth initial data is introduced and analyzed. In the proposed nonlocal model, the Caputo fractional in-time derivative and the fractional Laplacian replace the standard local operators. Piecewise linear finite elements and convolution quadratures are the basic tools involved in the presented numerical method. Error analysis and implementation issues are addressed together with the needed results of regularity for the continuous model. Also, the asymptotic behavior of solutions, for a vanishing fractional parameter and usual derivative in time, is discussed within the framework of the Gamma-convergence theory.