# Finite element approximations for fractional evolution problems

## TL;DR

This paper develops and analyzes a finite element scheme for fractional evolution problems involving both time and space derivatives, capturing memory and long-range dispersion effects, with numerical validation in 1D and 2D.

## Contribution

It introduces a novel finite element method combining convolution quadrature and linear elements for fractional evolution equations, with theoretical analysis and numerical experiments.

## Key findings

- The scheme is well-posed and stable.
- Numerical experiments confirm accuracy in 1D and 2D.
- Regularity estimates support the method's convergence.

## Abstract

This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discuss well-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linear elements for the space variable and a convolution quadrature for the time component. We illustrate the method's performance with numerical experiments in one- and two-dimensional domains.