# Optimal order finite element approximations for variable-order time-fractional diffusion equations

## TL;DR

This paper develops and analyzes a finite element method for variable-order time-fractional diffusion equations, achieving optimal convergence rates without requiring full regularity of solutions, validated by numerical experiments.

## Contribution

It introduces a fully discrete finite element scheme for variable-order fractional diffusion equations with proven optimal convergence rates.

## Key findings

- First-order accuracy in time and second-order in space achieved.
- Optimal convergence estimates proved without full regularity assumptions.
- Numerical experiments confirm theoretical results.

## Abstract

We study a fully discrete finite element method for variable-order time-fractional diffusion equations with a time-dependent variable order. Optimal convergence estimates are proved with the first-order accuracy in time (and second order accuracy in space) under the uniform or graded temporal mesh without full regularity assumptions of the solutions. Numerical experiments are presented to substantiate the analysis.