FEM for time-fractional diffusion equations, novel optimal error analyses

TL;DR

This paper develops a finite element method for time-fractional diffusion equations, providing optimal error bounds in spatial norms for both smooth and nonsmooth initial data using novel energy techniques.

Contribution

It introduces a semidiscrete Galerkin finite element approach with new energy arguments to achieve optimal error estimates for complex fractional diffusion problems.

Findings

Optimal error bounds in L2 and H1 norms for smooth initial data

Optimal error bounds in L2 and H1 norms for nonsmooth initial data

Method applicable to time-space dependent diffusivity on convex domains

Abstract

A semidiscrete Galerkin finite element method applied to time-fractional diffusion equations with time-space dependent diffusivity on bounded convex spatial domains will be studied. The main focus is on achieving optimal error results with respect to both the convergence order of the approximate solution and the regularity of the initial data. By using novel energy arguments, for each fixed time $t$, optimal error bounds in the spatial $L^2$- and $H^1$-norms are derived for both cases: smooth and nonsmooth initial data.