Error Analysis of Semidiscrete Finite Element Methods for Inhomogeneous Time-Fractional Diffusion

TL;DR

This paper analyzes the error behavior of semidiscrete finite element methods applied to inhomogeneous time-fractional diffusion equations, providing optimal error estimates and verifying results through numerical experiments.

Contribution

It offers the first detailed error analysis for semidiscrete Galerkin and lumped mass finite element schemes on inhomogeneous time-fractional diffusion problems.

Findings

Almost optimal error estimates for both schemes.

Lumped mass method requires symmetric meshes for optimal $L^2$ error.

Numerical experiments confirm theoretical error bounds.

Abstract

We consider the initial boundary value problem for the inhomogeneous time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and a nonsmooth right hand side data in a bounded convex polyhedral domain. We analyze two semidiscrete schemes based on the standard Galerkin and lumped mass finite element methods. Almost optimal error estimates are obtained for right hand side data $f(x,t)\in L^\infty(0,T;\dot H^q(\Omega))$, $-1< q \le 1$, for both semidiscrete schemes. For lumped mass method, the optimal $L^2(\Omega)$-norm error estimate requires symmetric meshes. Finally, numerical experiments for one- and two-dimensional examples are presented to verify our theoretical results.