Error estimates for a semidiscrete finite element method for fractional order parabolic equations

TL;DR

This paper analyzes the error estimates of semidiscrete finite element methods, specifically Galerkin and lumped mass Galerkin FEM, for solving fractional order parabolic equations with both smooth and nonsmooth initial data.

Contribution

It provides optimal error estimates for these methods applied to time-fractional diffusion equations, including cases with nonsmooth initial data.

Findings

Optimal error bounds established for Galerkin FEM.

Optimal error bounds established for lumped mass Galerkin FEM.

Effective for nonsmooth initial data.

Abstract

We consider the initial boundary value problem for the homogeneous time-fractional diffusion equation $\partial^\alpha_t u - \De u =0$ ($0< \alpha < 1$) with initial condition $u(x,0)=v(x)$ and a homogeneous Dirichlet boundary condition in a bounded polygonal domain $\Omega$. We shall study two semidiscrete approximation schemes, i.e., Galerkin FEM and lumped mass Galerkin FEM, by using piecewise linear functions. We establish optimal with respect to the regularity of the solution error estimates, including the case of nonsmooth initial data, i.e., $v \in L_2(\Omega)$.