Semidiscrete Finite Element Analysis of Time Fractional Parabolic Problems: A Unified Approach

TL;DR

This paper develops a unified finite element analysis framework for time-fractional parabolic problems, providing optimal error estimates for various spatial discretizations and initial data types, including extensions to multi-term models.

Contribution

It introduces a general, energy-based analysis method for semidiscrete finite element approximations of time-fractional parabolic problems, applicable to multiple spatial schemes and complex domains.

Findings

Optimal error estimates for smooth initial data

Error bounds for nonsmooth initial data

Extension to multi-term fractional models

Abstract

In this paper, we consider the numerical approximation of time-fractional parabolic problems involving Caputo derivatives in time of order $\alpha$, $0< \alpha<1$. We derive optimal error estimates for semidiscrete Galerkin FE type approximations for problems with smooth and nonsmooth initial data. Our analysis relies on energy arguments and exploits the properties of the inverse of the associated elliptic operator. We present the analysis in a general setting so that it is easily applicable to various spatial approximations such as conforming and nonconforming FEMs, and FEM on nonconvex domains. The finite element approximation in mixed form is also presented and new error estimates are established for smooth and nonsmooth initial data. Finally, an extension of our analysis to a multi-term time-fractional model is discussed.