Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview

TL;DR

This paper reviews numerical methods for solving time-fractional evolution equations with nonsmooth data, emphasizing their development, analysis, and application in various scientific fields.

Contribution

It provides a concise overview of numerical schemes for subdiffusion models with nonsmooth data, highlighting recent advances and key analytical techniques.

Findings

Finite element discretization effectively handles spatial variables.

Time-stepping schemes like convolution quadrature improve temporal accuracy.

Numerical experiments validate theoretical convergence rates.

Abstract

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.