High-order splitting finite element methods for the subdiffusion equation with limited smoothing property

TL;DR

This paper introduces a high-order finite element method for the subdiffusion equation with nonsmooth initial data, using a novel solution splitting approach to achieve high accuracy despite limited smoothing.

Contribution

A new splitting method separates the solution into smooth and nonsmooth parts, enabling high-order finite element approximations for subdiffusion equations with nonsmooth initial data.

Findings

Effective approximation of nonsmooth initial data including Dirac delta and interface measures.

Numerical experiments confirm theoretical accuracy and stability.

Method extends to equations with nonsmooth source data.

Abstract

In contrast with the diffusion equation which smoothens the initial data to $C^\infty$ for $t>0$ (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac--Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.