Two Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data

TL;DR

This paper introduces two fully discrete numerical schemes for fractional diffusion and diffusion-wave equations with nonsmooth initial data, demonstrating their accuracy, robustness, and efficiency through theoretical analysis and extensive numerical experiments.

Contribution

The paper develops two simple, fully discrete schemes using Galerkin finite elements and convolution quadrature, achieving optimal error estimates for nonsmooth data and providing a comprehensive comparison with existing methods.

Findings

First and second-order accuracy in time for nonsmooth initial data

Numerical experiments confirm optimal convergence and robustness

Proposed schemes are competitive with existing methods

Abstract

We consider the initial/boundary value problem for the fractional diffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two "simple" fully discrete schemes based on the Galerkin finite element method in space and convolution quadrature in time with the generating function given by the implicit backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of the initial data. These two schemes are first and second-order accurate in time for nonsmooth initial data. Extensive numerical experiments for one and two-dimensional problems confirm the convergence analysis. A detailed comparison with several popular time stepping schemes is also performed. The numerical results indicate that the proposed fully discrete schemes are accurate and robust for nonsmooth data, and competitive with existing schemes.