Analysis of a time-stepping discontinuous Galerkin method for fractional diffusion-wave equation with nonsmooth data

TL;DR

This paper investigates a discontinuous Galerkin method for fractional diffusion-wave equations, demonstrating nearly optimal convergence rates even with nonsmooth data, supported by theoretical analysis and numerical experiments.

Contribution

It provides a comprehensive convergence analysis of a time-stepping discontinuous Galerkin method for fractional diffusion-wave problems with nonsmooth data, including new error estimates.

Findings

Nearly optimal convergence rate established for nonsmooth data

Convergence proven without smooth initial conditions

Numerical experiments confirm theoretical results

Abstract

This paper analyzes a time-stepping discontinuous Galerkin method for fractional diffusion-wave problems. This method uses piecewise constant functions in the temporal discretization and continuous piecewise linear functions in the spatial discretization. Nearly optimal convergence rate with respect to the regularity of the solution is established when the source term is nonsmooth, and nearly optimal convergence rate $ \ln(1/\tau)(\sqrt{\ln(1/h)} h^2+\tau)$ is derived under appropriate regularity assumption on the source term. Convergence is also established without smoothness assumption on the initial value. Finally, numerical experiments are performed to verify the theoretical results.