Analysis of a Time-Stepping Discontinuous Galerkin Method for Fractional Diffusion-Wave Equations with Nonsmooth Data

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 with respect to the regularity of the solution is established when the source term is nonsmooth, and nearly optimal convergence rate ln(1/τ)(ln(1/h)h2+τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \scriptstyle \ln (1/\tau )(\sqrt{\ln (1/h)}h^2+\tau ) $$\end{document} 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.