Time-stepping discontinuous Galerkin methods for fractional diffusion problems

TL;DR

This paper develops and analyzes time-stepping discontinuous Galerkin methods for fractional subdiffusion problems, demonstrating stability, error estimates, and exponential convergence rates with numerical validation.

Contribution

It introduces and rigorously analyzes $hp$-version DG methods for fractional diffusion, including stability, error bounds, and exponential convergence results.

Findings

Error order of $O(k^{ ext{max}\{2,p\}+rac{ ext{ extalpha}}{2}})$ for $h$-version DG on graded meshes

Exponential convergence rates achieved with geometrically refined time-steps and increasing polynomial degrees

Numerical tests confirm theoretical error estimates and convergence rates

Abstract

Time-stepping $hp$-versions discontinuous Galerkin (DG) methods for the numerical solution of fractional subdiffusion problems of order $-\alpha$ with $-1<\alpha<0$ will be proposed and analyzed. Generic $hp$-version error estimates are derived after proving the stability of the approximate solution. For $h$-version DG approximations on appropriate graded meshes near$t=0$, we prove that the error is of order$O(k^{\max\{2,p\}+\frac{\alpha}{2}})$, where $k$ is the maximum time-step size and $p\ge 1$ is the uniform degree of the DG solution. For $hp$-version DG approximations, by employing geometrically refined time-steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are shown. Finally, some numerical tests are given.