Analysis of a time-stepping discontinuous Galerkin method for modified anomalous subdiffusion problems

TL;DR

This paper investigates a time-stepping discontinuous Galerkin method for modified anomalous subdiffusion equations with two fractional derivatives, establishing stability, accuracy, and verifying results through numerical experiments.

Contribution

It introduces a stable, high-accuracy discontinuous Galerkin method for complex fractional subdiffusion problems with singular solutions.

Findings

Method achieves temporal accuracy of O(τ^{m+1−β/2})

Stability of the method is proven

Numerical experiments confirm theoretical results

Abstract

This paper analyzes a time-stepping discontinuous Galerkin method for modified anomalous subdiffusion problems with two time fractional derivatives of orders $ \alpha $ and $ \beta $ ($ 0 < \alpha < \beta < 1 $). The stability of this method is established, the temporal accuracy of $ O(\tau^{m+1-\beta/2}) $ is derived, where $m$ denotes the degree of polynomials for the temporal discretization. It is shown that, even the solution has singularity near $ t = {0+} $, this temporal accuracy can still be achieved by using the graded temporal grids. Numerical experiments are performed to verify the theoretical results.