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

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.