Finite Element Method with Grünwald-Letnikov Type Approximation in Time for a Constant Time Delay Subdiffusion Equation

Abstract

In this work, a subdiffusion equation with constant time delay $\tau$ is considered. First, the regularity of the solution to the considered problem is investigated, finding that its first-order time derivative exhibits singularity at $t = 0^+$ and its second-order time derivative shows singularity at both $t = 0^+$ and $\tau^+$, while the solution can be decomposed into its singular and regular components. Then, we derive a fully discrete finite element scheme to solve the considered problem based on the standard Galerkin finite element method in space and the Grünwald-Letnikov type approximation in time. The analysis shows that the developed numerical scheme is stable. In order to discuss the error estimate, a new discrete Grönwall inequality is established. Under the above decomposition of the solution, we obtain a local error estimate in time for the developed numerical scheme. Finally, some numerical tests are provided to support our theoretical analysis.