Finite element method for a constant time delay subdiffusion equation with Riemann-Liouville fractional derivative

This work considers to numerically solve a subdiffusion equation involving constant time delay $\tau$ and Riemann-Liouville fractional derivative. First, a fully discrete finite element scheme is developed for the considered problem under the symmetric graded time mesh, where the Caputo fractional derivative is approximated via the L1 formula, while the Riemann-Liouville integral is discretized using the fractional right rectangular rule. Under the assumption that the exact solution has low regularities at $t=0$ and $\tau$, the local truncation errors of both the L1 formula and the fractional right rectangular rule are analyzed. It is worth noting that, by setting the mesh parameter $r=1$, the symmetric graded time mesh will degenerate to a uniform mesh. Consequently, we proceed to discuss the stability and convergence of the proposed numerical scheme under two scenarios. For the uniform time mesh, by introducing a discrete sequence $\{P_k\}$, the unconditional stability and local time error estimate for the developed scheme is established. Conversely, on the symmetric graded time mesh, through the introduction of a discrete fractional Gronwall inequality, the stability and globally optimal time error estimate can be obtained. Finally, some numerical tests are presented to validate the theoretical results.