Grünwald–Letnikov scheme for a multi-term time fractional reaction-subdiffusion equation

In this work, we establish the pointwise-in-time error estimate of a Grünwald–Letnikov scheme for a multi-term time fractional reaction-subdiffusion problem with initial singularity, where Legendre spectral Galerkin method is used for spatial discretization. The theoretical results show that the temporal accuracy is first order if t is away from 0, while, globally it is O(τα1), where α1 is the highest order of the multi-term temporal Caputo fractional derivatives. Numerical results are presented to show the sharpness of the error estimate.