High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation

In this paper, a high-order local discontinuous Galerkin (LDG) method combined with weighted and shifted Grünwald difference (WSGD) approximation is developed and discussed for a Caputo time-fractional subdiffusion equation. The time fractional derivative of order α, 0<α<1, is approximated by a third-order method based on the idea of WSGD operator, while the spatial operator is approximated by the LDG method. Some useful lemmas are first introduced and proved, then the analysis of stability and optimal error estimate O(Δt3+hk+1) are obtained for the LDG method. Extensive numerical results using Pk,k=0,1,2,3, elements are presented to validate our theoretical analysis.