A high-order stabilizer-free weak Galerkin finite element method on nonuniform time meshes for subdiffusion problems

We present a stabilizer-free weak Galerkin finite element method (SFWG-FEM) with polynomial reduction on a quasi-uniform mesh in space and Alikhanov's higher order L2-1 sigma scheme for discretization of the Caputo fractional derivative in time on suitable graded meshes for solving time-fractional subdiffusion equations. Typical solutions of such problems have a singularity at the starting point since the integer-order temporal derivatives of the solution blow up at the initial point. Optimal error bounds in H1 norm and L2 norm are proven for the semi-discrete numerical scheme. Furthermore, we have obtained the values of user-chosen mesh grading constant r, which gives the optimal convergence rate in time for the fully discrete scheme. The optimal rate of convergence of order O(hk+1 + M-2) in the L degrees degrees(L2)-norm has been established. We give several numerical examples to confirm the theory presented in this work.