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

Abstract

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 $ H^1 $ norm and $ L^2 $ 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 $ \mathcal{O}(h^{k+1}+M^{-2}) $ in the $ L^\infty(L^2) $-norm has been established. We give several numerical examples to confirm the theory presented in this work.

Keywords

• sub-diffusion problems
• graded temporal meshes
• stabilizer-free weak galerkin finite element method
• optimal rate of convergence