Superconvergence analysis of finite element methods for the variable-order subdiffusion equation with weakly singular solutions

Abstract

This paper investigates an efficient numerical method to solve the variable-order subdiffusion equation with weakly singular solutions, which uses the L1 scheme on graded meshes in time and the finite element method in space. To obtain the optimal error estimate, the truncation error of the nonuniform L1 scheme for the variable-order Caputo derivative is given. Combining this result with a novel discrete fractional Gronwall inequality, we derive an optimal error estimate in L∞(L2) norm and L∞(H1) norm. Furthermore, by using a simple postprocessing technique of the numerical solution, a higher convergence order in space is obtained. Finally, a numerical experiment is given to confirm the sharpness of our theoretical results.