An accurate, robust, and efficient weak Galerkin finite element scheme with graded meshes for the time-fractional quasi-linear diffusion equation

Abstract

A time-fractional quasi-linear diffusion equation with a Caputo time derivative of order 0 < α < 1 is considered. The weak Galerkin finite element method is used for the space discretization, the Caputo time derivative is discretized by L 1 method and Newton linear method is used for the quasi-linear term with graded meshes in the temporal direction. The stability and convergence for fully discrete weak Galerkin finite element scheme in L 2 -norm are proved. Numerical experiments are given to confirm the theoretical results.