A weak Galerkin finite element approximation of two-dimensional sub-diffusion equation with time-fractional derivative

Abstract

We develop a fully discrete weak Galerkin finite element method for the initial-boundary value problem of two-dimensional sub-diffusion equation with Caputo time-fractional derivative. A traditional $L_1$ discretization for the Caputo time-fractional derivative and a weak Galerkin scheme for the space integer differential operator are employed. We prove the stability of the numerical method and establish the error estimate in $L^2$ and discrete $H^1$ norms, respectively. Some numerical results are reported to confirm the theory.