A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation

In this paper, an implicit fully discrete direct discontinuous Galerkin (DDG) finite element method is considered for solving the time fractional diffusion equation. The scheme is based on the Gorenflo–Mainardi–Moretti–Paradisi (GMMP) scheme in time and direct discontinuous Galerkin method in space. Unlike the traditional local discontinuous Galerkin method, the DDG method is based on the direct weak formulation for solutions of parabolic equations in each computational cell, letting cells communicate via the numerical flux ux^ only. We prove that our scheme is stable and the energy norm error estimate is convergent with O((Δx)k+Δtα+1+Δtα2(Δx)k) by choosing admissible numerical flux. The DDG method has the advantage of easier formulation and implementation as well as the high order accuracy. Finally numerical experiments are presented to verify our theoretical findings.