Non-uniform L1/discontinuous Galerkin approximation for the time-fractional convection equation with weak regular solution

Abstract

In this paper, an efficient method seeking the numerical solution of a time-fractional convection equation whose solution is not smooth at the starting time is presented. The Caputo time-fractional derivative of order in (0,1) is discretized by the L1 finite difference method using non-uniform meshes; and, for the spatial derivative the discontinuous Galerkin (DG) finite element method is used. The stability and convergence of the method are analyzed for two-dimensional domains, using Cartesian and a particular class of unstructured grids. At last, several numerical examples are carried out which support the theoretical analysis.