Local Discontinuous Galerkin Methods with Novel Basis for Fractional Diffusion Equations with Non-smooth Solutions

Abstract

In this paper, we develop novel local discontinuous Galerkin (LDG) methods for fractional diffusion equations with non-smooth solutions. We consider such problems, for which the solutions are not smooth at boundary, and therefore the traditional LDG methods with piecewise polynomial solutions suffer accuracy degeneracy. The novel LDG methods utilize a solution information enriched basis, simulate the problem on a paired special mesh, and achieve optimal order of accuracy. We analyze the $L^2$ stability and optimal error estimate in $L^2$-norm. Finally, numerical examples are presented for validating the theoretical conclusions.