Direct discontinuous Galerkin method for solving nonlinear time fractional diffusion equation with weak singularity solution

Abstract

In this work, the nonlinear time fractional diffusion equation with Caputo fractional derivative of order α∈(0,1) is considered. By the well-known L1-type formula of Caputo derivative on a graded mesh in time, a direct discontinuous Galerkin (DDG) method on a uniform mesh in space , and the Newton linearization method approximation of the nonlinear term, a fully discrete DDG scheme is constructed. Its error at each time level tn is bounded in the L2(Ω) norm by means of a non-trivial projection of an unknown solution into the finite element space. Then, the optimal error estimate is proved by choosing a suitable graded mesh. Numerical experiments are presented to verify that our analysis is sharp.