A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition

摘要

A time-fractional reaction-diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Omega x [0, T], where Omega = (0,1) subset of R. The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t = 0. Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretisation on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L-2 (Omega) and H-1(Omega) norms at each discrete time level t(n) by means of a non-trivial projection of the unknown solution into the finite element space. The L-2 (Omega) bound is optimal for all t(n) ; the H-1 (Omega) bound is optimal for t n not close to t = 0. An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.