Error analysis of a finite element method with GMMP temporal discretisation for a time-fractional diffusion equation

摘要

A time-fractional initial-boundary value problem is considered, where the spatial domain has dimension d is an element of {1, 2, 3}, the spatial differential operator is a standard elliptic operator, and the time derivative is a Caputo derivative of order alpha is an element of (0, 1). To discretise in space we use a standard piecewise-polynomial finite element method, while for the temporal discretisation the GMMP scheme (a variant of the Grunwald-Letnikov scheme) is used on a uniform mesh. The analysis of the GMMP scheme for solutions that exhibit a typical weak singularity at the initial time t = 0 has not previously been considered in the literature. A global convergence result is derived in L-infinity (L-2), then a more delicate analysis of the error in this norm shows that, away from t = 0, the method attains optimal-rate convergence. Numerical results confirm the sharpness of the theoretical error bounds. (C) 2019 Elsevier Ltd. All rights reserved.