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

Abstract

A time-fractional initial–boundary value problem is considered, where the spatial domain has dimension d∈{1,2,3}, the spatial differential operator is a standard elliptic operator, and the time derivative is a Caputo derivative of order α∈(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 Grünwald-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∞(L2), 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.