Error analysis of a finite element method with GMMP temporal discretisation for a time-fractional diffusion equation. (1st May 2020) - Huang, Chaobao; Stynes, Martin
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 ∞ ( 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.