Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions

An initial-boundary value problem with a Caputo time derivative of fractional order α ∈ ( 0 , 1 ) α ∈ (0,1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the L ∞ L_∞ and L 2 L_2 norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.