Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions
It offers a unified error analysis framework for L1 methods applied to fractional PDEs, benefiting researchers working on numerical solutions of time-fractional problems.
The paper provides a framework for analyzing the error of L1-type discretizations for fractional-derivative problems with singular initial behavior, achieving error bounds in L∞ and L2 norms on graded and uniform meshes. Numerical experiments confirm the theory.
An initial-boundary value problem with a Caputo time derivative of fractional order $α\in(0,1)$ 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 $L_\infty$ and $L_2$ 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.