Optimal Error Analysis of a FEM for Fractional Diffusion Problems by Energy Arguments

In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, a priori optimal error bounds in $$L^2(\varOmega )$$L2(Ω)-, $$H^1(\varOmega )$$H1(Ω)-norms, and a quasi-optimal bound in $$L^{\infty }(\varOmega )$$L∞(Ω)-norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a $$t^m$$tm type of weights to take care of the singular behavior of the continuous solution at $$t=0$$t=0. The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.