Optimal error analysis of a FEM for fractional diffusion problems by energy arguments
This work provides rigorous error analysis for a numerical method solving fractional diffusion problems, which are important in modeling anomalous diffusion, but the contribution is incremental as it extends existing energy argument techniques.
The paper derives optimal a priori error bounds for the piecewise-linear FEM applied to time-fractional diffusion equations on convex domains, achieving optimal rates in L2 and H1 norms and a quasi-optimal bound in L∞ norm for both smooth and nonsmooth initial data.
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, {\it a priori} optimal error bounds in $L^2(Ω)$-, $H^1(Ω)$-norms, and a quasi-optimal bound in $L^{\infty}(Ω)$-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 $t^m$ type of weights to take care of the singular behavior of the continuous solution at $t=0.$ 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.