[Paper Review] A priori estimates of a finite element method for fractional diffusion problems by energy arguments

Samir Karaa, Kassem Mustapha+1 more|arXiv (Cornell University)|May 30, 2016

Fractional Differential Equations Solutions9 references3 citations

TL;DR

This paper develops a priori error estimates for a Galerkin piecewise-linear finite element method applied to time-fractional diffusion equations with variable diffusivity on convex domains. By employing a refined energy analysis with $t^m$-weighted norms and generalized Leibniz formulas for fractional derivatives, it establishes optimal $L^2$ and $H^1$ error bounds and quasi-optimal $L^∞$ bounds for both smooth and nonsmooth initial data.

ABSTRACT

In this article, the Galerkin piecewise-linear finite element (FE) method is applied to approximate the solution of time-fractional diffusion equations with variable diffusivity 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 $L^2(\Omega)$-, $H^1(\Omega)$-, and quasi-optimal in $L^{\infty}(\Omega)$-norms are derived for the semidiscrete FE scheme 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 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.

Motivation & Objective

• To derive a priori error bounds for the semidiscrete finite element method in time-fractional diffusion problems with variable diffusivity.
• To address the challenge of low solution regularity at $t=0$ that undermines standard energy techniques.
• To establish optimal $L^2$ and $H^1$ convergence rates and quasi-optimal $L^\infty$ bounds for both smooth and nonsmooth initial data.
• To develop a novel energy analysis framework incorporating $t^m$-weighted norms and fractional Leibniz rules to capture singular behavior at the initial time.

Proposed method

• Applies the Galerkin piecewise-linear finite element method to semidiscrete time-fractional diffusion problems on bounded convex domains.
• Uses a repeated application of an integral operator to handle the fractional derivative structure.
• Introduces $t^m$-type weights in the energy analysis to control the singular behavior of the solution at $t=0$.
• Employs the generalized Leibniz formula for fractional derivatives as a key analytical tool.
• Performs a delicate energy estimate that accounts for the low regularity of the exact solution.
• Derives error bounds in $L^2(\Omega)$, $H^1(\Omega)$, and $L^\infty(\Omega)$ norms through systematic integration by parts and norm control.

Experimental results

Research questions

• RQ1 Can standard energy arguments be adapted to achieve optimal error estimates in fractional diffusion problems with low-regularity solutions?
• RQ2 How can $t^m$-weighted norms improve the analysis of time-fractional PDEs with singular initial layers?
• RQ3 What role does the generalized Leibniz formula for fractional derivatives play in deriving sharp error bounds?
• RQ4 Can optimal $L^2$ and $H^1$ error estimates be achieved for both smooth and nonsmooth initial data in the semidiscrete finite element setting?
• RQ5 Is quasi-optimal $L^\infty$ convergence attainable for the finite element approximation of time-fractional diffusion equations?

Key findings

• Optimal $L^2(\Omega)$-norm error bounds are derived for the semidiscrete finite element scheme, valid for both smooth and nonsmooth initial data.
• Optimal $H^1(\Omega)$-norm error bounds are established through a refined energy analysis with $t^m$-weights.
• Quasi-optimal $L^\infty(\Omega)$-norm error bounds are obtained, indicating near-optimal convergence in the maximum norm.
• The generalized Leibniz formula for fractional derivatives is essential in handling the product rule for fractional derivatives in the energy estimates.
• The use of repeated integral operators and $t^m$-weighted norms effectively captures the initial layer singularity at $t=0$.
• Numerical experiments confirm the theoretical convergence rates in $L^2$, $H^1$, and $L^\infty$ norms.