Optimal error analysis of a FEM for fractional diffusion problems by\n energy arguments

2016/05/30 by Samir Karaa, Kassem Mustapha, Karaa, Samir +3
Engineering · Mathematics · #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Numerical methods in engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1605.09104

openalex publication_date 2016/05/30 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

In this article, the piecewise-linear finite element method (FEM) is applied\nto approximate the solution of time-fractional diffusion equations on bounded\nconvex domains. Standard energy arguments do not provide satisfactory results\nfor such a problem due to the low regularity of its exact solution. Using a\ndelicate energy analysis, it a priori optimal error bounds in\nL²(\Ω)-, H¹(\Ω)-norms, and a quasi-optimal bound in\nL^\∞(\Ω)-norm are derived for the semidiscrete FEM for cases with\nsmooth and nonsmooth initial data. The main tool of our analysis is based on a\nrepeated use of an integral operator and use of a t^m type of weights to take\ncare of the singular behavior of the continuous solution at t=0. The\ngeneralized Leibniz formula for fractional derivatives is found to play a key\nrole in our analysis. Numerical experiments are presented to illustrate some of\nthe theoretical results.\n

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