Error Analysis of a Finite Element Method for the Space-Fractional Parabolic Equation

Type
Article

Authors
Jin, Bangti
Lazarov, Raytcho
Pasciak, Joseph
Zhou, Zhi

KAUST Grant Number
KUS-C1-016-04

Date
2014-01

Abstract
© 2014 Society for Industrial and Applied Mathematics We consider an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann-Liouville type and order α ∈ (1, 2). We study a spatial semidiscrete scheme using the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and the Crank-Nicolson method. Error estimates in the L2(D)- and Hα/2 (D)-norm are derived for the semidiscrete scheme and in the L2(D)-norm for the fully discrete schemes. These estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data. Extensive numerical results are presented to illustrate the theoretical results.

Citation
Jin B, Lazarov R, Pasciak J, Zhou Z (2014) Error Analysis of a Finite Element Method for the Space-Fractional Parabolic Equation. SIAM J Numer Anal 52: 2272–2294. Available: http://dx.doi.org/10.1137/13093933X.

Acknowledgements
This work was supported by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).This author's work was supported by NSF grant DMS-1319052.The second author's work was supported in part by NSF grant DMS-1016525, and the third author's work was supported by NSF grant DMS-1216551.

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM Journal on Numerical Analysis

DOI
10.1137/13093933X

Permanent link to this record