Crank–Nicolson Finite Element Scheme and Modified Reduced-Order Scheme for Fractional Sobolev Equation. (18th November 2018)

• Record Type: Journal Article
• Authors: Liu, Jincun; Li, Hong; Liu, Yang
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 15(2018)
• Page Start: 1635
• Page End: 1655
• Publication Date: 2018-11-18
• Subjects: Crank–Nicolson finite element scheme -- error estimation -- reduced-order scheme -- stability -- time fractional Sobolev equation; 65N30 -- 65M60 -- 65M12 -- 35K99; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1488143
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: In this paper, a Crank–Nicolson finite difference/finite element method is considered to obtain the numerical solution for a time fractional Sobolev equation. Firstly, the classical finite element method is presented. Stability and error estimation for the fully discrete scheme are rigorously established. However, the amount of calculation and computing time are too large due to many degrees of freedom of classical finite element scheme and nonlocality of fractional differential operator. And then the modified reduced-order finite element scheme with low dimensions and sufficiently high accuracy, which is based on proper orthogonal decomposition technique, is provided. Stability and convergence for the reduced-order scheme are also studied. At last, numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions.