Crank–Nicolson Finite Element Scheme and Modified Reduced-Order Scheme for Fractional Sobolev Equation

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.

Keywords:

• Crank–Nicolson finite element scheme
• error estimation
• reduced-order scheme
• stability
• time fractional Sobolev equation

2000 Mathematics Subject Classification:

• 65N30
• 65M60
• 65M12
• 35K99

Additional information

Funding

Supported by National Natural Science Fund (11661058, 11761053), Natural Science Fund of Inner Mongolia Autonomous Region (2016MS0102, 2017MS0107), program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-17-A07), the Prairie Talent Project of Inner Mongolia Autonomous Region.