Mixed Finite Element Methods for Fractional Navier-Stokes Equations

Xiaocui Li & Xu You
2021 Journal of Computational Mathematics  
This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations. The proposed method is based on the mixed finite element method in space and a finite difference scheme in time. The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail. Furthermore, We give the convergence analysis for both semidiscrete and fully discrete schemes and then prove that the numerical solution converges the exact one
more » ... ith order O(h 2 + k), where h and k respectively denote the space step size and the time step size. Finally, numerical examples are presented to demonstrate the effectiveness of our numerical methods. Mathematics subject classification: 60N15, 65M60, 60N30, 75D05. 131 with I α being the temporal Riemann-Liouville fractional integral operator of order α. The above-mentioned problem has many physical applications in many areas such as heterogeneous flows and materials, turbulence, viscoelasticity and electromagnetic theory. Particularly when α = 1, the problem (1.1) reduces to the classical Navier-Stokes equations, numerical approximations of which have been studied by many authors [1-4, 8-19, 25, 32, 34, 37-44, 46, 47]. However, for the fractional Navier-Stokes equations (FNSE) which are nonlinear in character, most of them do not have exact analytical solutions. It is shown that very few cases in which the exact solution of fractional Navier-Stokes equations can be obtained, where it have to make certain assumptions about the state of the fluid and a simple configuration for the flow pattern is to be considered. Hence it is necessary to analyze and study the approximation and numerical techniques of FNSE. However, to our best knowledge, numerical analysis of such problem for fractional Navier-Stokes equations is missing except [27, 56] in the literature. Therefore, this article aims to fill the gap, study and obtain the strong convergence approximations of FNSE like (1.1).
doi:10.4208/jcm.1911-m2018-0153 fatcat:jn3uup7zknh5fio3chxrmngpwi