2018/09/12 by Hidesato Kuroki, Kohei Soga, Kuroki, Hidesato +1 · 2 citations
Computer Science · Engineering · Mathematics · #35D30 #35Q30 #65M06 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1809.04383
openalex publication_date 2018/09/12 · openalex created_date 2018/09/27 · openalex updated_date 2026/07/28
The projection method to solve the incompressible Navier-Stokes equations was first studied by Chorin [Math. Comp., 1969] in the framework of a finite difference method and Temam [Arch. Rational Mech. and Anal., 1969] in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with the periodic boundary condition assuming existence of a C5-solution, while Temam demonstrated an abstract argument to obtain a Leray-Hopf weak solution in problems on a bounded domain with the no-slip boundary condition. In the present paper, the authors extend Chorin's result with full details to obtain convergent finite difference approximation of a Leray-Hopf weak solution to the incompressible Navier-Stokes equations on an arbitrary bounded Lipschitz domain of ℝ3 with the no-slip boundary condition and an external force. We prove unconditional solvability of our implicit scheme and strong L2-convergence (up to subsequence) under the scaling condition h3-α≤τ (no upper bound is necessary), where h,τ are space, time discretization parameters, respectively, and α∈(0,2] is any fixed constant. The results contain a compactness method based on a new interpolation inequality for step functions.