TY - JOUR

T1 - On convergence of Chorin’s projection method to a Leray–Hopf weak solution

AU - Kuroki, Hidesato

AU - Soga, Kohei

N1 - Funding Information:
The second author is supported by JSPS Grant-in-aid for Young Scientists (B) #15K21369 and JSPS Grant-in-aid for Young Scientists #18K13443.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/10/1

Y1 - 2020/10/1

N2 - The projection method to solve the incompressible Navier–Stokes equations was first studied by Chorin (Math Comput, 1969) in the framework of a finite difference method and Temam (Arch Ration Mech Anal, 1969) in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with periodic boundary conditions 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 a 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.

AB - The projection method to solve the incompressible Navier–Stokes equations was first studied by Chorin (Math Comput, 1969) in the framework of a finite difference method and Temam (Arch Ration Mech Anal, 1969) in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with periodic boundary conditions 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 a 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.

UR - http://www.scopus.com/inward/record.url?scp=85089698986&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85089698986&partnerID=8YFLogxK

U2 - 10.1007/s00211-020-01144-w

DO - 10.1007/s00211-020-01144-w

M3 - Article

AN - SCOPUS:85089698986

VL - 146

SP - 401

EP - 433

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 2

ER -