A stabilized multi-level method for non-singular finite volume solutions of the stationary 3D Navier–Stokes equations Article

First Online: 14 March 2012 Received: 23 December 2010 Revised: 10 October 2011 DOI :
10.1007/s00211-012-0462-z

Cite this article as: Li, J., Chen, Z. & He, Y. Numer. Math. (2012) 122: 279. doi:10.1007/s00211-012-0462-z
Abstract This paper proposes and analyzes a stabilized multi-level finite volume method (FVM) for solving the stationary 3D Navier–Stokes equations by using the lowest equal-order finite element pair without relying on any solution uniqueness condition. This multi-level stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performing one Newton correction step on each subsequent mesh, thus only solving a large linear system. An optimal convergence rate for the finite volume approximations of nonsingular solutions is first obtained with the same order as that for the usual finite element solution by using a relationship between the stabilized FVM and a stabilized finite element method. Then the multi-level finite volume approximate solution is shown to have a convergence rate of the same order as that of the stabilized finite volume solution of the stationary Navier–Stokes equations on a fine mesh with an appropriate choice of the mesh size: \({ h_{j} ~ h_{j-1}^{2}, j = 1,\ldots, J}\) . Finally, numerical results presented validate our theoretical findings.

Mathematics Subject Classification 76D05 65M08 65M12 Supported in part by NCET-11-1041, the NSF of China (No. 11071193) and (No. 10971166), Natural Science New Star of Science and Technologies Research Plan in Shaanxi Province of China (No. 2011kjxx12), Research Program of Education Department of Shaanxi Province (No. 11JK0490), the project-sponsored by SRF for ROCS, SEM, the National Basic Research Program (No. 2005CB321703), and NSERC/AERI/Foundation CMG Chair and iCORE Chair Funds in Reservoir Simulation.

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CrossRef Authors and Affiliations 1. Department of Mathematics Baoji University of Arts and Sciences Baoji People’s Republic of China 2. Department of Chemical and Petroleum Engineering, Schulich School of Engineering University of Calgary Alberta Canada 3. Center for Computational Geosciences, College of Mathematics and Statistics Xian Jiaotong University Xian People’s Republic of China