We propose two algorithms of two-level methods for resolving the nonlinearity in the stabilized finite volume approximation of the Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. A macroelement condition is introduced for constructing the local stabilized finite volume element formulation. Moreover the two-level methods consist of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. The error analysis shows that the two-level stabilized finite volume element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite volume element solution solving the Navier-Stokes equations on a fine mesh for a related choice of mesh widths. 1. Introduction We consider a two-level method for the resolution of the nonlinear system arising from finite volume discretizations of the equilibrium, incompressible Navier-Stokes equations: where is the velocity vector, is the pressure, is the body force, is the viscosity of the fluid, and , the flow domain, is assumed to be bounded, to have a Lipschitz-continuous boundary , and to satisfy a further condition stated in (H1). Finite volume method is an important numerical tool for solving partial differential equations. It has been widely used in several engineering fields, such as fluid mechanics, heat and mass transfer, and petroleum engineering. The method can be formulated in the finite difference framework or in the Petrov-Galerkin framework. Usually, the former one is called finite volume method [1, 2], MAC (marker and cell) method [3], or cell-centered method [4], and the latter one is called finite volume element method (FVE) [5–7], covolume method [8], or vertex-centered method [9, 10]. We refer to the monographs [11, 12] for general presentations of these methods. The most important property of FVE is that it can preserve the conservation laws (mass, momentum, and heat flux) on each control volume. This important property, combined with adequate accuracy and ease of implementation, has attracted more people to do research in this field. On the other hand, the two-level finite element strategy based on two finite element spaces on one coarse and one fine mesh has been widely studied for steady semilinear elliptic equations [13, 14] and the Navier-Stokes equations [15–22]. For the finite volume element method, Bi and Ginting [23] have studied two-grid finite volume element method for linear and nonlinear elliptic problems; Chen et al. [24] have applied two-grid methods for
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