A finite volume method based on stabilized finite element for the two-dimensional stationary Navier-Stokes equations is analyzed. For the element, we obtain the optimal error estimates of the finite volume solution and . We also provide some numerical examples to confirm the efficiency of the FVM. Furthermore, the effect of initial value for iterative method is analyzed carefully. 1. Introduction In paper [1], G. He and Y. He introduce a finite volume method (FVM) based on the stabilized finite element method for solving the stationary Navier-Stokes problem and obtain the optimal error estimates for discretization velocity, however, to our dismay, without the optimal error estimate. It is inspiring that the following further numerical examples tell us that it has nearly second-order convergence rate. So, in this paper, we introduce a new technique to prove the optimal error of a generalized bilinear form and then gain the optimal error estimates of the stabilized finite volume method for the stationary Navier-Stokes problem. For the convenience of analysis, we introduce the following useful notations. Let be a bounded domain in assumed to have a Lipschitz continuous boundary and to satisfy a further smooth condition to ensure the weak solution's existence and regularity of Stokes problem. (For more information, see the A1 assumption stated in [1, 2].) We consider the stationary Navier-Stokes equations where represents the velocity vector, the pressure, the prescribed body force, and the viscosity. For the mathematical setting of problem (1.1), we introduce the following Hilbert spaces: The spaces ( ) are endowed with the usual -scalar product and norm , as appropriate. The space is equipped with the scalar product and norm . Define , which is the operator associated with the Navier-Stokes equations. It is positive self-adjoint operator from onto , so, for , the power of is well defined. In particular, , , and for all . We also introduce the following continuous bilinear forms and on and , respectively, by a generalized bilinear form on by and a trilinear form on by Under the above notations, the variational formulation of the problem (1.1) reads as follows: find such that for all : The following existence and uniqueness results are classical (see [3]). Theorem 1.1. Assume that and satisfy the following uniqueness condition: where Then the problem (1.7) admits a unique solution such that 2. FVM Based on Stabilized Finite Element Approximation In this section, we consider the FVM for two-dimensional stationary incompressible Navier-Stokes equations (1.1).
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