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Under the necessary compatibility condition and some mild regularity assumptions on the interior and the boundary data, we prove the existence, uniqueness, and

Abstract. We consider two-level finite element discretization methods for the stream function formulation of the Navier-Stokes equations. The two-level method consists of solving a small nonlinear system on the coarse mesh, then solving a linear system on the fine mesh. The basic result states that the errors between the coarse and fine meshes are related superlinearly. This paper demonstrates that the two-level method can be implemented to approximate efficiently solutions to the Navier-Stokes equations. Two fluid flow calculations are considered to test problems which have a known solution and the driven cavity problem. Stream function contours are displayed showing the main features of the flow. Key words. Two-level method, Navier-Stokes equations, finite element, stream function formulation, Reynolds number AMS subject classifications. 65N35, 76M30, 76D051. Introduction. The numerical treatment of nonlinear problems that arise in areas such as fluid mechanics often requires solving large systems of nonlinear equations. Many methods have been proposed that attempt to solve these systems efficiently; one such class of methods are two-level methods that reduce the computational time. The computational attractions of the methods are that they require the solution of only a small system of nonlinear equations on coarse mesh and one linear system of equations on fine mesh. Apparently, the two-level method was proposed first in [17,16,15] and used for semilinear elliptic problems. The method was implemented for the velocity-pressure formulation of the Navier-Stokes equations in [11,12,13] and for the stream function formulation of the Navier-Stokes equations in [8,18,7].The Navier-Stokes equations may be solved using either the primitive variable or stream function formulation. Here we use the stream function formulation. The attractions of the stream function formulation are that the incompressibility constraint is automatically satisfied, the pressure is not present in the weak form, and there is only one scalar unknown to solve for. The standard weak formulation of the stream function version first appeared in 1979 in [10]. In this direction, Cayco and Nicolaides [5,4,3] studied a general analysis of convergence for this standard weak formulation.The goal of this paper is to demonstrate that the two-level method can be implemented to approximate solutions for incompressible viscous flow problems with high Reynolds number.

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Abstract. We consider two-level finite element discretization methods for the stream function formulation of the Navier-Stokes equations. The two-level method consists of solving a small nonlinear system on the coarse mesh, then solving a linear system on the fine mesh. It is shown in [8] that the errors between the coarse and fine meshes are related superlinearly. This paper presents an algorithm for pressure recovery and a general analysis of convergence for the algorithm. The numerical example for the 2D driven cavity fluid is considered. Streamfunction contours are displayed showing the main features of the flow.