PROJECTION METHOD .1. CONVERGENCE AND NUMERICAL BOUNDARY-LAYERS

This is the first of a series of papers on the subject of projection m ethods for viscous incompressible flow calculations. The purpose of th ese papers is to provide a thorough understanding of the numerical phe nomena involved in the projection methods, particularly when boundarie s are present, and point to ways of designing more efficient, robust, and accurate numerical methods based on the primitive variable formula tion. This paper contains the following topics: 1. convergence and opt imal error estimates for both velocity and pressure up to the boundary ; 2. explicit characterization of the numerical boundary layers in the pressure approximations and the intermediate velocity fields; 3. the effect of choosing different numerical boundary conditions at the proj ection step. We will show that a different choice of boundary conditio ns gives rise to different boundary layer structures. In particular, t he straightforward Dirichlet boundary condition for the pressure leads to O(1) numerical boundary layers in the pressure and deteriorates th e accuracy in the interior; and 4. postprocessing the numerical soluti ons to get more accurate approximations for the pressure.