Projection Method I: Convergence and Numerical Boundary Layers

This is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations. The purpose of these papers is to provide a thorough understanding of the numerical phenomena involved in the projection methods, particularly when boundaries are present, and point to ways of designing more efficient, robust, and accurate numerical methods based on the primitive variable formulation. This paper contains the following topics: 1. convergence and optimal 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 projection step. We will show that a different choice of boundary conditions gives rise to different boundary layer structures. In particular, the straightforward Dirichlet boundary condition for the pressure leads to O (1) numerical boundary layers in the pressure and deteriorates the accuracy in the interior; and 4. postprocessing the numerical solutions to get more accurate approximations for the pressure.