This dissertation focuses on the development, analysis, and implementation of numerical methods for both steady and unsteady incompressible Stokes and Navier-Stokes equations. Unlike existing methods based on the velocity-pressure formulation, I investigate a new, accurate, robust, and highly efficient numerical methods based on a novel pseudostress-velocity formulation. The well-posedness of the systems is established and error bonds for both time and space discretization are obtained. The pseudostress and the velocity are approximated by a stable pair of finite elements: Raviart-Thomas (RT) elements of index k ≥ 0 and discontinuous piecewise polynomials of degree k ≥ 0, respectively. The pseudostress system from the discretization is solved by the H (div) type of multigrid method, and the velocity is then calculated explicitly. Some numerical examples and comparison are presented too.
