A new approach in analysis and computation of Navier-Stokes equation in bounded domain For incompressible Navier-Stokes equations in a bounded domain, I will present a formula for the pressure that involves the commutator of the Laplacian and Leray-Helmholtz projection operators. And I will then show that this commutator and hence the pressure is strictly dominated by the viscous term at leading order. As a consequence, I will present a simple well-posedness theorem for an extended Navier-Stokes dynamics unconstrained by the divergence-free condition. The main talk will be on application of this new approach in analyzing and designing efficient numerical methods for incompressible flow, particularly high order projection methods and numerical methods based on pressure Poisson equation.