Keywords: Time-dependent incompressible viscous flow, stable discretization, time splitting, Stokes pressure, Leray projection. Abstract: How to properly specify boundary conditions for pressure for no-slip incompressible viscous flow has been a longstanding issue in analysis and computation. A recent analytical resolution of this issue is based on a local well-posedness theorem for an extended Navier-Stokes dynamics, in which the zero-divergence condition is replaced by a pressure formula that involves the commutator of the Laplacian and Leray projection operators. I'll indicate progress on some related analytical questions (domains with corners, MHD), but will focus on improvements in numerical schemes that involve projection methods in time and finite elements in space. We find schemes that involve simple kinds of finite elements (Lagrange of equal order for velocity and pressure, including piecewise-linear, for example) that are stable in tests with large time steps at low Reynolds number, with up to 3rd-order accuracy in time for both velocity and pressure. Notably, these schemes do not include projection methods that update the pressure from previous steps.