  We consider a projection method for time-dependent incompressible
Navier-Stokes equations with a total pressure boundary condition. The
projection method is one of the numerical calculation methods for
incompressible viscous fluids often used in engineering. In general, the
projection method needs additional boundary conditions to solve a
pressure-Poisson equation, which does not appear in the original Navier-Stokes
problem. On the other hand, many mechanisms generate flow by creating a
pressure difference, such as water distribution systems and blood circulation.
We propose a new additional boundary condition for the projection method with a
Dirichlet-type pressure boundary condition and no tangent flow. We demonstrate
stability for the scheme and establish error estimates for the velocity and
pressure under suitable norms. A numerical experiment verifies the theoretical
convergence results. Furthermore, the existence of a weak solution to the
original Navier-Stokes problem is proved by using the stability.
