Convergence of a Ramshaw-Mesina iteration

In 1991 Ramshaw and Mesina introduced a clever synthesis of penalty methods and artificial compression methods. Its form makes it an interesting option to replace the pressure update in the Uzawa iteration. The result, for the Stokes problem, is

{Step1 : - Delta u(n+1) + del p(n) = f(x), in Omega, u(n+1)vertical bar(partial derivative Omega) = 0,

Step2 : p(n+1) - p(n) + beta del center dot (u(n+1) - u(n)) + alpha(2) del center dot u(n+1) = 0. (1)

For saddle point problems, including Stokes, this iteration converges under a condition similar to the one required for Uzawa iteration.