Convergence of a Ramshaw-Mesina Iteration
Aytekin Çibik, William Layton
TL;DR
The paper analyzes a Ramshaw–Mesina type iteration that replaces the Uzawa pressure update in saddle-point formulations with a discretized penalty/artificial-compression scheme for the Stokes problem. By casting the method in a Schur-complement framework and leveraging energy estimates, it proves convergence under the condition $β+α^{2} < 1/M$, where $M$ bounds the Schur operator, and demonstrates this with a numerical study on a lid-driven cavity. The findings indicate the RM approach behaves similarly to Uzawa in terms of convergence, while suggesting that any dramatic benefits may stem from explicit time stepping rather than the core iteration alone. The work points to future exploration of Step 1 modifications, such as a first-order Richardson step, to better understand potential improvements in Step 2 dynamics.
Abstract
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 \begin{equation} \left\{ \begin{array} [c]{cc} Step\ 1: & -\triangle u^{n+1}+\nabla p^{n}=f(x),\ {\rm in}\ Ω,\ u^{n+1}|_{\partialΩ}=0,\\ Step\ 2: & p^{n+1}-p^{n}+β\nabla\cdot(u^{n+1}-u^{n})+α^{2}\nabla\cdot u^{n+1}=0. \end{array} \right. \end{equation} For saddle point problems, including Stokes, this iteration converges under a condition similar to the one required for Uzawa iteration.