Convergence of a Ramshaw-Mesina Iteration
<|reference_start|>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 \begin{equation} \left\{ \begin{array} [c]{cc} Step\ 1: & -\triangle u^{n+1}+\nabla p^{n}=f(x),\ {\rm in}\ \Omega,\ u^{n+1}|_{\partial\Omega}=0,\\ Step\ 2: & p^{n+1}-p^{n}+\beta\nabla\cdot(u^{n+1}-u^{n})+\alpha ^{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.<|reference_end|>
@article{çibik2024convergence,
    title={Convergence of a Ramshaw-Mesina Iteration},
    author={Aytekin c{C}ibik and William Layton},
    journal={arXiv preprint arXiv:2403.04702},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.04702},
    primaryClass={math.NA cs.NA}
}