# Optimization of the Hermitian and Skew-Hermitian Splitting Iteration for Saddle-Point Problems

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## Abstract

We study the asymptotic rate of convergence of the alternating Hermitian/skew-Hermitian iteration for solving saddle-point problems arising in the discretization of elliptic partial differential equations. By a careful analysis of the iterative scheme at the continuous level we determine optimal convergence parameters for the model problem of the Poisson equation written in div-grad form. We show that the optimized convergence rate for small mesh parameter *h* is asymptotically 1−O(*h*^{1/2}). Furthermore we show that when the splitting is used as a preconditioner for a Krylov method, a different optimization leading to two clusters in the spectrum gives an optimal, *h*-independent, convergence rate. The theoretical analysis is supported by numerical experiments.

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