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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Benzi, M., Gander, M.J. & Golub, G.H. Optimization of the Hermitian and Skew-Hermitian Splitting Iteration for Saddle-Point Problems. BIT Numerical Mathematics 43, 881–900 (2003). https://doi.org/10.1023/B:BITN.0000014548.26616.65
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DOI: https://doi.org/10.1023/B:BITN.0000014548.26616.65