On the role of commutator arguments in the development of parameter-robust preconditioners for Stokes control problems
On the role of commutator arguments in the development of parameter-robust preconditioners for Stokes control problems
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关于换向器参数在斯托克斯控制问题的参数鲁棒预条件子开发中的作用
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发表时间：
2013
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通讯作者：
J. Pearson
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作者：
J. Pearson
用于偏微分方程约束优化问题的预处理器的开发是最近引起人们广泛关注的数值分析领域。特别研究的一类问题是斯托克斯控制问题，即以斯托克斯（或纳维-斯托克斯）方程作为约束来最小化泛函的问题。在这篇手稿中，我们提出了一种使用泊松控制问题的预处理器来预处理斯托克斯控制问题的方法，最重要的是，应用了换向器参数。这种方法导致了该问题的两个块对角预处理器，其中一个是由 W. Zulehner 于 2011 年使用非标准范数参数针对该鞍点问题导出的（SIAM. J. Matrix Anal. & Appl., v.32），另一个我们认为是新的。我们还使用相同的方法推导了两个相关的块三角预处理器，并给出了数值结果来证明四种预处理器在实践中的性能。
The development of preconditioners for PDE-constrained optimization problems is a field of numerical analysis which has recently generated much interest. One class of problems which has been investigated in particular is that of Stokes control problems, that is the problem of minimizing a functional with the Stokes (or Navier-Stokes) equations as constraints. In this manuscript, we present an approach for preconditioning Stokes control problems using preconditioners for the Poisson control problem and, crucially, the application of a commutator argument. This methodology leads to two block diagonal preconditioners for the problem, one of which was previously derived by W. Zulehner in 2011 (SIAM. J. Matrix Anal. & Appl., v.32) using a nonstandard norm argument for this saddle point problem, and the other of which we believe to be new. We also derive two related block triangular preconditioners using the same methodology, and present numerical results to demonstrate the performance of the four preconditioners in practice.