Multiple preconditioners for saddle-point and other problems

在科学和工程中，有许多问题涉及两种完全不同的现象，但这两种现象相互影响（或一个影响另一个）。这种情况的一个例子是流体沿着两种不同的介质流动。一种流动是自由的，如河流中的水，而另一种流动则受到一些多孔介质的限制，如河床下面渗出的流体。当人们试图描述这些物理现象时，除了要考虑它们的间期发生的情况外，还需要对这两个现象分别使用非常不同的方程。在这个项目中，PI将研究刚刚描述的一类方程组的解。PI应该结合为这两种现象开发的求解技术，以便最终的组合为我们提供有效的计算工具。PI希望对所谓的耦合斯托克斯-达西流现象提供更好的建模和求解技术，这种现象的微分方程被有限元离散，特别是在达西区域被不连续的伽辽金离散。PI建议使用新的精确和不精确约束类型（不定）前置条件。PI将研究谱（和值场）等效边界，这将为这些鞍点型问题的GMRES收敛提供网格无关边界。多预处理是PI想要开发的另一个解决鞍点等问题的工具。在多预条件GMRES的情况下，PI建议通过使用多元多项式提供收敛分析。在这种方法中，在每次迭代中，该方法以隐式方式计算两个前置条件的最优组合。对于某些鞍点问题，预期这种动态组合预处理比两种原始预处理中的任何一种都更有效。

There are many problems in science and engineering where one deals with two completely different phenomena, but these two phenomena influence one another (or one influences the other). One example of this situation is fluid flow along two different kinds of media. One flow is free, as in water in a river, and the other is constrained by some porous media, as the fluid seeping through below the river bed. When one tries to describe these physical phenomena, one needs to use very different equations for each of the two, in addition to taking into account what happens in their interphase. In this project, the PI shall study the solution of systems of equations of the kind just described. The PI shall combine solution techniques developed for each of the two phenomena, so that the resulting combination gives us an efficient computational tool.Among the phenomena to which the PI wants to contribute better modeling and solution techniques is the so-called coupled Stokes-Darcy flow, with the differential equations discretized by finite elements, and in particular by discontinuous Galerkin in the Darcy region. The PI proposes to use new exact and inexact constraint-type (indefinite) preconditioners. The PI will study spectral (and field-of-values) equivalence bounds, which should provide mesh independence bounds on the GMRES convergence for these problems of saddle-point type. Multipreconditioning is another tool that the PI wants to develop for the solution of saddle-point and other problems. In the case of Multipreconditioned GMRES, the PI proposes to provide convergence analysis through the use of multivariate polynomials. In this approach, at each iteration, the method computes in an implicit manner the optimal combination of two preconditioners. For certain saddle-point problems this dynamical combination preconditioning is expected to be more efficient than either of the two original preconditioners.