Multiple preconditioners for saddle-point and other problems

在科学和工程中有许多问题，其中一个人处理两个完全不同的现象，但这两个现象相互影响(或一个影响另一个)。这种情况的一个例子是流体沿两种不同类型的介质流动。一种流动是自由的，就像在河流中的水一样，另一种流动受到一些多孔介质的限制，因为流体在河床下渗入。当人们试图描述这些物理现象时，除了考虑到它们的相间发生的情况外，还需要对这两种现象中的每一种使用非常不同的方程。在这个项目中，PI将研究刚才描述的那种方程组的解。PI应该结合为这两个现象中的每一个开发的求解技术，以便所产生的组合给我们一个有效的计算工具。在PI希望对其做出更好建模和求解技术的现象中，是所谓的耦合Stokes-Darcy流，微分方程组由有限元离散，特别是通过Darcy区域中的不连续Galerkin离散。PI建议使用新的精确和不精确约束型(不定)预条件子。PI将研究谱(和值域)等效界，这将为这些鞍点型问题的GMRES收敛提供网格无关界。多重预处理是PI为解决鞍点等问题而开发的另一种工具。在多预条件GMRES的情况下，PI建议通过使用多元多项式来提供收敛分析。在这种方法中，在每一次迭代中，该方法以隐式方式计算两个预条件的最优组合。对于某些鞍点问题，这种动态组合预处理法有望比两种原始预处理器中的任何一种更有效。

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.