Multiple preconditioners for saddle-point and other problems

在科学和工程中有许多问题，其中一个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沿着两种不同介质流动。一种流动是自由的，如河流中的水，另一种则受到某些多孔介质的限制，如渗透到河床下面的流体。当人们试图描述这些物理现象时，除了考虑到在它们的相间发生的事情之外，还需要对两者中的每一个使用非常不同的方程。在这个项目中，PI将研究刚刚描述的那种方程组的解。PI应结合联合收割机的解决方案技术开发的两种现象中的每一个，使所产生的组合给我们一个有效的计算工具。在现象，其中PI要贡献更好的建模和解决方案技术是所谓的耦合斯托克斯-达西流，与微分方程离散的有限元，特别是不连续Galerkin在达西区。PI建议使用新的精确和不精确的约束类型（不定）预条件。PI将研究谱（和值域）等价界，这应该为这些鞍点类型问题的GMRES收敛提供网格独立界。多重预处理是PI希望开发的另一种工具，用于解决鞍点和其他问题。在多重预处理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预处理器的最佳组合。对于某些鞍点问题，这种动态组合预处理，预计将是更有效的比原来的两个预条件。

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.