User-Friendly Solvers and Solver-Friendly Discretizations

在本建议中，将开发各种算法和理论，以部分实现开发用户友好的求解器和求解器友好的离散化的以下四个阶段的策略：（1）为少数基本求解器友好的系统（即离散泊松方程及其变体）开发用户友好的最佳求解器和相关理论;（2）扩展了求解器友好的偏微分方程列表（如离散的斯托克斯和麦克斯韦方程），通过减少他们的解决方案的一小部分基本的解决方案友好的系统（可应用最佳和用户友好的求解器）;（3）为更复杂的偏微分方程（系统）开发求解友好的离散化技术，使得离散化系统将加入求解友好的系统列表（例如用于Navier-Stokes方程、Johnson-Segalman方程和磁流体动力学方程的欧拉-拉格朗日方法）;以及（4）通过使用求解器友好的离散化来从一般离散化求解离散化系统（如果它本身不是一个令人满意的离散化来获得数值解）作为辅助离散化，可以用作预处理器或用于获得线性或非线性迭代的良好初始猜测的手段。这些技术将被开发，目的是使它们有效地解决复杂的问题，如非牛顿模型和燃料电池模型方程。并行实现将是这些算法设计中的一个主要考虑因素。理论问题--在过去的几十年里，研究人员花费了大量的精力来开发有效的迭代方法来求解离散偏微分方程。虽然这些努力已经产生了许多数学上的最优解，如多重网格方法，不幸的是，多重网格方法在实际应用中并没有得到太多的应用。这种理论与实践之间的显著差距主要是由于传统多重网格方法的脆弱性及其实现的复杂性。该提案旨在开发缩小这一差距的理论和技术，特别是通过开发在实践中强大且易于使用的数学最优解算器。拟议的研究将集中在一个综合的应用程序的用户友好的求解器和求解器友好的离散化的各种基本偏微分方程，在许多应用中出现的，因此，这个建议的结果预计将直接适用于计算和应用数学的许多领域。我们生产的求解器和离散化技术，包括数学算法，分析和软件，将为探索物理，化学和工程中的多尺度模型提供强大的工具。 通过附带的矩阵求解社区项目（http：//www.multigrid.org/solvers/），该提案的结果将产生及时和广泛的影响。拟议的项目也将产生强大的教育影响，因为它侧重于培养现代计算科学和跨学科应用的理论和实践方面的研究生。

In this proposal, various algorithms and theories will be developed to partially realize the following four-stage strategy for developing user-friendly solvers and solver-friendly discretizations: (1) develop user-friendly optimal solvers and relevant theories for a small number of basic solver-friendly systems, namely the discrete Poisson's equation and its variants; (2) extend the list of solver-friendly partial differential equations (such as the discrete Stokes and Maxwell equations) by reducing them to the solution of a handful of basic solver-friendly systems (for which optimal and user-friendly solvers can be applied); (3) develop solver-friendly discretization techniques for more complicated PDEs (systems) such that the discretized systems will join the list of solver-friendly systems (such as the Eulerian-Lagrangian method for the Navier-Stokes equation, the Johnson-Segalman equations, and the magnetohydrodynamics equations); and (4) solve the discretized system from a general discretization by using a solver-friendly discretization (if it is not a satisfactory discretization to obtain the numerical solution by itself) as an auxiliary discretization that can be used as a preconditioner or a means for obtaining a good initial guess for a linear or nonlinear iteration. These techniques will be developed with the purpose of making them effective for solving complicated problems such as non-Newtonian models and fuel cell model equations. Parallel implementations will be one major consideration in the design of these algorithms. Theoretical issues---such as the most fundamental open problem concerning the optimal convergence of algebraic multigrid methods---will be carefully investigated.Many problems in scientific and engineering computing can be reduced to the numerical solution of certain partial differential equations. Over the last few decades, researchers have expended significant effort on developing efficient iterative methods for solving discretized partial differential equations. Though these efforts have yielded many mathematically optimal solvers such as the multigrid method, the unfortunate reality is that multigrid methods have not been much used in practical applications. This marked gap between theory and practice is mainly due to the fragility of traditional multigrid methodology and the complexity of its implementation. This proposal aims to develop theories and techniques that will narrow this gap, specifically by developing mathematically optimal solvers that are robust and easy to use in practice. The proposed study will focus on an integrated application of user-friendly solvers and solver-friendly discretizations for various basic partial differential equations that arise in many applications; therefore, the results of this proposal are expected to be directly applicable in many areas of computational and applied mathematics. The solver and discretization techniques we produce, including mathematical algorithms, analyses, and software, will provide powerful tools for exploring multiscale models in physics, chemistry, and engineering. Through the accompanying Matrix-Solver Community Project (http://www.multigrid.org/solvers/), the results of this proposal will lead to timely and broad impacts. The proposed project will have a strong educational impact as well, as it focuses on training graduate students in theoretical and practical aspects of modern computational science and interdisciplinary applications.