Multigrid Methods for a Class of Saddle Point Problems

一类鞍点问题的多重网格方法

• 批准号: 1418934
• 负责人: Long Chen
• 金额: $ 20.5万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418934&HistoricalAwards=false
• 关键词: Multigrid; Methods; Class; Saddle; Point

The fast multigrid methods developed and studied in this work are expected to have a broader impact on the numerical solutions of a large class of practical problems. Important applications include: vector (Hodge) Laplacian, Maxwell equations, Stokes equations, Oseen and Navier-Stokes equations, and Magnetohydrodynamics (MHD) etc. MHD, in particular, has important applications in the development of fusion technology and casting processes. In these applications, since no experimentation is nowadays possible, the numerical simulation of the corresponding partial differential equations is indispensable. These simulations are very challenging, requiring large computational resources. The multigrid solvers developed in this project offer the potential for increasingly accurate models to be solved. In addition, our improvements in algorithm developments will have impact on many other areas, such as image processing, and computer graphics.This project is divided into two parts: algorithmic development and theoretical analysis. For the algorithmic development, multigrid solvers will be developed for mixed finite element discretization based on Finite Element Exterior Calculus (FEEC). In our study, effective smoothers, which are the key of multigrid methods, will be developed by using existing effective preconditioners or splitting schemes. One such example is a distributive smoother proposed in this project which is highly related to the well-known projection methods used in computational fluid dynamic. In addition to the algorithmic development, a more completed convergence theory of multigrid methods for saddle point problems will be developed. This theory aims to relax the strong regularity assumption in existing work. Consequently our theory can be applied to more realistic problems especially for solutions with singularities. Our theoretical investigation will also provide insight for the algorithmic development, e.g., the construction of approximated distributive smoothers and Schwarz smoothers, and optimal choice of relaxation parameters used in several smoothers.

预计在这项工作中开发和研究的快速多机方法将对大量实际问题的数值解决方案产生更大的影响。重要的应用包括：矢量（Hodge）拉普拉斯，麦克斯韦方程，stokes方程，Oseen和Navier-Stokes方程以及磁水动力学（MHD）等。MHD在融合技术和铸造过程的开发中具有重要的应用。在这些应用中，由于如今没有实验，因此对相应的部分微分方程的数值模拟是必不可少的。这些模拟非常具有挑战性，需要大量的计算资源。该项目中开发的多机求解器为越来越精确的模型提供了解决。此外，我们在算法开发方面的改进将对许多其他领域（例如图像处理和计算机图形）产生影响。该项目分为两个部分：算法开发和理论分析。对于算法开发，将基于有限元外观演算（FEEC）开发多族求解器，以用于混合有限元离散化。在我们的研究中，将通过使用现有有效的预处理或拆分方案来开发有效的Smoothorts，这是多方方法的关键。一个这样的例子是该项目中提出的分布式更光滑，该分布与计算流体动态中使用的众所周知的投影方法高度相关。除了算法开发外，还将开发出更完整的鞍点问题的收敛理论。该理论旨在放松现有工作中强有力的规律性假设。因此，我们的理论可以应用于更现实的问题，尤其是用于具有奇异性的解决方案。我们的理论研究还将为算法开发提供洞察力，例如，构建近似的分布型Smoother和Schwarz Smoothers，以及在几种Smoothers中使用的放松参数的最佳选择。