Robust Multilevel Preconditioning Techniques for Elliptic PDE with Variable Coefficients

变系数椭圆偏微分方程的鲁棒多级预处理技术

• 批准号: 1319110
• 负责人: Yunrong Zhu
• 金额: $ 8.56万
• 依托单位: Idaho State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1319110&HistoricalAwards=false
• 关键词: Robust; Multilevel; Preconditioning; Techniques; Elliptic

The proposed research is to design robust preconditioners for solving the finite element discretization of second order elliptic PDEs with variable coefficients and then apply these preconditioners in developing efficient iterative solvers for Maxwell's equations and H(div) equations. The PI shall develop, analyze and implement nearly optimal iterative solvers for elliptic PDE with variable coefficients, especially multiscale elliptic PDEs. Based on several successful preliminary investigations, the PI will design robust multilevel preconditioners for various types of finite element discretization, such as conforming, nonconforming, discontinuous Galerkin discretization and mixed formulation of second order elliptic PDEs with general variable coefficients on both structured and unstructured bisection grids. The approach will be based on the auxiliary space preconditioning framework. This technique will be proved to be robust with respect to both the variations in the coefficients and the grid size for solving general multiscale elliptic PDEs. The research will allow one to choose different coarse grid problems other than the standard variational coarse grid problems in the preconditioners. Upon obtaining robust preconditioners for various finite element discretizations, the PI will then use these preconditioners to develop robust iterative solvers for Maxwell's equations and H(div) equations with variable coefficients using the auxiliary space preconditioning techniques. The algorithms will be implemented as open source software packages which will be used in collaborations with domain-specific scientists. This project has broad impact in education and other areas of mathematics, engineering, and physics through software development. The methods to be developed will contribute to the advancement of numerical methods for both linear and nonlinear systems. The research results will provide powerful tools for the exploration of important models such as reservoir simulations and electromagnetic computation. The project enriches the graduate program in the Department of Mathematics at ISU, especially in the area of partial differential equations and numerical analysis. The research shall involve undergraduate and graduate students, and excellent mentoring shall be provided by the PI.

拟议的研究是设计可靠的预处理程序，以解决具有可变系数的二阶椭圆PDE的有限元离散化，然后将这些预调节器应用于Maxwell的方程式和H（DIV）方程式的有效迭代求解器。 PI应开发，分析和实施具有可变系数的椭圆PDE，尤其是多尺度椭圆PDE的最佳迭代求解器。基于几项成功的初步研究，PI将为各种有限元的离散化设计可靠的多级预处理器，例如符合，不合格的，不连续的Galerkin离散化，以及对二阶椭圆形PDE的混合配方，并具有与结构和非结构性双级远距离电网上的一般可变系数。该方法将基于辅助空间预处理框架。对于求解一般多尺度椭圆PDES的系数和网格大小的变化，该技术将被证明是可靠的。这项研究将使人们可以选择不同的粗网格问题，而不是预处理器中的标准变化粗网格问题。在为各种有限元离散化获得可靠的预处理后，PI将使用这些预调节器为Maxwell的方程式和H（div）方程开发可变系数的稳健迭代求解器，并使用辅助空间预处理技术具有可变系数。该算法将作为开源软件包实现，该软件包将与特定于领域的科学家合作使用。该项目通过软件开发对教育以及数学，工程和物理的其他领域产生了广泛的影响。要开发的方法将有助于线性和非线性系统的数值方法的发展。研究结果将为探索重要模型（例如储层模拟和电磁计算）提供强大的工具。该项目丰富了ISU数学系的研究生课程，尤其是在部分微分方程和数值分析领域。该研究应涉及本科生和研究生，PI应提供出色的指导。