Robust Multilevel Preconditioning Techniques for Elliptic PDE with Variable Coefficients

本论文的研究目的是设计一种用于求解二阶变系数椭圆型偏微分方程有限元离散化的鲁棒预条件子，并将其应用于麦克斯韦方程和H（div）方程的迭代求解。PI应开发，分析和实现变系数椭圆偏微分方程，特别是多尺度椭圆偏微分方程的近似最优迭代求解器。基于几个成功的初步调查，PI将设计鲁棒的多级预处理器的各种类型的有限元离散，如符合，不连续的Galerkin离散和混合制定的二阶椭圆型偏微分方程与一般变系数的结构和非结构化二分法网格。该方法将基于辅助空间预处理框架。这种技术将被证明是鲁棒的系数和网格大小的变化，解决一般的多尺度椭圆偏微分方程。该研究将允许一个选择不同的粗网格问题，而不是标准的变分粗网格问题的预处理。在获得各种有限元离散化的鲁棒预处理器后，PI将使用这些预处理器，使用辅助空间预处理技术开发变系数麦克斯韦方程和H（div）方程的鲁棒迭代求解器。这些算法将作为开源软件包实现，并将与特定领域的科学家合作使用。该项目通过软件开发在教育和数学、工程和物理等领域产生了广泛的影响。待开发的方法将有助于线性和非线性系统的数值方法的进步。研究成果将为油藏模拟、电磁计算等重要模型的探索提供有力的工具。该项目丰富了ISU数学系的研究生课程，特别是在偏微分方程和数值分析领域。研究应涉及本科生和研究生，PI应提供出色的指导。

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.