Robust Multilevel Preconditioning Techniques for Elliptic PDE with Variable Coefficients

本文的研究目的是设计用于求解二阶变系数椭圆型偏微分方程组有限元离散的稳健预条件算子，并将其应用于麦克斯韦方程和H(Div)方程的高效迭代求解器的开发。PI应为变系数椭圆型偏微分方程组，特别是多尺度椭圆型偏微分方程组开发、分析和实现近乎最优的迭代求解器。在几个成功的初步研究的基础上，PI将为各种类型的有限元离散设计稳健的多层预条件，例如协调、非协调、间断Galerkin离散以及结构和非结构二等分网格上具有普通变系数的二阶椭圆型偏微分方程组的混合格式。该方法将基于辅助空间预适应框架。对于求解一般的多尺度椭圆型偏微分方程组，这种方法对于系数的变化和网格大小都是稳健的。这项研究将允许人们选择不同的粗网格问题，而不是预条件中的标准变分粗网格问题。在获得各种有限元离散的稳健预条件后，PI将使用这些预条件来利用辅助空间预条件技术来开发变系数Maxwell方程和H(Div)方程的稳健迭代求解器。这些算法将以开源软件包的形式实施，将用于与特定领域的科学家的合作。该项目通过软件开发在教育以及数学、工程和物理的其他领域产生了广泛的影响。所发展的方法将有助于推进线性和非线性系统的数值方法。研究成果将为油藏模拟、电磁计算等重要模型的勘探提供有力的工具。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式和数值分析领域。研究应包括本科生和研究生，并由PI提供优秀的指导。

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.