New Approaches in Solving Saddle Point Problems

这一研究项目为开发、分析和实现偏微分方程(PDE)的更快、更有效的计算方法提供了计划，包括二阶椭圆型问题、Stokes和Navier-Stokes系统、弹性问题和Maxwell方程。偏微分方程组在数学、物理、化学、生物和工程等领域有着广泛的应用。求解Stokes和Navier-Stokes系统的新方法将有助于对规模和复杂性越来越大的流动问题进行建模，例如对平面附近的气流进行建模或对浸没物体附近的流动进行建模。麦克斯韦方程的新方法在光子学、器件计算模式和雷达散射中都有实际应用。研究将集中在两个方面：鞍点问题的求解和非匹配网格上的离散化。为了建立新的高效算法，PI将他在解决鞍点问题方面的新想法与来自不同数值分析领域的已知方法相结合，如迭代方法、多层方法和椭圆型偏微分方程组的自适应方法。主要技术将基于PI发现的鞍点系统的新光谱结果。所提出的求解鞍点系统的工作在最优化、最优控制、计算流体力学、线弹性、电磁学、电网络、统计学中的线性模型和图像恢复等方面具有科学和技术应用。第二个研究领域将是在基于单位划分方法的非匹配网格离散的背景下，研究多级离散和多级预适应技术。它的应用是模拟复杂物体附近的流体和气体流动以及多孔介质中的流体流动。

This research project provides a plan for the development, analysis, and implementation of faster and more efficient computational methods for partial differential equations (PDEs), including second-order elliptic problems, Stokes and Navier-Stokes systems, elasticity problems, and Maxwell's equations. PDEs have applications to many fields such as mathematics, physics, chemistry, biology, and engineering. The new methods for solving Stokes and Navier-Stokes systems will contribute to modeling flow problems of increasing size and complexity, e.g., modeling the air flow near a plane or modeling the flow near immersible objects. The new approach for Maxwell's equations has practical applications to photonics in computing modes for devices and in radar scattering. The research will focus on two areas: solving saddle-point problems and discretization on non-matching grids. To build new and efficient algorithms, the PI will combine his new ideas on solving saddle-point problems with already known methods from distinctive fields of numerical analysis such as iterative methods, multilevel methods, and adaptive methods for elliptic PDEs. The main technique will be based on the new spectral results for saddle-point systems found by the PI. The proposed work for solving saddle-point systems has scientific and technical applications in optimization, optimal control, computational fluid dynamics, linear elasticity, electromagnetism, electrical networks, linear models in statistics, and image restoration. A second area of research will be to investigate multilevel discretizations and multilevel preconditioning techniques in the context of discretizations on nonmatching grids based on the Partition of Unity method. The applications are to modeling fluid and gas flow near complex objects and fluid flow in porous media.