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Robust Least Squares Discretization for Mixed Variational Formulations

混合变分公式的稳健最小二乘离散化

基本信息

批准号：
2011615
负责人：
Constantin Bacuta
金额：
$ 21.5万
依托单位：
University of Delaware
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2011615&HistoricalAwards=false
关键词：
Robust Least Squares Discretization Mixed

项目摘要

This project aims to enable more reliable, fast, and accurate simulation methods for a variety of applications in science and engineering, such as electromagnetism, elasticity and acoustics, fluid flow, and diffusion through heterogenous porous media. The project focuses on efficient computational methods based on finite element analysis. Specific applications include modeling compressible gas dynamics and computational fluid mechanics for atmospheric prediction and ocean fluid flow behavior, as well as computational solution of the time-harmonic Maxwell model with applications in nano-optics and analog signal packages. This project will provide interdisciplinary applied mathematics training and research experiences for students.The project will develop, analyze, and implement robust and efficient numerical algorithms for solving partial differential equations (PDEs) that admit variational formulations with different types of test and trial spaces. When approximating the solutions of these PDEs, it is desirable to obtain robust estimates of all physical quantities in the presence of parameters, such as diffusion coefficient or frequency. It is also important to obtain good approximations of the solution, even in the case of low regularity near boundaries or along material discontinuities, or in the case of low data regularity. The focus of the project is on approximating PDE models with parameters and discontinuous coefficients. The project develops a general discretization and algorithm development method that bridges between the field of symmetric saddle point problems and the field of preconditioning elliptic symmetric problems. The project aims to: introduce a new saddle point least-squares theory that allows non-conforming trial spaces to approximate discontinuous solutions of PDEs; construct locally smooth projection type of trial spaces that lead to higher order of approximation for the solution or related quantities of interest; introduce optimal test spaces and balanced norms that allow robust approximation for parametric problems; and construct efficient preconditioning techniques for general mixed variational formulations. The research will broaden the mathematical theory and the range of applications of the mixed finite-element approximation field and will create new connections among mixed variational formulations, adaptive and multilevel techniques, and preconditioning parametric or fractional norms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目旨在为科学和工程中的各种应用提供更可靠、更快速和更准确的模拟方法，如电磁学、弹性和声学、流体流动和在非均匀多孔介质中的扩散。该项目专注于基于有限元分析的高效计算方法。具体应用包括为大气预报和海洋流体流动行为模拟可压缩气体动力学和计算流体力学，以及在纳米光学和模拟信号包中应用时间谐和麦克斯韦模型的计算解。该项目将为学生提供跨学科的应用数学培训和研究经验。该项目将开发、分析和实施稳健和高效的数值算法来解决偏微分方程(PDE)，这些偏微分方程允许使用不同类型的测试和测试空间的变分公式。当近似这些偏微分方程组的解时，希望在存在参数的情况下获得所有物理量的稳健估计，例如扩散系数或频率。获得解的良好近似也很重要，即使在边界附近或沿材料不连续的低规则性的情况下，或者在数据规则性低的情况下也是如此。该项目的重点是用参数和不连续系数来逼近PDE模型。该项目开发了一种通用的离散化和算法开发方法，在对称鞍点问题领域和预条件椭圆对称问题领域之间架起了桥梁。该项目的目的是：引入一个新的鞍点最小二乘理论，允许非协调试探空间逼近偏微分方程组的间断解；构造局部光滑投影型试探空间，从而获得解或相关感兴趣的量的高阶逼近；引入最优测试空间和平衡范数，允许稳健逼近参数问题；以及为一般混合变分公式构造有效的预条件技术。这项研究将拓宽混合有限元近似领域的数学理论和应用范围，并将在混合变分公式、自适应和多水平技术以及预条件参数或分数规范之间建立新的联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。