Robust Least Squares Discretization for Mixed Variational Formulations

该项目旨在为科学和工程中的各种应用提供更可靠，快速和准确的模拟方法，例如电磁学，弹性和声学，流体流动以及通过非均匀多孔介质的扩散。该项目侧重于基于有限元分析的高效计算方法。具体的应用包括模拟可压缩气体动力学和计算流体力学的大气预测和海洋流体流动行为，以及计算解决方案的时间谐波麦克斯韦模型与纳米光学和模拟信号包中的应用。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，分析和实现用于解决偏微分方程（PDE）的鲁棒和高效的数值算法，这些偏微分方程允许具有不同类型的测试和试验空间的变分公式。当近似这些偏微分方程的解决方案，它是可取的，以获得所有物理量的参数，如扩散系数或频率的存在下，鲁棒估计。即使在边界附近或沿着材料不连续性的低规则性的情况下，或者在低数据规则性的情况下，获得解的良好近似也是重要的。该项目的重点是近似偏微分方程模型的参数和不连续系数。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处理椭圆对称问题领域之间架起了桥梁。该项目旨在：引入一个新的鞍点最小二乘理论，允许非协调的试验空间近似不连续解的偏微分方程;构造局部光滑投影型的试验空间，导致更高阶的近似解或相关的感兴趣的量;引入最佳测试空间和平衡范数，允许参数问题的鲁棒近似;构造了一般混合变分方程的有效预处理技术。该研究将拓宽数学理论和混合有限元近似领域的应用范围，并将在混合变分公式、自适应和多级技术以及预处理参数或分数范数之间建立新的联系。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.