Advances in Multilevel Methods for Saddle Point Problems

尽管当今计算机的计算能力快速增长，但发展更快的方程模型算法对于当今在科学和工程中实际应用中所研究的现象的模拟是至关重要的。对于科学世界中的大范围情况，计算机模拟必须取代实验模拟和数据收集。对于计算机模拟，为了获得感兴趣的物理量的准确近似，必须将强大的计算系统与快速可靠的算法相结合来测试和确认所设计的模型。该项目将有助于构建和实施新的、有效的算法来解决具有数值挑战性的问题。特别是，拟议的研究将专注于构建快速和健壮的算法来解决计算流体力学和电磁问题。该项目的突破性方法是基于数值分析的最新结果、新算法的设计和实施以及对新计算工具的科学测试和验证。该项目中的时间谐和麦克斯韦方程的方法在纳米光学和模拟信号封装中有广泛的应用。求解变分问题的工作在电网优化和图像恢复中具有科学和技术应用价值。这项研究将研究和开发可靠和高效的数值算法来求解可以描述为变分鞍点系统或混合变分公式的偏微分方程组。该项目的目标是建立稳健的算法，在由于数据或系数的不连续而导致解的正则性较低的情况下求解此类方程。这项研究将对一大类鞍点问题进行严格和系统的分析，重点是高效的算法开发和测试。这些方法将基于在多层次和自适应选择近似空间的背景下的有限元离散算法。PI方法对偏微分方程组使用了一种新的鞍点最小二乘离散化方法，该方法充分利用了解的正则性，并采用了一种有效的水平变换准则来最小化全局迭代过程的运行时间。这项研究将为有限元方法在科学和工程领域的各种应用提供更可靠的方法，例如那些对弹性、电磁、摩擦和计算流体动力学感兴趣的人。研究结果将在该领域内分享，并与包括数学和科学高中教师在内的更广泛的受众分享。

In spite of the fast increase in computational power of today's computers, the development of faster algorithms for equational models is paramount for the simulation of phenomena investigated nowadays in practical applications in science and engineering. For a large spectrum of situations in the scientific world, a computer simulation has to replace experimental simulation and data collection. For a computer simulation, in order to obtain accurate approximation of the physical quantities of interest, a powerful computational system has to be combined with a fast and reliable algorithm to test and to confirm the designed models. The project will contribute to the construction and implementation of new and efficient algorithms for solving numerically challenging problems. In particular, the proposed research will focus on constructing fast and robust algorithms for solving computational fluid dynamics and electromagnetic problems. The breakthrough approach of the project is based on recent results in numerical analysis, on new algorithm designing and implementation, and on scientific testing and validation of the new computational tools. The methodology from this project for the time-harmonic Maxwell's equations has a broad range of applications in nano-optics and analog signal packages. The work for solving variational problems has scientific and technical applications in optimization of electrical networks and image restoration. The research will study and develop reliable and efficient numerical algorithms for solving partial differential equations that can be described as variational saddle point systems or mixed variational formulations. The goal of the project is to build robust algorithms for solving such equations in the presence of low regularity of solutions due to discontinuities in data or coefficients. The research will produce a rigorous and systematic analysis of a large class of saddle point problems, focusing on efficient algorithm development and testing. The methods will be based on finite element discretization algorithms in the context of a multilevel and adaptive choice of approximation spaces. The PI's approach uses a new saddle point least-squares type of discretization for systems of PDEs, that takes full advantage of the regularity of the solution and involves an efficient level change criterion that minimizes the running time of the global iterative process. This study will lead to more reliable methods for a variety of applications of the finite element method to science and engineering communities, such as those interested in elasticity, electromagnetism, friction, and computational fluid dynamics. The research findings will be shared within the field and with a more general audience including mathematics and science high-school teachers.