Close Evaluation of Layer Potentials

本研究计画针对边界积分方程法在数值求解边值问题时所面临的几个突出问题。除了推进计算数学领域的知识，这项研究的目的是导致创新，将提供一个有效和准确的研究计算平台，补充正在进行的实验研究，在流体力学和电磁学。该项目包括一个跨学科的，以团队为基础的方法与研究生培训和教育的组成部分，也将作为招聘工具，以进一步促进妇女和代表性不足的少数民族研究生在科学和工程领域。该项目的目的是介绍，分析，并实施新的高阶数值方法，以密切评估层的潜力。在边界积分方程方法中，层势被用来表示线性椭圆型偏微分方程的解。封闭评估问题是指高阶求积规则在评估域边界附近的点处的层电位时产生的大误差，尽管在域中的其他地方高度准确。解决密切评价问题是重要的许多应用，包括斯托克斯流动问题和等离子体激元领域。目前的一个挑战是制定有效的，易于实施的方法，解决密切评价问题的三个方面。这个项目将通过使用渐近分析来识别和分析这些积分算子的近奇异行为来解决这个挑战。本项目的主要目标是（1）开发新的方法来解决封闭评估问题，（2）分析误差并研究效率和计算问题，以及（3）将方法扩展到其他线性椭圆型偏微分方程。该项目的成果有望开发出新颖、准确、高效的计算方法，以解决近距离评估问题。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知识价值和更广泛的影响审查标准进行评估来支持。

This research project addresses several outstanding questions in the numerical solution of boundary value problems using boundary integral equation methods. In addition to advancing knowledge within the field of computational mathematics, this research aims to lead to innovations that will provide an efficient and accurate research computing platform complementing ongoing experimental research in fluid mechanics and electromagnetics. The project includes an interdisciplinary, team-based approach with graduate training and education components that will also serve as recruiting tools to further promote women and underrepresented minority graduate students in science and engineering fields.The purpose of this project is to introduce, analyze, and implement new high-order numerical methods for the close evaluation of layer potentials. Layer potentials are used to represent solutions of linear elliptic partial differential equations in boundary integral equation methods. The close-evaluation problem refers to large errors incurred by high-order quadrature rules when evaluating layer potentials at points near the boundary of the domain despite being highly accurate elsewhere in the domain. Addressing the close-evaluation problem is important for many applications, including Stokes flow problems and the field of plasmonics. A current challenge is to develop efficient, easily-implementable methods that address the close-evaluation problem in three dimensions. This project will address the challenge by identifying and analyzing the nearly-singular behavior of these integral operators using asymptotic analysis. The main objectives of this project are to (1) develop new methods to address the close-evaluation problem, (2) analyze the error and investigate efficiency and computational issues, and (3) extend the methods to other linear elliptic partial differential equations. Results from this project are expected to enable development of novel, accurate, and efficient computational methods that address the close-evaluation problem.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.