Close Evaluation of Layer Potentials

本研究项目利用边界积分方程方法解决了边值问题数值解中的几个突出问题。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，补充正在进行的流体力学和电磁学的实验研究。该项目包括一个跨学科、以团队为基础的方法，其中包括研究生培训和教育内容，这也将作为招聘工具，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的发展。本课题的目的是介绍、分析和实现新的高阶数值方法来近距离评价层电位。边界积分方程法采用层势表示线性椭圆型偏微分方程的解。闭合求值问题是指高阶正交规则在计算区域边界附近点的层势时产生的较大误差，尽管在区域的其他地方是非常精确的。解决闭合评估问题对于许多应用都很重要，包括斯托克斯流问题和等离子体领域。当前的挑战是开发有效的、易于实现的方法来解决三维的接近评估问题。本项目将通过使用渐近分析识别和分析这些积分算子的近奇异行为来解决这一挑战。本项目的主要目标是：(1)开发新的方法来解决接近求值问题；(2)分析误差并研究效率和计算问题；(3)将方法推广到其他线性椭圆型偏微分方程。该项目的结果有望开发出新颖、准确、高效的计算方法来解决接近评估问题。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.