Efficient and highly accurate solvers for integral equations on surfaces with edges and corners

这个项目涉及物理现象的计算机建模。特别是，PI寻求准确地建模电磁波和声波的行为。这个主题有着悠久的历史，以前已经取得了许多重要的成果，但我们目前缺乏准确建模涉及复杂几何形状的情况的能力。该项目旨在通过将纯数学的几个观察结果与新的工程方法相结合来解决这些困难。该项目的最终成果将是允许电磁波和声波精确建模的工具。这些工具将适用于许多问题，但PI特别感兴趣的是将它们应用于集成电路分析和生物力学模拟（例如，囊泡流）。这种方法在电动力学、流体力学和弹性力学中有重要的应用。然而，大多数应用涉及具有奇异性的域，并且众所周知，在求解此类域上的积分方程时难以实现高精度和高效率。在原则上，似乎应该没有困难，解决一大类积分方程的表面上与边缘和角落奇异的蛮力的方式。数学物理中许多重要的边值问题都可以用平方积分函数空间上的可逆的、良好条件的积分算子来表示。Galerkin离散这样的运营商收敛，以及作为基础运营商的条件。从这些观察可以得出，假设离散化的所有方面都得到了正确的处理，在足够密集的网格上用多项式基函数简单地局部表示解将导致高度精确的近似。然而，在实际应用中，这种方法会产生一些困难，其中主要的问题是：（1）密集网格会导致线性系统过于庞大，即使是现代的O（N）快速求解器也不足以解决这些问题;（2）表示边缘附近的奇异性最好用高度各向异性的网格，这给现有的离散化技术和快速求解器带来了困难;（3）以高精度评估系数矩阵的条目（其涉及估计奇异和“近似”奇异积分）通常是相当具有挑战性的，并且在拐角和边缘区域附近实质上更是如此。本项目的目标是开发高精度的快速鲁棒求解器，以克服这些困难，实现高精度和高效率的边缘和角点奇异曲面上的积分方程。它将在不使用先验渐近估计（在许多感兴趣的情况下不可用）的情况下这样做。该项目由三个阶段的方法组成：（1）PI将实施一个高精度和非常强大的“蛮力”程序;（2）将部署几个工具，包括本地快速自适应网格生成器和操作员压缩技术，以加速蛮力求解器;（3）有效的数值预计算求积公式，它刻画了解的奇异性，并作为先验渐近估计的替代，将使用加速求解器构造。

This project concerns the computer modeling of physical phenomena. In particular, the PI seeks to accurately model the behavior of electromagnetic and acoustic waves. This topic has a long history, and many important results have been previously achieved, but we currently lack the ability to accurately model situations involving complex geometry. This project seeks to address these difficulties by combing several observations from pure mathematics with new engineering approaches. The end result of this project will be tools which allow for the accurate modeling of electromagnetic and acoustic waves. These tools will be applicable to many problems, but the PI is particularly interested in applying them to integrated circuit analysis and biomechanical simulations (for instance, vesicle flows).Many of the partial differential equations of mathematical physics can be profitably reformulated as integral equations. Such methods have important applications to problems in electrodynamics, fluid dynamics and elasticity. However, most applications involve domains with singularity, and it is notoriously difficult to achieve high-accuracy and efficiency when solving integral equations on such domains. In principle, it appears that there should be no difficulty in solving a large class of integral equations given on surfaces with edge and corner singularities in a brute-force fashion. Many important boundary value problems in mathematical physics can be formulated using integral operators which are invertible and well-conditioned on spaces of square integral functions. Galerkin discretizations of such operators converge and are as well-conditioned as the underlying operator. It follows from these observations that, assuming all aspects of discretization are correctly handled, simply representing solutions locally with polynomial basis functions on a sufficiently dense mesh will result in highly accurate approximations. However, several difficult problems arise in practice with this brute-force approach; chief among them are: (1) dense meshes lead to excessively large linear systems that even modern O(N) fast solvers are inadequate to address; (2) representing singularities near edges is best done with highly anisotropic meshes which cause difficulties for currently available discretization techniques and fast solvers; (3) evaluating the entries of coefficient matrices to high accuracy, which involves estimating singular and "nearly" singular integrals, is quite challenging in general and substantially more so near corner and edge regions. The goal of this project is to develop highly-accurate fast robust solvers for integral equations on surfaces with edge and corner singularities which overcome these difficulties and achieve high-accuracy and efficiency. It will do so without the use of a priori asymptotic estimates (which are not available in many cases of interest). The project consists of a three-phased approach: (1) The PI will implement a highly-accurate and very robust "brute-force" procedure; (2) several tools, including local fast adaptive mesh generators and operator compression techniques will be deployed in order to accelerate the brute-force solver; (3) finally, efficient numerically precomputed quadrature formulae, which characterize the singularities of solutions and serve as a substitutes for a priori asymptotic estimates, will be constructed using the accelerated solver.