Fast and accurate numerical algorithms for boundary value problems of elliptic partial differential equations on open surfaces in three dimensions

工作的重点是发展快速、准确和健壮的计算技术，用于解决当散射面由一组开放曲面(即有边界的曲面)组成时的大规模散射问题。虽然这类问题最初通常被描述为椭圆型偏微分方程组的边值问题，但积分方程组已被用作数值求解散射问题的主要工具之一，特别是对于外部问题。从历史上看，使用的大多数积分方程组都是第一类，因为与这类方程相关的数值不稳定性对于当时可以处理的相对较小规模的问题并不是至关重要的。改进的硬件与最近在“快速”算法设计方面的进步相结合，极大地改变了这种情况。势论积分方程组的离散化所产生的线性代数方程组的条件数变得至关重要，而限制这种条件数的最简单方法是从第二类积分方程组开始。因此，将散射问题归结为散射体边界上的第二类积分方程组的研究受到了越来越多的关注。研究人员提出利用位势理论和奇异积分的工具来构造开放表面问题的第二类积分方程组(SKIE)(特别是对于那些控制偏微分方程组是Laplace或Helmholtz方程的问题)。在获得SKIE公式后，研究人员计划将其应用于开发和实施针对开放表面问题的高效和准确的数值算法，使用迭代求解器和快速多极子方法的组合。由于拉普拉斯方程和亥姆霍兹方程在应用数学中普遍存在，而且许多实际问题都涉及开放表面，因此本文的研究将对许多活跃的研究领域产生广泛的影响，包括声学和电磁散射问题、流体力学、弹性问题和逆散射问题。预计这项拟议的研究还将对反射天线和集成电路等关键技术产生长期影响。

The focus of the proposed work is directed toward the development of fast, accurate, and robust computational techniques for solving large-scale scattering problems when the scattering surfaces consist of a collection of open surfaces (i.e., surfaces with boundary). Though such problems are often initially formulated as boundary-value problems of elliptic partial differential equations, integral equations have been employed as one of principal tools for the numerical solution of scattering problems, particularly for exterior problems. Historically, most of the integral equations used have been of the first kind, since numerical instabilities associated with such equations have not been critically important for the relatively small-scale problems that could be handled at the time. The combination of improved hardware with the recent progress in the design of "fast" algorithms has changed the situation dramatically. Condition numbers of systems of linear algebraic equations resulting from the discretization of integral equations of potential theory have become critical, and the simplest way to limit such condition numbers is by starting with second-kind integral equations. Hence, there is increasing interest in reducing scattering problems to systems of second-kind integral equations on the boundaries of scatterers.The investigator proposed to apply tools from potential theory and singular integrals to construct second-kind integral-equation (SKIE) formulations for open-surface problems (especially for those whose governing PDEs are the Laplace or Helmholtz equations). After SKIE formulations have been obtained, the investigator plans to apply them to develop and implement efficient and accurate numerical algorithms for open-surface problems, using a combination of iterative solvers and the fast multipole method. Since the Laplace equation and the Helmholtz equation are ubiquitous in applied mathematics and many practical problems involve open surfaces, the proposed research will have broad impacts on many active research fields including acoustic and electromagnetic scattering problems, fluid mechanics, elasticity problems, and inverse-scattering problems. The proposed research is also expected to have long-term impacts on key technologies such as reflecting antennas and integrated circuits.