Mathematical Sciences: Numerical Analysis and Software for Integral Equations in Three Dimensions

9403589阿特金森，建议的重点是三维拉普拉斯方程的边界积分方程组的数值解。这项工作将包括进一步发展关于所使用的数值方法的理论知识，并进一步开发一个大型计算机程序包(BIEPACK)来求解三维表面上的边界元问题。要研究的数值分析问题包括曲面逼近理论的进一步发展，曲面上奇异函数和非奇异函数的数值积分，使用梯度网格来逼近曲面边角附近表现不佳的函数，以及离散边界积分方程组时产生的大型稠密线性方程组的迭代求解。此外，还将考虑涉及边界积分流的使用和数值解的新课题。这些方法包括：(1)使用第一类边界积分器来求解拉普拉斯方程的Dirichlet和Neumann问题；(2)超奇异边界积分器的逼近和求解；(3)使用“聚类法”来降低大多数数值方法中的数值积分和矩阵-向量乘法的成本。这些研究的许多结果将被纳入目前的包BIEPACK，以改进其现有的方案，并将其扩展到新的BIE。还计划将为BIE开发的数值方法应用于计算机图形学中产生的积分方程式的求解，特别是“渲染方程式”。许多工程问题都涉及到一个称为“拉普拉斯方程”的数学方程的解或对它的摄动；而“位势理论”是用来研究和应用拉普拉斯方程及其密切相关的方程的。拉普拉斯方程本身存在于流体力学(无旋不可压缩流体流动)、传热学、引力场、静电场等应用中。拉普拉斯方程在流体力学(斯托克斯流)、线弹性理论、电磁波和声波传播以及其他主题中都有变化。拉普拉斯方程的解是对空间某一区域的所有点定义的一个函数，其目的是求这个函数。在三个方面，这可能是一个非常耗时的过程。拉普拉斯方程及相关方程的求解采用了许多解析和数值方法，包括有限差分法、有限元方法和边界积分方程法。后者的优点之一是将问题重新定义为在定义拉普拉斯兴趣方程的原始区域的边界上的新方程，这通常可以在整个求解过程中节省时间。这个新问题被称为“边界积分方程式”或BIE。本研究的目的是改进边界积分方程组数值解的数值方法，并在理论上加深对边界积分方程组数值求解方法的理解。这些结果将被用来改进一个称为BIEPACK的程序包，该程序包已被开发用于许多BIE的拉普拉斯方程的数值解。它还计划将为BIE开发的数值方法应用于计算机图形学中出现的积分方程式的求解。

9403589 Atkinson The focus of the proposal is the numerical solution of boundary integral equation (BIE) reformulations of Laplace's equation in three dimensions. The work will involve both the development of further theoretical knowledge on the numerical methods being used and the further development of a large computer package (BIEPACK) to solve BIE on surfaces in three dimensions. The numerical analysis problems to be studied include the further development of theory for the approximation of surfaces, the numerical integration of singular and nonsingular functions over surfaces, the use of graded meshes to approximate functions that are ill-behaved around edges and corners of the surface, and the iterative solution of the large dense linear systems of equations which arise when discretizing the BIE. In addition, new topics involving the use and numerical solution of BIE are to be considered. These include: (1) the use of BIE of the first kind in solving the Dirichlet and Neumann problems for Laplace's equation; (2) the approximation and solution of hypersingular BIE; and (3) the use of "clustering methods" to reduce the cost of the numerical integration and matrix-vector multiplications that are a part of most numerical methods. Many of the results of these studies will be incorporated into the present package BIEPACK, to both improve its present programs and to extend it to new BIE. It is also planned to apply numerical methods that were developed for BIE to the solution of integral equations arising in computer graphics, and in particular, the "rendering equation". Many engineering problems involve the solution of a mathematical equation called "Laplace's equation" or perturbations of it; and the name "potential theory" is given to the study and application of Laplace's equation and closely related equations. The Laplace equation itself occurs in applications such as fluid mechanics (irrotational incompressible fluid flows), heat transfer, gravit ational fields, electrostatic fields, and others. Variations on Laplace's equation occur in fluid mechanics (Stokes' flows), linear elasticity theory, electromagnetic and acoustic wave propagation, and other topics. The solution of Laplace's equation is a function defined for all points of some region in space, and the objective is to find this function. In three dimensions, this can be a very time-consuming process. Many analytical and numerical techniques are used to solve Laplace's equation and related equations, including finite difference methods, finite element methods, and boundary integral equation methods. One of the advantages of the latter is that the problem is re-defined as a new equation over the boundary of the original region on which the Laplace equation of interest is defined, and this can often lead to time savings in the overall solution process. The new problem is called a "boundary integral equation" or BIE. The intention of the present study is to improve numerical methods for the numerical solution of BIE, as well as to develop a further theoretical understanding of numerical methods for solving BIE. These results will be used to improve a package, called BIEPACK, that has been developed for the numerical solution of many BIE for Laplace's equation. It is also planned to apply numerical methods that were developed for BIE to the solution of integral equations arising in computer graphics.