The Solution of Partial Differential Equations on Realistic Geometries

许多有用的物理量(电容、应力、电磁散射等)通过在具有包含角点、边和圆锥点的边界的真实几何图形上求解椭圆型偏微分方程(PDE)来计算。即使对于简单的边界(即立方体)，相应区域上的偏微分方程组的解通常在这些特征附近具有奇异性。这种非光滑行为是椭圆型偏微分方程组数值解及其数学理论中的一个主要症结。从数值上讲，非平滑行为可能会出现以下问题。虽然光滑函数通常可以用短的有限傅立叶级数(或切比雪夫多项式或勒让德多项式的短级数)来表示高精度，但奇异函数可以呈现出令人困惑的各种行为。例如，解析函数可以有极点、分支或基本奇点(在这些奇点附近，它可以采用所有复数值，可能只有一个！)。通常这类函数可以用嵌套的切比雪夫离散化或Gauss-Legendre离散化来表示，但通常需要大量的自由度来高精度地捕捉所有可能的未知行为。然而，事实证明，在具有角和边的几何上遇到的许多奇异函数都可以非常详细地刻画出来。例如，在具有角点的二维区域上的拉普拉斯方程的情况下，角点附近的奇异解可以用已知奇异幂的初等渐近级数来表示。有了这样的表示，偏微分方程组的解的行为变得更加受限，因此可以为奇异解构造有效的特殊目的离散化。因此，许多历史上数值难解的偏微分方程组可以快速地求解，基本上达到机器精度。在对椭圆型偏微分方程组进行高精度数值求解时，常常需要利用经典位势理论将问题转化为第二类积分方程组。在二维中，几个椭圆型偏微分方程角附近的积分方程解已被刻画，然而更详细(也更有用)的三维情形在很大程度上仍未被探索。我们建议构造一个分析工具来精确地刻画与各种椭圆型偏微分方程组(拉普拉斯、亥姆霍兹、斯托克斯和最终的麦克斯韦)有关的积分方程解在三维边角和圆锥点附近的行为。我们将构建一个利用这些分析信息的数值设备，从而消除嵌套离散的需要。这种基于边界积分的三维格式的创建，如果消除了长期存在的(和数值上难以解决的)边缘问题，将是工程学和应用科学的一大进步。

Many useful physical quantities (capacitance, stresses, electromagnetic scattering, etc.) are computed by solving elliptic partial differential equations (PDEs) on realistic geometries with boundaries containing corners, edges, and conical points. Even for simple boundaries (i.e. a cube), the solutions to the PDEs on the corresponding regions usually have singularities near such features. This non-smooth behavior is a major sticking point in both the numerical solution of elliptic PDEs, and their mathematical theory. Numerically, non-smooth behavior can present the following problem. While smooth functions are usually representable to high precision by short finite Fourier series (or short series of Chebyshev or Legendre polynomials), singular functions can take on a bewildering variety of behaviors. For example, an analytic function can have poles, branches, or essential singularities (in the neighborhood of which it takes on every complex value except possibly one!). Often such functions can be represented by nested Chebyshev or Gauss-Legendre discretizations, but a large number of degrees of freedom is usually required to capture all of the possible unknown behavior to high precision. It turns out, however, that many of the singular functions encountered on geometries with corners and edges can be characterized in great detail. For instance, in the case of Laplace's equation on a two-dimensional domain with corners, the singular solutions near corners are representable by elementary asymptotic series of known singular powers. With such a representation in hand, the behavior of the solutions to the PDEs becomes significantly more circumscribed, and so efficient special-purpose discretizations can be constructed for the singular solutions. As a result, many historically numerically refractory PDEs involving domains with corners can be solved rapidly and to essentially machine precision. When solving elliptic PDEs numerically to high precision, it is often necessary to reformulate the problems as second kind integral equations using classical potential theory. In two dimensions, the solutions to the associated integral equations near corners have been characterized for several elliptic PDEs, however the much more detailed (and useful) case of three dimensions remains largely unexplored. We propose to construct an analytical apparatus characterizing precisely the behavior of the solutions to the integral equations associated with various elliptic PDEs (Laplace, Helmholtz, Stokes, and eventually Maxwell) in the vicinity of edges, corners, and conical points in three dimensions. We will construct a numerical apparatus exploiting this analytical information, obviating the need for nested discretizations. The creation of such boundary integral based schemes in three dimensions, if eliminating the longstanding (and numerically intractable) issues surrounding edges, would constitute a major advance in engineering and applied sciences.