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）是通过求解具有边角、边缘和圆锥形点的实际几何形状的椭圆偏微分方程（PDEs）来计算的。即使对于简单的边界（如立方体），相应区域上偏微分方程的解通常在这些特征附近具有奇点。这种非光滑性是椭圆偏微分方程数值解及其数学理论的一个主要难点。在数值上，非光滑行为会带来以下问题。虽然光滑函数通常可以用有限的短傅立叶级数（或切比雪夫多项式或勒让德多项式的短级数）高精度地表示，但奇异函数可以表现出令人眼花缭乱的各种行为。例如，解析函数可以具有极点、分支或本质奇点（在其邻域中，它取除可能的一个之外的所有复值）。通常这样的函数可以用嵌套的Chebyshev或Gauss-Legendre离散化来表示，但是通常需要大量的自由度来高精度地捕获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可以非常详细地表征。例如，对于二维角域上的拉普拉斯方程，其角附近的奇异解可以用已知奇异幂的初等渐近级数表示。有了这样的表示，偏微分方程解的行为变得更加受限，因此可以为奇异解构造有效的专用离散化。因此，许多历史上数值难解的偏微分方程涉及角域，可以快速求解，并基本达到机器精度。在对椭圆偏微分方程进行高精度数值求解时，往往需要用经典势理论将问题重新表述为第二类积分方程。在二维中，一些椭圆偏微分方程在角附近的相关积分方程的解已经被表征，然而更详细（和有用）的三维情况在很大程度上仍未被探索。我们建议建立一个解析装置，精确地描述与各种椭圆偏微分方程（拉普拉斯、亥姆霍兹、斯托克斯和最终麦克斯韦）在三维边缘、角落和圆锥点附近的积分方程的解的行为。我们将利用这些分析信息构建一个数值装置，从而避免嵌套离散化的需要。这种基于三维边界积分的方案的创建，如果消除了围绕边缘的长期（和数值上棘手的）问题，将构成工程和应用科学的重大进步。

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.