The Solution of Partial Differential Equations on Realistic Geometries

现实几何上偏微分方程的解

• 批准号: RGPIN-2020-06022
• 负责人: Serkh, Kirill
• 金额: $ 1.31万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740980
• 关键词: Solution; Partial; Differential; Equations; Realistic

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.

许多有用的物理量（电容、应力、电磁散射等）是通过求解椭圆偏微分方程（PDE）的实际几何形状与边界包含角，边缘，和圆锥点。即使对于简单的边界（即立方体），相应区域上的偏微分方程的解通常在这些特征附近具有奇点。这种非光滑行为是椭圆偏微分方程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里叶级数（或切比雪夫或勒让德多项式的短级数）表示为高精度，但奇异函数可以呈现出令人困惑的各种行为。例如，一个解析函数可以有极点、分支或本质奇点（在其邻域内，它可以取除了一个之外的所有复值！）。通常，此类函数可以用嵌套的切比雪夫或高斯-勒让德离散化来表示，但通常需要大量的自由度才能高精度地捕获所有可能的未知行为。然而，事实证明，在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描述。例如，在二维角点区域上的拉普拉斯方程，角点附近的奇异解可用已知奇异幂的初等渐近级数表示。有了这样的表示，偏微分方程的解的行为变得更加严格，因此可以为奇异解构造有效的专用离散化。其结果是，许多历史上数值难处理的偏微分方程，涉及域的角落，可以快速解决，基本上机器精度。在高精度数值求解椭圆型偏微分方程时，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。在二维空间中，角点附近相关积分方程的解已经被描述为几个椭圆偏微分方程的特征，但是三维空间中更详细（和有用）的情况仍然在很大程度上未被探索。我们建议构造一个分析仪器，精确地表征与各种椭圆偏微分方程（拉普拉斯，亥姆霍兹，斯托克斯，并最终麦克斯韦）在三维的边缘，角落和圆锥点附近的积分方程的解的行为。我们将构建一个数值装置，利用这些分析信息，避免嵌套离散化的需要。在三维中创建这种基于边界积分的方案，如果消除围绕边缘的长期存在的（和数值上难以处理的）问题，将构成工程和应用科学的重大进步。