论文标题
一种用于弯曲表面上奇异和高超边界积分算子的统一梯形正交法
A Unified Trapezoidal Quadrature Method for Singular and Hypersingular Boundary Integral Operators on Curved Surfaces
论文作者
论文摘要
本文介绍了一种梯形正交方法，用于在解决椭圆形偏微分方程的边界值问题时出现的奇异和超边界积分算子（BIOS）的离散化。正交基于参数空间中的均匀网格以及标准穿刺梯形规则。一个关键的观察结果是，可以使用涉及2D中的Riemann Zeta函数的广义Euler-Maclaurin公式来精确表示内核中的误差，而3D中的Epstein Zeta函数。利用这些扩展以通过新型的系统矩拟合方法在奇异点通过本地模板来纠正错误。这种新方法提供了所有常见BIOS（Laplace，Helmholtz，Stokes等）的统一处理。我们提出了数值示例，这些示例表明，相对于网格大小，在2D和第9阶中最多32阶收敛。
This paper describes a trapezoidal quadrature method for the discretization of singular and hypersingular boundary integral operators (BIOs) that arise in solving boundary value problems for elliptic partial differential equations. The quadrature is based on a uniform grid in parameter space coupled with the standard punctured Trapezoidal rule. A key observation is that the error incurred by the singularity in the kernel can be expressed exactly using generalized Euler-Maclaurin formulae that involve the Riemann zeta function in 2D and the Epstein zeta functions in 3D. These expansions are exploited to correct the errors via local stencils at the singular point using a novel systematic moment-fitting approach. This new method provides a unified treatment of all common BIOs (Laplace, Helmholtz, Stokes, etc.). We present numerical examples that show convergence of up to 32nd-order in 2D and 9th-order in 3D with respect to the mesh size.
