论文标题
Zeta校正：一种构造奇异整体操作员校正后梯形正交规则的新方法
Zeta Correction: A New Approach to Constructing Corrected Trapezoidal Quadrature Rules for Singular Integral Operators
论文作者
论文摘要
引入了平面上封闭平滑轮廓上边界积分方程（BIE）离散化的高阶准确正交规则。该正交可以看作是Kress（1991）的光谱正交的杂种，以及Kapur and Rokhlin（1997）的局部校正的梯形正交。新技术结合了两种方法的优势，并获得了高阶收敛，数值稳定性，易于实现以及与“快速”算法的兼容性（例如快速多极方法或快速直接求解器）。引入了穿刺梯形规则与Riemann Zeta函数之间的重要连接，从而实现完整的收敛分析，并导致非常简单的程序来构建正交校正。本文报告了Kress，Kapur和Rokhlin和Alpert（1999）的Kress方法和方法之间的详细比较。
A high-order accurate quadrature rule for the discretization of boundary integral equations (BIEs) on closed smooth contours in the plane is introduced. This quadrature can be viewed as a hybrid of the spectral quadrature of Kress (1991) and the locally corrected trapezoidal quadrature of Kapur and Rokhlin (1997). The new technique combines the strengths of both methods, and attains high-order convergence, numerical stability, ease of implementation, and compatibility with the "fast" algorithms (such as the Fast Multipole Method or Fast Direct Solvers). Important connections between the punctured trapezoidal rule and the Riemann zeta function are introduced, which enable a complete convergence analysis and lead to remarkably simple procedures for constructing the quadrature corrections. The paper reports a detailed comparison between the new method and the methods of Kress, of Kapur and Rokhlin, and of Alpert (1999).
