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首页> 外文期刊>Numerische Mathematik >Corrected trapezoidal rules for boundary integral equations in three dimensions
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Corrected trapezoidal rules for boundary integral equations in three dimensions

机译：修正了三维边界积分方程的梯形规则

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摘要

The manuscript describes a quadrature rule that is designed for the high order discretization of boundary integral equations (BIEs) using the Nystrom method. The technique is designed for surfaces that can naturally be parameterized using a uniform grid on a rectangle, such as deformed tori, or channels with periodic boundary conditions. When a BIE on such a geometry is discretized using the Nystrom method based on the Trapezoidal quadrature rule, the resulting scheme tends to converge only slowly, due to the singularity in the kernel function. The key finding of the manuscript is that the convergence order can be greatly improved by modifying only a very small number of elements in the coefficient matrix. Specifically, it is demonstrated that by correcting only the diagonal entries in the coefficient matrix, O(h(3)) convergence can be attained for the single and double layer potentials associated with both the Laplace and the Helmholtz kernels. A nine-pointcorrection stencil leads to an O(h(5)) scheme. The method proposed can be viewed as a generalization of the quadrature rule of Duan and Rokhlin, which was designed for the 2D Lippmann-Schwinger equation in the plane. The techniques proposed are supported by a rigorous error analysis that relies on Wigner-type limits involving the Epstein zeta function and its parametric derivatives.
机译：该手稿描述了一种正交规则，该规则设计用于使用 Nystrom 方法对边界积分方程 （BIE） 进行高阶离散化。该技术设计用于可以使用矩形上的均匀网格自然参数化的曲面，例如变形的 tori 或具有周期性边界条件的通道。当使用基于梯形正交规则的 Nystrom 方法离散化这种几何上的 BIE 时，由于核函数中的奇异性，所得到的方案往往收敛缓慢。该手稿的主要发现是，通过仅修改系数矩阵中极少量的元素，可以大大提高收敛顺序。具体而言，通过仅校正系数矩阵中的对角线条目，可以实现与拉普拉斯核和亥姆霍兹核相关的单层和双层势的O（h（3））收敛性。九点校正模板导致 O（h（5）） 方案。所提出的方法可以看作是Duan和Rokhlin正交法则的推广，该正交法则是为平面中的二维Lippmann-Schwinger方程设计的。所提出的技术得到了严格的误差分析的支持，该分析依赖于涉及 Epstein zeta 函数及其参数导数的 Wigner 型极限。

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