论文标题
在三维中校正边界积分方程的梯形规则
Corrected Trapezoidal Rules for Boundary Integral Equations in Three Dimensions
论文作者
论文摘要
手稿描述了使用NyStröm方法为边界积分方程（BIE）高阶离散化设计的正交规则。该技术是为可以自然可以使用矩形上的均匀网格（例如变形Tori或具有周期性边界条件的通道）进行参数化的表面设计的。当使用基于梯形正交规则的Nyström方法离散地将这种几何形状上的Bie离散时，由于内核函数的奇异性，结果方案倾向于缓慢收敛。手稿的主要发现是，通过仅修改系数矩阵中的少量元素，可以大大改善收敛顺序。具体而言，可以证明，通过仅纠正系数矩阵中的对角线条目，$ O（H^{3}）$收敛可以用于与Laplace和Helmholtz内核相关的单层和双层电位的融合。九点校正模具导致$ O（H^5）$方案。提出的方法可以看作是Duan和Rokhlin的正交规则的概括，该规则是为平面中2D Lippmann-Schinginger方程设计的。提出的技术由严格的误差分析支持，该分析依赖于涉及爱泼斯坦Zeta函数及其参数衍生物的Wigner型极限。
The manuscript describes a quadrature rule that is designed for the high order discretization of boundary integral equations (BIEs) using the Nyström 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 Nyström 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-point correction 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.
