使用RBFs的当地稳定集成法,用于赫尔姆霍尔茨等式,并用于波的混乱和电磁显微反应器 (A stabilized local integral method using RBFs for the Helmholtz equation with applications to wave chaos and dielectric microresonators)

Most problems in electrodynamics do not have an analytical solution so much effort has been put in the development of numerical schemes, such as the finite-difference method, volume element methods, boundary element methods, and related methods based on boundary integral equations. In this paper we introduce a local integral boundary domain method with a stable calculation based on Radial Basis Functions (RBF) approximations, in the context of wave chaos in acoustics and dielectric microresonator problems. RBFs have been gaining popularity recently for solving partial differential equations numerically, becoming an extremely effective tool for interpolation on scattered node sets in several dimensions with high-order accuracy and flexibility for nontrivial geometries. One key issue with infinitely smooth RBFs is the choice of a suitable value for the shape parameter which controls the flatness of the function. It is observed that best accuracy is often achieved when the shape parameter tends to zero. However, the system of discrete equations obtained from the interpolation matrices becomes ill-conditioned, which imposes severe limits to the attainable accuracy. A few numerical algorithms have been presented that are able to stably compute an interpolant, even in the increasingly flat basis function limit. We present the recently developed Stabilized Local Boundary Domain Integral Method in the context of boundary integral methods that improves the solution of the Helmholtz equation with RBFs. Numerical results for small shape parameters that stabilize the error are shown. Accuracy and comparison with other methods are also discussed for various case studies. Applications in wave chaos, acoustics and dielectric microresonators are discussed to showcase the virtues of the method, which is computationally efficient and well suited to the kind of geometries with arbitrary shape domains.

翻译：电子动力学中的大多数问题都没有分析解决方案。因此,在开发基于边界整体方程式的定量差异法、量元素法、边界元件方法和相关方法等数字方法时, 已经投入了大量努力。在本文件中, 我们引入了本地整体边界域法方法, 以辐射基础函数近似值为基础, 稳定计算。在声学和电磁微反射器问题中, 电流混乱的波状混乱中, 电流中的大多数问题都没有分析。 REBF最近越来越受欢迎, 解决部分偏差方数字化, 成为以高顺序精确度和灵活性对非三角地貌进行分散节点组合的极有效的相互调工具。使用无限平滑的 RBFS 的关键问题之一是选择一个合适的形状参数, 控制着函数的平坦度。观察到, 在声波变的音频参数中, 离子方程式的离子方程式系统变得不妥, 使可实现的精确度受到严格限制。少数数字算法被展示了, 在平坦的平面上, 也显示平坦平的平的 RBRF 方法。