The Green's functions for the Laplace equation respectively satisfying the Dirichlet and Neumann boundary conditions on the upper side of an infinite plane with a circular hole are introduced and constructed. These functions enables solution of the boundary value problems in domains where the hole is closed by any surface. This approach enables accounting for arbitrary positive and negative ground elevations inside the domain of interest, which, generally, is not possible to achieve using the regular method of images. Such problems appear in electrostatics, however, the methods developed apply to other domains where the Laplace or Poisson equations govern. Integral and series representations of the Green's functions are provided. An efficient computational technique based on the boundary element method with fast multipole acceleration is developed. A numerical study of some benchmark problems is presented.
In this paper, we prove the exponential convergence of the multipole and local expansions, shifting and translation operators used in fast multipole methods (FMMs) for 3-dimensional Laplace equations in layered media. These theoretical results ensure the exponential convergence of the FMM which has been shown by the numerical results recently reported in [9]. As the free space components are calculated by the classic FMM, this paper will focus on the analysis for the reaction components of the Green's function for the Laplace equation in layered media. We first prove that the density functions…
A method for analytical computation of the double surface integrals for all layer potential kernels associated with the Laplace Green's function, in the Galerkin boundary element method (BEM) in $\mathbb{R}^3$ using piecewise constant flat elements is presented. The method uses recursive dimensionality reduction from 4D ($\mathbb{R}^2\times\mathbb{R}^2$) based on Gauss' divergence theorem. Computable analytical expressions for all cases of relative location of the source and receiver triangles are covered for the single and double layer potentials and their gradients with analytical treatment …
In this paper, we present a generalization of the classical error bounds of Greengard-Rokhlin for the Fast Multipole Method (FMM) for Laplace potentials in three dimensions, extended to the case of local expansion (instead of point) targets. We also present a complementary, less sharp error bound proven via approximation theory whose applicability is not restricted to Laplace potentials. Our study is motivated by the GIGAQBX FMM, an algorithm for the fast, high-order accurate evaluation of layer potentials near and on the source layer. GIGAQBX is based on the FMM, but unlike a conventional FMM…
A parallel method for solving Laplace equations with Dirichlet data using local boundary integral equations and random walks
State Key Lab. of ASIC & System Fudan Univ.（复旦大学ASIC与系统国家重点实验室）；Dept. of Mathematics & Statistics University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校数学与统计系）
Accurate computation of Galerkin double surface integrals in the 3-D boundary element method
Department of Computer Science, University of Maryland, College Park（马里兰大学学院公园分校计算机科学系）；Fantalgo, LLC（Fantalgo有限责任公司）；Institute for Advanced Computer Studies, University of Maryland, College Park（马里兰大学学院公园分校高级计算机研究所）
We develop walk-on-sphere for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk-on-sphere method is based on probabilistic represen tation of the fractional Poisson equation. We propose effcient quadrature rules to evaluate integral representation in the ball and apply rejection sampling method to drawing from the computed probabilities in general domains. Moreover, we provide an estimate of the number of walks in the mean value for the method when the domain is a ball. We show that the number of walks is increasing in the fractional order and the di…
Discrete transparent boundary conditions for the linearized Green-Naghdi system of equations
Institut de Mathématiques de Toulouse（图卢兹数学研究所；UMR5219，图卢兹大学；法国国家科学研究中心，UPS IMT）；UMR5219, Université de Toulouse（图卢兹数学研究所；UMR5219，图卢兹大学；法国国家科学研究中心，INSA）；CNRS, UPS IMT；Institut de Mathématiques de Toulouse；UMR5219, Université de Toulouse；CNRS, INSA
In this paper, a perfectly matched layer (PML) method is proposed to solve the time-domain electromagnetic scattering problems in 3D effectively. The PML problem is defined in a spherical layer and derived by using the Laplace transform and real coordinate stretching in the frequency domain. The well-posedness and the stability estimate of the PML problem are first proved based on the Laplace transform and the energy method. The exponential convergence of the PML method is then established in terms of the thickness of the layer and the PML absorbing parameter. As far as we know, this is the fi…
Kernel Methods for Some Transport Equations with Application to Learning Kernels for the Approximation of Koopman Eigenfunctions: A Unified Approach via Variational Methods, Green's Functions and the Method of Characteristics
Department of Computing and Mathematical Sciences, Caltech（加州理工学院计算与数学科学系）；The Alan Turing Institute（艾伦·图灵研究所）；Department of Mechanical Engineering, Clemson University（克莱姆森大学机械工程系）
