Outline Easy technique for computing integrals Piecewise constant approach Gaussian Quadrature Convergence properties Essential role of orthogonal polynomials Multidimensional Integrals Techniques for singular kernels Adaptation and variable transformation Singular quadrature

Outline Laplace Problems Exterior Radiation Condition Green's function Ansatz or Indirect Approach Single and Double Layer Potentials First and Second Kind Equations Greens Theorem Approach First and Second Kind Equations

1 Model problem 1.1 Formulations 1.1.1 Strong formulation LIDE Find a such that for Q a polygonal domain Generalizat ion We look here at a particularly simple but nevertheless illustrative problem

1 Outline for this Module Slide 1 Overview of Integral Equation Methods Important for many exterior problems (Fluids, Electromagnetics, Acoustics) Quadrature and Cubature for computing integrals One and Two dimensional basics Dealing with Singularities 1st and 2nd Kind Integral Equations

Goals Theory A priori A priori error estimates N1 bound various“ measures” of u exact]-un [approximate] in terms of C(n, problem parameters h [mesh diameter, and u

Formulations Model problem Strong Formulation Find u such that Vu=f in n2 a =0 on I for a polygonal domain

Outline Reminder about 1-D 1st and 2nd Kind egns Three-D Laplace Problems Interior Neumann Problem Null space issue First Kind Theory for 3-D Laplace Informal Convergence Theory

Laplace Problems Exterior Radiation Condition Green’s function Ansatz or Indirect Approach Single and Double Layer Potentials First and Second Kind Equations Greens Theorem Approach First and Second Kind Equations

麻省理工学院：偏微分方程式数字方法（英文版）_lec26

第一节GPS控制网的技术设计 1、精度等级 精度的主要指标是基线长度中误差1b 《全球定位系统(GPS)测量规范)》2001分六个等级 《全球定位系统城市测量规程》1997分二等、三等、四等、一级和二 级等五个等级。 2、基准设计 （1)坐标系:WGS-84、北京54/80,地方独立坐标系 （2)位置基准、方位基准、尺度基准