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

课前提示:法律的功能与价值的实现,需 要通过法律由纸面降临尘世这一中间环节, 而这仰赖于守法、执法、司法和法律监督 等多个方式的运行。其中,守法是最为重 要的一种方式,因为法律的实现主要依靠 人们的自觉遵守,一部为大多数人所反对 和违抗的法律,即使依靠强制力获得一时 的遵守,其效力也不可能持久