lec20_notes

lec20_notes - Numerical Methods for PDEs Integral Equation...

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Numerical Methods for PDEs Integral Equation Methods, Lecture 1 Discretization of Boundary Integral Equations Notes by Suvranu De and J. White April 23, 2003
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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 1 st and 2 nd Kind Integral Equations Collocation, Galerkin and Nystrom theory Alternative Integral Formulations Ansatz approach and Green’s theorem Fast Solvers Fast Multipole and FFT-based methods. 2 Outline Slide 2 Integral Equation Methods Exterior versus interior problems Start with using point sources Standard Solution Methods Collocation Method Galerkin Method Some issues in 3D Singular integrals 3 Interior Vs Exterior Problems Slide 3 Interior Exterior Temperature known on surface 2 0 T ∇= inside 2 0 T outside Temperature known on surfac "Temperature in a tank" "Ice cube in a bath" What is the heat flow? Heat flow = Thermal conductivity R surface ∂T ∂n Note 1 Why use integral equation methods? For both of the heat conduction examples in the above ±gure, the temperature, T , is a function of the spatial coordinate, x , and satis±es 2 T ( x )=0 .Inbo th 1
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problems T ( x ) is given on the surface, deFned by Γ , and therefore both problems are Dirichlet problems. ±or the “temperature in a tank” problem, the problem domain, is the interior of the cube, and for the “ice cube in a bath” problem, the problem domain is the inFnitely extending region exterior to the cube. ±or such an exterior problem, one needs an additional boundary condition to specify what happens sufficiently far away from the cube. Typically, it is assumed there are no heat sources exterior to the cube and therefore lim k x k→∞ T ( x ) 0 . ±or the cube problem, we might only be interested in the net heat flow from the surface. That flow is given by an integral over the cube surface of the normal derivative of temperature, scaled by a thermal conductivity. It might seem inefficient to use the Fnite-element or Fnite-di²erence methods discussed in previous sections to solve this problem, as such methods will need to compute the temperature everywhere in . Indeed, it is possible to write an integral equation which relates the temperature on the surface directly to its surface normal, as we shall see shortly. In the four examples below, we try to demonstrate that it is quite common in applications to have exterior problems where the known quantities and the quantities of interest are all on the surface. 4E x a m p l e s 4.1 Computation of Capacitance Slide 4 v + - 2 0 Outsi ∇Ψ= is given on S Ψ potential What is the capacitance?
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lec20_notes - Numerical Methods for PDEs Integral Equation...

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