麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 24 Outline Laplace Problems

Numerical Methods for PDEs Integral Equation Methods, Lecture 4 Formulating Boundary Integral Equations Notes by Suvranu De and J. White Apri30,2003

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 SMA+HPC⊙2003M Formulating Equations 1

3-D Laplace Differential Equation Problems Laplace's equation in 3-D 02(x)82u(x),82u() VAu(i) 0 where =,y,z∈ and n is bounded by r SMA+HPC⊙2003M Formulating Equations 2

3-D Laplace Boundary Conditions Problems Dirichlet condition ()=ur()∈ OR Neumann Condition 0u(d)8ur() c∈T an PLUS A Radiation Condition SMA+HPC⊙2003M Formulating Equations 3

3-D Laplace Boundary Conditions Problems Radiation Condition The Radiation Condition 10m|d→0()→>0 not specific enough! Need →。 ()→O(列-) OR 12mix(6)→O(|l-2 SMA+HPC⊙2003M Formulating Equations 4

Greens Function 3-D Laplace Problems Laplace's Equation Greens Function V2G(d)=4丌6(d) 6(0)≡ impulse in3D Defined by its behavior in an integral 6()f()d=f(0) Not too hard to show G() SMA+HPC⊙2003M Formulating Equations 5

Ansatz(Indirect) Single Layer Potential Formulations Consider () (a')dr |-l u(d) automatically satisfies Vu=0 on Q Must now enforce boundary conditions SMA+HPC⊙2003M Formulating Equations 6

Ansatz(Indirect) Single Layer Potential Formulations Boundary Conditions Dirichlet problem 1 r() o0()dr∈r lai-/ Neumann Problem Our(i) a an eo()dri∈r SMA+HPC⊙2003M Formulating Equations 7

Ansatz(Indirect) Single Layer Potential Formulations Care Evaluating Integrals On a smooth surface m o E)dr →0mT 1 2丌()+ r oni lt-o()dr SMA+HPC⊙2003M Formulating Equations 8

Ansatz(Indirect) Single Layer Potential Formulations Neumann Problem 2nd Kind! 0ur() 1 27(x)+ (e)dr ana r onca- a SMA+HPC⊙2003M Formulating Equations 9