Motivation The Poisson problem has a strong formulation; a minimization formulation; and a weak formulation. The minimization/weak formulations are more general than the strong formulation in terms of reqularity and admissible data

Scalar Definitions Conservation Laws

1 Scalar Conservation laws 1.1 Definit 1.1.1 Conservative form

First Order Wave Equation INITIAL BOUNDARY VALUE PROBLEM(IBVP

Integral Equation Methods Reminder about galerkin and Collocation Example of convergence issues in 1D First and second kind integral equations Develop some intuition about the difficulties Convergence for second kind equations Consistency and stability issues

Easy technique for computing integrals Piecewise constant approach sian Quadra Convergence pI ssential role of orthogonal polynomials Multidimensional Integra Techniques for singular kernels Adapt ation and variable transformation Singular quadrature

Integral Equation Methods Reminder about Galerkin and collocation Example of convergence issues in 1D First and second kind integral equations Develop some intuition about the difficulties Convergence for second kind equations Consistency and stability issues Nystrom Method

Outline for this Module 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

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