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

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

Finite Difference Problem Definition Formulas

Motivation Consider a standard second order finite difference discretization

Course Outline Overview of pdes(1) o Finite differences methods(6) Finite volume methods(3) Finite element methods(7) Boundary integral methods(6)

1 Background Brandt(1973)published first paper SLIDE 1 Offers the possibility of solving a problem with work and storage propor tional to the number of unknowns Well developed for elliptic proble

1 Motivation SLIDE 1 Consider a standard second order finite difference discretization of V-u= on a regular g 1.2. and 3 dimensions 1.1 1D Finite differences

Governing Equation Stability Analysis 3 Examples Relationship between σ and λh Implicit Time-Marching Scheme

Outline Governing Equation Stability Analysis 3 Examples Relationship between σ and λh Implicit Time-Marching Scheme

1 Finite difference formulas 1.1 Problem definition We have seen that one of the necessary ingredients in devising finite differe