Sumários
Maximum/Minimum Discrete Principles
17 março 2008, 14:30 • Carlos Alves
Theorem of the Mean for the Laplace Equation. Maximum/Minimum Principles for the Poisson Equation.
Proofs of the Discrete Versions for the Laplacian second order approximation.
Well posedness in the Hadamard sense. Uniqueness and existence of the discrete FDM solution for the Poisson equation.
Discretization of the Laplace operator
13 março 2008, 11:30 • Carlos Alves
Discretization of the Laplace operator using centered differences in both directions.
Application to the Poisson equation - Finite Difference Method (FDM).
FDM for the Dirichlet problem. System of equations.
Matrix structure in the case of a rectangular shape.
Problems
11 março 2008, 10:00 • Carlos Alves
Exercises: Uniqueness for Dirichlet, Neumann (up to a constant), Robin (positive Z) problems.
Exercises: Particular solutions using cartesian and polar coordinates separation of variables.
Poisson equation. Boundary conditions.
10 março 2008, 14:30 • Carlos Alves
Classical differential operators - Laplacian, gradient, curl.
Poisson/Laplace equation. Harmonic functions and polynomials.
Boundary conditions of Dirichlet, Neumann, Robin or mixed type.
Review of integral theorems - Gauss (Divergence, Stokes) theorems.
Green's formulas.
Start of the course in English.
6 março 2008, 11:30 • Carlos Alves
Review of previous notions in English.
Operator approximation - relation between degree and order.
Introduction to the three cases of PDEs to be considered in the course: Poisson, Heat, and Wave equations.