PGD Solution of the Poisson Equation

Abstract

This chapter describes the main features of the PGD technique, in particular the one related to the construction of a separated representation of the unknown field involved in a partial differential equation. For this purpose, we consider the solution of the two-dimensional Poisson equation in a square domain. The solution is sought as a finite sum of terms, each one involving the product of functions of each coordinate. The solution is then calculated by means of a sequence of one-dimensional problems. The chapter starts with the simplest case, that is later extended to cover more complex problems: non-constant source terms, non-homogeneous Dirichlet and Neumann boundary conditions, and high-dimensional problems. Carefully solved numerical examples are discussed to illustrate the theoretical developments.