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Erschienen in:
2014 | OriginalPaper | Buchkapitel
2. PGD Solution of the Poisson Equation
verfasst von : Francisco Chinesta, Roland Keunings, Adrien Leygue
Erschienen in: The Proper Generalized Decomposition for Advanced Numerical Simulations
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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.