The Discontinuous Petrov-Galerkin Method for Wave Propagation Problems.
Part I: Introduction and Preliminary Results.
David Pardo, Leszek Demowicz, Jay Gopalakrishnan, Jeff Zitelli, Ignacio Muga, Victor M. Calo.,
Jueves 27 de enero 2011
We introduce a new Petrov-Galerkin formulation for solving systems of first-order partial differential equations (PDE's). The method constructs via solution of local problems a set of "optimal" test basis functions intended to provide maximum stability to the resulting formulation. Thus, discrete solutions become best solutions over a given discrete subspace, eliminating the so-called dispersion (pollution) error. In addition, the method delivers stiffess matrices that are always symmetric and positive definite, even for frequency-domain wave propagation problems where traditional finite element methods provide indefinite matrices. Moreover, the method also delivers an exact error representation function that can be used for automatic grid-refinements.
In this presentation, we will introduce the method starting from basic finite element concepts (Cea's lemma), and we will show 1D and 2D numerical applications in wave propagation problems. Then, we will have an open discussion to illustrate how the method can be combined with iso-geometric analysis and/or other type of trial basis functions that may be of interest to the audience.