Talk on Jun 13, 2013

Thursday, Jun 13, 2013 02:00 pm

On the Stability of the Discontinuous Petrov-Galerkin Method

Felix Gruber, M.Sc., Angela Klewinghaus, M.Sc. (RWTH Aachen)

Abstract:
In many Finite Element applications establishing inf-sup stability (with constant bounded away from zero) is a mayor challenge. Recently, this problem has been addressed by Dahmen et al. [1], Demkowicz et al. [2] and Stevenson et al. [3] by constructing optimal test spaces that guarantee an inf-sup constant of one. In general, these optimal test spaces cannot be computed exactly. The existing theory for the finite dimensional approximation of those test spaces is limited to abstract concepts, which have so far only been proven for some simple example problems (e.g. Laplace). Nonetheless, Demkowicz et al. have successfully applied such ideas in the context of Discontinuous Petrov Galerkin (DPG) methods to a vast range of equations. In this talk, we will present the theoretical background of DPG methods and apply it to convection problems and to the multidimensional Helmholtz equation. Furthermore, we will point out and discuss remaining open questions.

[1] Dahmen, W.; Huang, C.; Schwab, C. & Welper, G. Adaptive Petrov-Galerkin Methods for First Order Transport Equations, SIAM Journal on Numerical Analysis, 2012, 50, 2420-2445
[2] Demkowicz, L. & Gopalakrishnan, J. An Overview of the DPG Method, ICES Report 13-02, ICES, UT Austin, 2013 [3] Broersen, D. & Stevenson, R. A Petrov-Galerkin Discretization with Optimal Test Space of a Mild-Weak Formulation of Convection-Diffusion Equations in Mixed Form, preprint, 2012

Time: 02:00 pm

Location: Room 149, Hauptgebäude, RWTH Aachen, Templergraben 55, 52056 Aachen