Flux-upwind stabilization of the discontinuous Petrov–Galerkin formulation with Lagrange multipliers for advection-diffusion problems

¹ INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt BP 105, 78153 Le Chesnay Cedex, France
² Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy. riccardo.sacco@mate.polimi.it
³ D. Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Dr., 30332 Atlanta GA, USA

Received: 12 January 2005
Revised: 10 March 2005

Abstract

In this work we consider the dual-primal Discontinuous Petrov–Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous, a static condensation procedure can be carried out, leading to a single-field nonconforming discretization scheme. For this latter formulation, we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum principle under standard geometrical assumptions on the computational grid. A convergence analysis is developed, proving first-order accuracy of the method in a discrete H¹-norm, and the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers.