Flux-upwind stabilization of the discontinuous Petrov-Galerkin formulation with Lagrange multipliers for advection-diffusion problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 6, p. 1087-1114

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 ${H}^{1}$-norm, and the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers.

DOI : https://doi.org/10.1051/m2an:2005050
Classification:  65N99
Keywords: finite element methods, mixed and hybrid methods, discontinuous Galerkin and Petrov-Galerkin methods, nonconforming finite elements, stabilized finite elements, upwinding, advection-diffusion problems
@article{M2AN_2005__39_6_1087_0,
author = {Causin, Paola and Sacco, Riccardo and Bottasso, Carlo L.},
title = {Flux-upwind stabilization of the discontinuous Petrov-Galerkin formulation with Lagrange multipliers for advection-diffusion problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {39},
number = {6},
year = {2005},
pages = {1087-1114},
doi = {10.1051/m2an:2005050},
zbl = {1084.65105},
language = {en},
url = {http://www.numdam.org/item/M2AN_2005__39_6_1087_0}
}

Causin, Paola; Sacco, Riccardo; Bottasso, Carlo L. Flux-upwind stabilization of the discontinuous Petrov-Galerkin formulation with Lagrange multipliers for advection-diffusion problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 6, pp. 1087-1114. doi : 10.1051/m2an:2005050. http://www.numdam.org/item/M2AN_2005__39_6_1087_0/

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