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

Paola Causin; Riccardo Sacco; Carlo L. Bottasso

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 6, page 1087-1114
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Causin, Paola, Sacco, Riccardo, and 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 39.6 (2010): 1087-1114. <http://eudml.org/doc/194298>.

@article{Causin2010,
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. },
author = {Causin, Paola, Sacco, Riccardo, Bottasso, Carlo L.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite element methods; mixed and hybrid methods; discontinuous Galerkin and Petrov–Galerkin methods; nonconforming finite elements; stabilized finite elements; upwinding; advection-diffusion problems.; finite element methods; numerical examples; convergence; discontinuous Galerkin and Petrov-Galerkin methods; nonconforming finite elements; stabilized finite elements; advection-diffusion problems},
language = {eng},
month = {3},
number = {6},
pages = {1087-1114},
publisher = {EDP Sciences},
title = {Flux-upwind stabilization of the discontinuous Petrov–Galerkin formulation with Lagrange multipliers for advection-diffusion problems},
url = {http://eudml.org/doc/194298},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Causin, Paola
AU - Sacco, Riccardo
AU - Bottasso, Carlo L.
TI - Flux-upwind stabilization of the discontinuous Petrov–Galerkin formulation with Lagrange multipliers for advection-diffusion problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 6
SP - 1087
EP - 1114
AB - 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.
LA - eng
KW - Finite element methods; mixed and hybrid methods; discontinuous Galerkin and Petrov–Galerkin methods; nonconforming finite elements; stabilized finite elements; upwinding; advection-diffusion problems.; finite element methods; numerical examples; convergence; discontinuous Galerkin and Petrov-Galerkin methods; nonconforming finite elements; stabilized finite elements; advection-diffusion problems
UR - http://eudml.org/doc/194298
ER -

References

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