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

Paola Causin; Riccardo Sacco; Carlo L. Bottasso

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

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topCausin, 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 - Modélisation Mathématique et Analyse Numérique 39.6 (2005): 1087-1114. <http://eudml.org/doc/245416>.

@article{Causin2005,

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 $H^1$-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 - Modélisation Mathématique et Analyse Numérique},

keywords = {finite element methods; mixed and hybrid methods; discontinuous Galerkin and Petrov–Galerkin methods; nonconforming finite elements; stabilized finite elements; upwinding; advection-diffusion problems; numerical examples; convergence; discontinuous Galerkin and Petrov-Galerkin methods},

language = {eng},

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/245416},

volume = {39},

year = {2005},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2005

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 $H^1$-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; numerical examples; convergence; discontinuous Galerkin and Petrov-Galerkin methods

UR - http://eudml.org/doc/245416

ER -

## References

top- [1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). Zbl0314.46030MR450957
- [2] D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19 (1985) 7–32. Zbl0567.65078
- [3] D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Discontinuous Galerkin methods. Lect. Notes Comput. Sci. Engrg. 11, Springer-Verlag (2000) 89–101. Zbl0948.65127
- [4] I. Babuska and J. Osborn, Generalized finite element methods, their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510–536. Zbl0528.65046
- [5] J. Baranger, J.F. Maitre and F. Oudin, Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 445–465. Zbl0857.65116
- [6] C.L. Bottasso, S. Micheletti and R. Sacco, The Discontinuous Petrov-Galerkin method for elliptic problems. Comput. Methods Appl. Mech. Engrg. 191 (2002) 3391–3409. Zbl1010.65050
- [7] C.L. Bottasso, S. Micheletti and R. Sacco, A multiscale formulation of the Discontinuous Petrov–Galerkin method for advective-diffusion problems. Comput. Methods Appl. Mech. Engrg. 194 (2005) 2819–2838. Zbl1093.76030
- [8] F. Brezzi, L.D. Marini and P. Pietra, Numerical simulation of semiconductor devices. Comput. Meths. Appl. Mech. Engrg. 75 (1989) 493–514. Zbl0698.76125
- [9] F. Brezzi, L.D. Marini and P. Pietra, Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26 (1989) 1342–1355. Zbl0686.65088
- [10] P. Causin, Mixed-hybrid Galerkin and Petrov-Galerkin finite element formulations in fluid mechanics. Ph.D. Thesis, Università degli Studi di Milano (2003).
- [11] P. Causin and R. Sacco, Mixed-hybrid Galerkin and Petrov-Galerkin finite element formulations in continuum mechanics. in Proc. of the Fifth World Congress on Computational Mechanics (WCCM V), Vienna, Austria. H.A. Mang, F.G. Rammerstorfer and J. Eberhardsteiner Eds., Vienna University of Technology, Austria, http://wccm.tuwien.ac.at, July 7–12 (2002).
- [12] P. Causin and R. Sacco, A Discontinuous Petrov–Galerkin method with Lagrangian multipliers for second order elliptic problems. SIAM J. Numer. Anal. 43 (2005) 280–302. Zbl1087.65105
- [13] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978). Zbl0383.65058MR520174
- [14] B. Cockburn and J. Gopalakhrisnan, A characterization of hybridized mixed methods for second order elliptic problems. SIAM Jour. Numer. Anal. 42 (2003) 283–301. Zbl1084.65113
- [15] M. Crouzeix and P.A. Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO, R-3 (1973) 33–76. Zbl0302.65087
- [16] C. Dawson, Godunov mixed methods for advection-diffusion equations in multidimensions. SIAM J. Numer. Anal. 30 (1993) 1315–1332. Zbl0791.65062
- [17] C. Dawson and V. Aizinger, Upwind-mixed methods for transport equations. Comp. Geosc. 3 (1999) 93–110. Zbl0962.65084
- [18] J. Gopalakhrisnan and G. Kanschat, A multilevel discontinuous galerkin method. Numer. Math. 95 (2003) 527–550. Zbl1044.65084
- [19] J. Jaffré, Décentrage et éléments finis mixtes pour les équations de diffusion-convection. Calcolo 2 (1984) 171–197. Zbl0562.65077
- [20] J.W. Jerome, Analysis of Charge Transport. Springer-Verlag, Berlin, Heidelberg (1996). Zbl0835.65151MR1437143
- [21] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1968). Zbl0165.10801
- [22] L.D. Marini, An inexpensive method for the evaluation of the solution of the lower order Raviart–Thomas method. SIAM J. Numer. Anal. 22 (1985) 493–496. Zbl0573.65082
- [23] P.A. Markowich, The Stationary Semiconductor Device Equations. Springer-Verlag, Wien, New York (1986). MR821965
- [24] S. Micheletti, R. Sacco and F. Saleri, On some mixed finite element methods with numerical integration. SIAM J. Sci. Comput. 23 (2001) 245–270. Zbl0992.65126
- [25] J.J. Miller and S. Wang, A new non-conforming Petrov–Galerkin finite element method with triangular elements for an advection-diffusion problem. IMA J. Numer. Anal. 14 (1994) 257–276. Zbl0806.65111
- [26] A. Mizukami and T.J.R. Hughes, A Petrov-Galerkin finite element method for convection–dominated flows: an accurate upwinding technique satisfying the discrete maximum principle. Comput. Meth. Appl. Mech. Engrg. 50 (1985) 181–193. Zbl0553.76075
- [27] K. Ohmori and T. Ushijima, A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations. RAIRO 3 (1984) 309–332. Zbl0586.65080
- [28] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, New York, Berlin (1994). Zbl0803.65088MR1299729
- [29] P.A. Raviart and J.M. Thomas, Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comp. 31-138 (1977) 391–413. Zbl0364.65082
- [30] J.E. Roberts and J.M. Thomas, Mixed and hybrid methods. In Finite Element Methods, Part I. P.G. Ciarlet and J.L. Lions (Eds.), North-Holland, Amsterdam 2 (1991). Zbl0875.65090MR1115239
- [31] H.G. Roos, M. Stynes and L. Tobiska, Numerical methods for singularly perturbed differential equations. Springer-Verlag, Berlin, Heidelberg (1996). Zbl0844.65075MR1477665
- [32] R. Sacco, E. Gatti and L. Gotusso, The patch test as a validation of a new finite element for the solution of convection-diffusion equations. Comp. Meth. Appl. Mech. Engrg. 124 (1995) 113–124. Zbl0948.78013
- [33] P. Siegel, R. Mosé, Ph. Ackerer and J. Jaffré, Solution of the advection-diffusion equation using a combination of discontinuous and mixed finite elements. Inter. J. Numer. Methods Fluids 24 (1997) 593–613. Zbl0894.76041
- [34] R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam (1977). Zbl0383.35057MR769654

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