A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations

Gabriel R. Barrenechea; Volker John; Petr Knobloch

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 5, page 1335-1366
  • ISSN: 0764-583X

Abstract

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An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.

How to cite

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Barrenechea, Gabriel R., John, Volker, and Knobloch, Petr. "A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.5 (2013): 1335-1366. <http://eudml.org/doc/273303>.

@article{Barrenechea2013,
abstract = {An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.},
author = {Barrenechea, Gabriel R., John, Volker, Knobloch, Petr},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite element method; local projection stabilization; crosswind diffusion; convection-diffusion-reaction equation; well posedness; time dependent problem; stability; error estimates},
language = {eng},
number = {5},
pages = {1335-1366},
publisher = {EDP-Sciences},
title = {A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations},
url = {http://eudml.org/doc/273303},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Barrenechea, Gabriel R.
AU - John, Volker
AU - Knobloch, Petr
TI - A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 5
SP - 1335
EP - 1366
AB - An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.
LA - eng
KW - finite element method; local projection stabilization; crosswind diffusion; convection-diffusion-reaction equation; well posedness; time dependent problem; stability; error estimates
UR - http://eudml.org/doc/273303
ER -

References

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  1. [1] M. Augustin, A. Caiazzo, A. Fiebach, J. Fuhrmann, V. John, A. Linke and R. Umla, An assessment of discretizations for convection-dominated convection-diffusion equations. Comput. Methods Appl. Mech. Engrg.200 (2011) 3395–3409. Zbl1230.76021MR2844065
  2. [2] R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo38 (2001) 173–199. Zbl1008.76036MR1890352
  3. [3] R. Becker and M. Braack, A two-level stabilization scheme for the Navier-Stokes equations, Proc. of ENUMATH 2003, Numerical Mathematics and Advanced Applications, edited by M. Feistauer, V. Dolejıš´, P. Knobloch and K. Najzar. Springer-Verlag, Berlin (2004) 123–130. Zbl1198.76062MR2121360
  4. [4] M. Braack and E. Burman, Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal.43 (2006) 2544–2566. Zbl1109.35086MR2206447
  5. [5] M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg.196 (2007) 853–866. Zbl1120.76322MR2278180
  6. [6] A.N. Brooks and T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg.32 (1982) 199–259. Zbl0497.76041MR679322
  7. [7] E. Burman and A. Ern, Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence. Math. Comput.74 (2005) 1637–1652. Zbl1078.65088MR2164090
  8. [8] E. Burman and M.A. Fernández, Finite element methods with symmetric stabilization for the transient convection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg.198 (2009) 2508–2519. Zbl1228.76081MR2536082
  9. [9] E. Burman and P. Hansbo, Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg.193 (2004) 1437–1453. Zbl1085.76033MR2068903
  10. [10] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). Zbl0511.65078MR520174
  11. [11] R. Codina, A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation. Comput. Methods Appl. Mech. Engrg.110 (1993) 325–342. Zbl0844.76048MR1256324
  12. [12] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Springer-Verlag, New York (2004). Zbl1059.65103MR2050138
  13. [13] L.P. Franca, S.L. Frey and T.J.R. Hughes, Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Methods Appl. Mech. Engrg.95 (1992) 253–276. Zbl0759.76040MR1155924
  14. [14] L.P. Franca and F. Valentin, On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation. Comput. Methods Appl. Mech. Engrg.190 (2000) 1785–1800. Zbl0976.76038MR1807478
  15. [15] S. Ganesan and L. Tobiska, Stabilization by local projection for convection-diffusion and incompressible flow problems. J. Sci. Comput.43 (2010) 326–342. Zbl1203.76138MR2639639
  16. [16] T.J.R. Hughes, L.P. Franca and G.M. Hulbert, A new finite element formulation for computational fluid dynamics. VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Engrg. 73 (1989) 173–189. Zbl0697.76100MR1002621
  17. [17] V. John and P. Knobloch, On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part I – A review. Comput. Methods Appl. Mech. Engrg.196 (2007) 2197–2215. Zbl1173.76342MR2302890
  18. [18] V. John and P. Knobloch, On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part II – Analysis for P1 and Q1 finite elements. Comput. Methods Appl. Mech. Engrg.197 (2008) 1997–2014. Zbl1194.76122MR2417168
  19. [19] V. John, P. Knobloch and S.B. Savescu, A posteriori optimization of parameters in stabilized methods for convection-diffusion problems – Part I. Comput. Methods Appl. Mech. Engrg.200 (2011) 2916–2929. Zbl1230.76026MR2824164
  20. [20] V. John, J.M. Maubach and L. Tobiska, Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math.78 (1997) 165–188. Zbl0898.65068MR1485996
  21. [21] V. John, T. Mitkova, M. Roland, K. Sundmacher, L. Tobiska and A. Voigt, Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Engrg. Sci.64 (2009) 733–741. 
  22. [22] V. John and J. Novo, Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations. SIAM J. Numer. Anal.49 (2011) 1149–1176. Zbl1233.65065MR2812562
  23. [23] V. John and E. Schmeyer, Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion. Comput. Methods Appl. Mech. Engrg.198 (2008) 475–494. Zbl1228.76088MR2479278
  24. [24] P. Knobloch, A generalization of the local projection stabilization for convection-diffusion-reaction equations. SIAM J. Numer. Anal.48 (2010) 659–680. Zbl1251.65159MR2670000
  25. [25] P. Knobloch, Local projection method for convection-diffusion-reaction problems with projection spaces defined on overlapping sets. Proc. of ENUMATH 2009, Numerical Mathematics and Advanced Applications, edited by G. Kreiss, P. Lötstedt, A. M?lqvist and M. Neytcheva. Springer-Verlag, Berlin (2010) 497–505. Zbl1216.65157
  26. [26] P. Knobloch and G. Lube, Local projection stabilization for advection-diffusion-reaction problems: One-level vs. two-level approach. Appl. Numer. Math.59 (2009) 2891–2907. Zbl1180.65139MR2560823
  27. [27] T. Knopp, G. Lube and G. Rapin, Stabilized finite element methods with shock capturing for advection-diffusion problems. Comput. Methods Appl. Mech. Engrg.191 (2002) 2997–3013. Zbl1001.76058MR1903196
  28. [28] O.A. Ladyzhenskaya, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them. Tr. Mat. Inst. Steklova102 (1967) 85–104. Zbl0202.37802
  29. [29] G. Lube and G. Rapin, residual-based stabilized higher-order FEM for advection-dominated problems. Comput. Methods Appl. Mech. Engrg. 195 (2006) 4124–4138. Zbl1125.76042MR2229836
  30. [30] G. Matthies, P. Skrzypacz and L. Tobiska, A unified convergence analysis for local projection stabilizations applied to the Oseen problem. Math. Model. Numer. Anal.41 (2007) 713–742. Zbl1188.76226MR2362912
  31. [31] H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems, 2nd ed. Springer-Verlag, Berlin (2008). Zbl1155.65087MR2454024
  32. [32] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis North-Holland, Amsterdam (1977). Zbl0568.35002MR609732

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