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

Gabriel R. Barrenechea; Volker John; Petr Knobloch

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

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topBarrenechea, 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

top- [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] 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] 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] 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] 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] 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] 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] 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] 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] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). Zbl0511.65078MR520174
- [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] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Springer-Verlag, New York (2004). Zbl1059.65103MR2050138
- [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis North-Holland, Amsterdam (1977). Zbl0568.35002MR609732

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