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

Published online by Cambridge University Press:  30 July 2013

Gabriel R. Barrenechea ,
Volker John  and
Petr Knobloch
Gabriel R. Barrenechea
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, Scotland. gabriel.barrenechea@strath.ac.uk
Volker John
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany and Free University of Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany; john@wias-berlin.de
Petr Knobloch
Affiliation:
Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic; knobloch@karlin.mff.cuni.cz
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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.

Keywords

Finite element methodlocal projection stabilizationcrosswind diffusionconvection-diffusion-reaction equationwell posednesstime dependent problemstabilityerror estimates
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1335 - 1366
Copyright
© EDP Sciences, SMAI, 2013

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References

Augustin, M., Caiazzo, A., Fiebach, A., Fuhrmann, J., John, V., Linke, A. and Umla, R., An assessment of discretizations for convection-dominated convection-diffusion equations. Comput. Methods Appl. Mech. Engrg. 200 (2011) 3395–3409. Google Scholar
Becker, R. and Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001) 173–199. Google Scholar
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.
Braack, M. and Burman, E., Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43 (2006) 2544–2566. Google Scholar
Braack, M., Burman, E., John, V. and Lube, G., Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853–866. Google Scholar
Brooks, A.N. and Hughes, T.J.R., 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. Google Scholar
Burman, E. and Ern, A., Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence. Math. Comput. 74 (2005) 1637–1652. Google Scholar
Burman, E. and Fernández, M.A., Finite element methods with symmetric stabilization for the transient convection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg. 198 (2009) 2508–2519. Google Scholar
Burman, E. and Hansbo, P., Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437–1453. Google Scholar
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
Codina, R., A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation. Comput. Methods Appl. Mech. Engrg. 110 (1993) 325–342. Google Scholar
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Springer-Verlag, New York (2004).
Franca, L.P., Frey, S.L. and Hughes, T.J.R., Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Methods Appl. Mech. Engrg. 95 (1992) 253–276. Google Scholar
Franca, L.P. and Valentin, F., On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation. Comput. Methods Appl. Mech. Engrg. 190 (2000) 1785–1800. Google Scholar
Ganesan, S. and Tobiska, L., Stabilization by local projection for convection-diffusion and incompressible flow problems. J. Sci. Comput. 43 (2010) 326–342. Google Scholar
Hughes, T.J.R., Franca, L.P. and Hulbert, G.M., 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. Google Scholar
John, V. and Knobloch, P., 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. Google Scholar
John, V. and Knobloch, P., On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part II – Analysis for P 1 and Q 1 finite elements. Comput. Methods Appl. Mech. Engrg. 197 (2008) 1997–2014. Google Scholar
John, V., Knobloch, P. and Savescu, S.B., A posteriori optimization of parameters in stabilized methods for convection-diffusion problems – Part I. Comput. Methods Appl. Mech. Engrg. 200 (2011) 2916–2929. Google Scholar
John, V., Maubach, J.M. and Tobiska, L., Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math. 78 (1997) 165–188. Google Scholar
John, V., Mitkova, T., Roland, M., Sundmacher, K., Tobiska, L. and Voigt, A., Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Engrg. Sci. 64 (2009) 733–741. Google Scholar
John, V. and Novo, J., Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations. SIAM J. Numer. Anal. 49 (2011) 1149–1176. Google Scholar
John, V. and Schmeyer, E., Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion. Comput. Methods Appl. Mech. Engrg. 198 (2008) 475–494. Google Scholar
Knobloch, P., A generalization of the local projection stabilization for convection-diffusion-reaction equations. SIAM J. Numer. Anal. 48 (2010) 659–680. Google Scholar
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.
Knobloch, P. and Lube, G., Local projection stabilization for advection-diffusion-reaction problems: One-level vs. two-level approach. Appl. Numer. Math. 59 (2009) 2891–2907. Google Scholar
Knopp, T., Lube, G. and Rapin, G., Stabilized finite element methods with shock capturing for advection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 191 (2002) 2997–3013. Google Scholar
Ladyzhenskaya, O.A., 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. Steklova 102 (1967) 85–104. Google Scholar
Lube, G. and Rapin, G., residual-based stabilized higher-order FEM for advection-dominated problems. Comput. Methods Appl. Mech. Engrg. 195 (2006) 4124–4138. Google Scholar
Matthies, G., Skrzypacz, P. and Tobiska, L., A unified convergence analysis for local projection stabilizations applied to the Oseen problem. Math. Model. Numer. Anal. 41 (2007) 713–742. Google Scholar
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).
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis North-Holland, Amsterdam (1977).