American Institute of Mathematical Sciences

October  2015, 8(5): 901-911. doi: 10.3934/dcdss.2015.8.901

Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations

 1 Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic

Received  February 2014 Revised  August 2014 Published  July 2015

A recently proposed local projection stabilization (LPS) finite element method containing a nonlinear crosswind diffusion term is analyzed for a transient convection-diffusion-reaction equation using a one-step $\theta$-scheme as temporal discretization. Both the fully nonlinear method and its semi-implicit variant are considered. Solvability of the discrete problem is established and a priori error estimates in the LPS norm are proved. Uniqueness of the discrete solution is proved for the semi-implicit approach or for sufficiently small time steps.
Citation: Petr Knobloch. Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 901-911. doi: 10.3934/dcdss.2015.8.901
References:
 [1] G. Barrenechea, V. John and P. Knobloch, A nonlinear local projection stabilization for convection-diffusion-reaction equations, in Numerical Mathematics and Advanced Applications 2011, Proceedings of ENUMATH 2011 (eds. A. Cangiani, R. Davidchack, E. Georgoulis, A. Gorban, J. Levesley and M. Tretyakov), Springer-Verlag, Berlin, 2013, 237-245. doi: 10.1007/978-3-642-33134-3_26. [2] S. Ganesan and L. Tobiska, Stabilization by local projection for convection-diffusion and incompressible flow problems, J. Sci. Comput., 43 (2010), 326-342. doi: 10.1007/s10915-008-9259-8. [3] 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. doi: 10.1016/j.cma.2006.11.013. [4] P. Knobloch, A generalization of the local projection stabilization for convection-diffusion-reaction equations, SIAM J. Numer. Anal., 48 (2010), 659-680. doi: 10.1137/090767807. [5] P. Knobloch, Local projection method for convection-diffusion-reaction problems with projection spaces defined on overlapping sets, in Numerical Mathematics and Advanced Applications 2009, Proceedings of ENUMATH 2009 (eds. G. Kreiss, P. Lötstedt, A. Målqvist and M. Neytcheva), Springer-Verlag, Berlin, 2010, 497-505. doi: 10.1007/978-3-642-11795-4_53. [6] G. Matthies, P. Skrzypacz and L. Tobiska, A unified convergence analysis for local projection stabilizations applied to the Oseen problem, M2AN Math. Model. Numer. Anal., 41 (2007), 713-742. doi: 10.1051/m2an:2007038. [7] 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.

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References:
 [1] G. Barrenechea, V. John and P. Knobloch, A nonlinear local projection stabilization for convection-diffusion-reaction equations, in Numerical Mathematics and Advanced Applications 2011, Proceedings of ENUMATH 2011 (eds. A. Cangiani, R. Davidchack, E. Georgoulis, A. Gorban, J. Levesley and M. Tretyakov), Springer-Verlag, Berlin, 2013, 237-245. doi: 10.1007/978-3-642-33134-3_26. [2] S. Ganesan and L. Tobiska, Stabilization by local projection for convection-diffusion and incompressible flow problems, J. Sci. Comput., 43 (2010), 326-342. doi: 10.1007/s10915-008-9259-8. [3] 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. doi: 10.1016/j.cma.2006.11.013. [4] P. Knobloch, A generalization of the local projection stabilization for convection-diffusion-reaction equations, SIAM J. Numer. Anal., 48 (2010), 659-680. doi: 10.1137/090767807. [5] P. Knobloch, Local projection method for convection-diffusion-reaction problems with projection spaces defined on overlapping sets, in Numerical Mathematics and Advanced Applications 2009, Proceedings of ENUMATH 2009 (eds. G. Kreiss, P. Lötstedt, A. Målqvist and M. Neytcheva), Springer-Verlag, Berlin, 2010, 497-505. doi: 10.1007/978-3-642-11795-4_53. [6] G. Matthies, P. Skrzypacz and L. Tobiska, A unified convergence analysis for local projection stabilizations applied to the Oseen problem, M2AN Math. Model. Numer. Anal., 41 (2007), 713-742. doi: 10.1051/m2an:2007038. [7] 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.
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