# 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 & 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, (2011), 237. 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. 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. 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. 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, (2009), 497. 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. 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, (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, (2011), 237. 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. 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. 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. 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, (2009), 497. 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. 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, (2008).
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