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A Nonlinear Local Projection Stabilization for Convection-Diffusion-Reaction Equations

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Numerical Mathematics and Advanced Applications 2011

Abstract

We propose a new local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations. The discretization contains a crosswind diffusion term which depends on the unknown discrete solution in a nonlinear way. Consequently, the resulting method is nonlinear. Solvability of the nonlinear problem is established and an a priori error estimate in the LPS norm is proved. Numerical results show that the nonlinear crosswind diffusion term leads to a reduction of spurious oscillations.

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References

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Acknowledgements

The work of P. Knobloch is a part of the research project MSM 0021620839 financed by MSMT and it was partly supported by the Grant Agency of the Czech Republic under the grant No. 201/08/0012.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, Scotland

    G. R. Barrenechea

  2. Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117, Berlin, Germany

    V. John

  3. Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 6, 14195, Berlin, Germany

    V. John

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

    P. Knobloch

Authors
  1. G. R. Barrenechea
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  2. V. John
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  3. P. Knobloch
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Corresponding author

Correspondence to P. Knobloch .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Leicester, Leicester, LE!1 7RH, United Kingdom

    Andrea Cangiani

  2. , Department of Mathematics, University of Leicester, Leicester, LE1 7RL, United Kingdom

    Ruslan L. Davidchack

  3. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Emmanuil Georgoulis

  4. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  5. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Jeremy Levesley

  6. , School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

    Michael V. Tretyakov

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Barrenechea, G.R., John, V., Knobloch, P. (2013). A Nonlinear Local Projection Stabilization for Convection-Diffusion-Reaction Equations. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_26

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