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Adaptive Discontinuous Galerkin Methods for Nonlinear Diffusion-Convection-Reaction Equations

  • Bulent Karasözen
  • Murat Uzunca
  • Murat Manguoǧlu
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)

Abstract

In this work, we apply the adaptive discontinuous Galerkin method (DGAFEM) to the convection dominated nonlinear, quasi-steady state convection diffusion reaction equations. We propose an efficient algorithm to solve the sparse linear systems iteratively arising from the discretized nonlinear equations. Numerical examples demonstrate the effectiveness of the DGAFEM to damp the spurious oscillations for the convection dominated nonlinear equations.

Keywords

Posteriori Error Discontinuous Galerkin Method Spurious Oscillation Sparse Linear System Nonlinear Convection 
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References

  1. 1.
    D. Arnold, F. Brezzi, B. Cockburn, L. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    B. Ayuso, L.D. Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 47, 1391–1420 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    M. Bause, Stabilized finite element methods with shock-capturing for nonlinear convection-diffusion-reaction models, in Numerical Mathematics and Advanced Applications, ed. by G. Kreiss, P. Lötstedt, A. Møalqvist, M. Neytcheva (Springer, Berlin, 2010), pp. 125–134Google Scholar
  4. 4.
    M. Bause, K. Schwegler, Higher order finite element approximation of systems of convection-diffusion-reaction equations with small diffusion. J. Comput. Appl. Math. 246, 52–64 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    L. Chen, iFEM: an innovative finite element method package in MATLAB. Technical report, Department of Mathematics, University of California, Irvine, 2008Google Scholar
  6. 6.
    D. Schötzau, L. Zhu, A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations. Appl. Numer. Math. 59, 2236–2255 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    O. Tarı, M. Manguoğlu, A new sparse matrix reordering scheme using the largest eigenvector of the graph Laplacian. Numerical Linear Algebra with Applications, in review. http://people.maths.ox.ac.uk/ekertl/PRECON13/talks/Talk_Tari_Manguoglu.pdf
  8. 8.
    R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods (Oxford University Press, Oxford, 2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    H. Yücel, M. Stoll, P. Benner, Discontinuous Galerkin finite element methods with schock-capturing for nonlinear convection dominated models. Comput. Chem. Eng. 58, 278–287 (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Bulent Karasözen
    • 1
  • Murat Uzunca
    • 2
  • Murat Manguoǧlu
    • 3
  1. 1.Department of Mathematics and Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey
  3. 3.Department of Computer Engineering, Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey

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