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An Efficient Hybrid Numerical Scheme for Singularly Perturbed Problems of Mixed Parabolic-Elliptic Type

  • Kaushik Mukherjee
  • Srinivasan Natesan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)

Abstract

This article is dealt with the study of a hybrid numerical scheme for a class of singularly perturbed mixed parabolic-elliptic problems possessing both boundary and interior layers. The domain under consideration is partitioned into two subdomains. In the first subdomain, the given problem takes the form of parabolic reaction-diffusion type, whereas in the second subdomain elliptic convection-diffusion-reaction types of problems are posed. To solve these problems, the time derivative is discretized by the backward-Euler method, while for the spatial discretization the classical central difference scheme is used on the first subdomain and a hybrid finite difference scheme is proposed on the second subdomain. The proposed method is designed on a layer resolving piecewise-uniform Shishkin mesh and computationally it is shown that the method converges ε-uniformly with almost second-order spatial accuracy in the discrete supremum norm.

Keywords

Spatial Discretization Hybrid Scheme Uniform Mesh Interior Layer Rectangular Mesh 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Braianov, I.A.: Numerical solution of a mixed singularly perturbed parabolic-elliptic problem. J. Math. Anal. and Appl. 320, 361–380 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Braianov, I.A.: Uniformly convergent difference scheme for singularly perturbed problem of mixed type. Electron. Trans. Numer. Anal. 23, 288–303 (2006)MathSciNetGoogle Scholar
  3. 3.
    Mukherjee, K., Natesan, S.: Uniform convergence analysis of hybrid numerical scheme for singularly perturbed problems of mixed type, working paper (2013)Google Scholar
  4. 4.
    Mukherjee, K., Natesan, S.: ε-Uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with interior layers. Numer. Alogrithms 58(1), 103–141 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd edn. Springer, Berlin (2008)zbMATHGoogle Scholar
  6. 6.
    Stiemer, M.: A Galarkin method for mixed parabolic-elliptic partial differential equations. Numer. Math. 116, 435–462 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Kaushik Mukherjee
    • 1
  • Srinivasan Natesan
    • 2
  1. 1.Department of MathematicsIndian Institute of Space Science and TechnologyThiruvananthapuramIndia
  2. 2.Department of MathematicsIndian Institute of Technology GuwahatiGuwahatiIndia

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