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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Braianov, I.A.: Numerical solution of a mixed singularly perturbed parabolic-elliptic problem. J. Math. Anal. and Appl. 320, 361–380 (2006)
Braianov, I.A.: Uniformly convergent difference scheme for singularly perturbed problem of mixed type. Electron. Trans. Numer. Anal. 23, 288–303 (2006)
Mukherjee, K., Natesan, S.: Uniform convergence analysis of hybrid numerical scheme for singularly perturbed problems of mixed type, working paper (2013)
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)
Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd edn. Springer, Berlin (2008)
Stiemer, M.: A Galarkin method for mixed parabolic-elliptic partial differential equations. Numer. Math. 116, 435–462 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mukherjee, K., Natesan, S. (2013). An Efficient Hybrid Numerical Scheme for Singularly Perturbed Problems of Mixed Parabolic-Elliptic Type. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2012. Lecture Notes in Computer Science, vol 8236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41515-9_46
Download citation
DOI: https://doi.org/10.1007/978-3-642-41515-9_46
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41514-2
Online ISBN: 978-3-642-41515-9
eBook Packages: Computer ScienceComputer Science (R0)