Exponentially Refined Mesh for Singularly Perturbed Interior Layer Problem

Abstract

The main aim of this article is to demonstrate the efficiency of the exponentially refined a priori mesh \((H(\ell )\)) on capturing the interior layer. Upwind scheme on the proposed \(H(\ell )\) mesh is considered for a class of singularly perturbed differential equation with discontinuous convection coefficient. We have estimated that the algorithm is perturbation parameter (\(\epsilon \)) uniformly convergent with error asymptotic to \(O(N^{-1}(g^{\ell }(N))^2)\) where \(g^\ell \) is an arithmetic function and \(\ell \in {\mathbb {N}}\). The numerical estimates based on \(H(\ell )\) mesh and other layer adapted meshes like B-mesh, Shishkin mesh and \(S(\ell )\) mesh are compared demonstrating the efficiency of the \(H(\ell )\) mesh on capturing the interior layer.

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Acknowledgements

We thank the referees and the editor for all their comments and suggestions to improve the presentation.

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Correspondence to V. P. Ramesh.

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Ramesh, V.P., Kadalbajoo, M.K., Prithvi, M. et al. Exponentially Refined Mesh for Singularly Perturbed Interior Layer Problem. Int. J. Appl. Comput. Math 6, 82 (2020). https://doi.org/10.1007/s40819-020-00839-w

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Keywords

  • Singular perturbation problem (SPP)
  • Interior layer
  • Generalized Shishkin
  • \(H(\ell )\) mesh
  • Finite difference scheme
  • B-type mesh

Mathematics Subject Classification

  • 34E15
  • 34K28