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# International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board Member: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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ISSN
2191-0294
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### Volume 18, Issue 1 (Feb 2017)

30,00 € / \$42.00 / £23.00

# A Finite Element Domain Decomposition Approximation for a Semilinear Parabolic Singularly Perturbed Differential Equation

Sunil Kumar
• Department of Mathematics and Statistics, IIT Kanpur, Kanpur-208016, India
• :
/ B. V. Rathish Kumar
• Department of Mathematics and Statistics, IIT Kanpur, Kanpur-208016, India
• :
Published Online: 2017-01-20 | DOI: https://doi.org/10.1515/ijnsns-2015-0156

## Abstract

In this paper, we propose a Monotone Schwarz Iterative Method (MSIM) under the framework of Domain Decomposition Strategy for solving semilinear parabolic singularly perturbed partial differential equations (SPPDEs). A three-step Taylor Galerkin Finite Element (3TGFE) approximation of semilinear parabolic SPPDE is carried out during each of the stages of the MSIM. Appropriate Interface Problems are introduced to update the subdomain boundary conditions in the Monotone Iterative Domain Decomposition (MIDD) method. The convergence of the MIDD method has been established. In addition, the stability and $ϵ$-uniform convergence of 3TGFE based MIDD has been discussed. Further, by using maximum principle and induction hypothesis, the convergence of the proposed MSIM has been established. Also, the proposed 3TGFE based MIDD has been successfully implemented on a couple of test problems.

PACS: 34K10; 34K28; 35K20; 35K58; 65M12; 65M55; 65M60