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

Issues

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
  • Email:
/ B. V. Rathish Kumar
  • Corresponding author
  • Department of Mathematics and Statistics, IIT Kanpur, Kanpur-208016, India
  • Email:
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.

Keywords: singularly perturbed problems; Taylor Galerkin method; monotone Schwarz iterative method; maximum principle

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

References

  • [1] H. G. Roos,M. Stynes and L. Tobiska, Numericalmethods for singularly perturbed differential equations, convectiondiffusion and flow problems, Springer, Berlin, 1996.

  • [2] N. S. Bakhvalov, On the optimization of methods of solving boundary value problems with boundary layers, USSR Comp. Mat. and Mat. Phys. 9 (1969), 139–166.

  • [3] G. I. Shishkin, Approximation of the solutions of singularly perturbed boundary-value problems with a parabolic boundary layer, USSR Comp. Mat. andMat. Phys. 29 (1989), 01–10.

  • [4] E. P. Doolan, J. H. H.Miller andW. H. A. Schilders, Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dublin, 1980.

  • [5] M. Stynes and E. O’Riorden, L1 and L uniform convergence of a difference scheme for a semilinear singular perturbation problem, Numer. Math. 50 (1987), 519–531.

  • [6] M. Stynes and E. O’Riorden, Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points, Numer. Math. 55 (1989), 521–544.

  • [7] W. Guo and M. Stynes, Finite element analysis of exponentially fitted Lumped schemes for time-dependent convection diffusion problems, Numer. Math. 66 (1993), 347–371.

  • [8] C. E. Pearson, On a differential equation of boundary layer type, J. Math. Phys. Sci. 47 (1968), 134–154.

  • [9] J. J. H. Miller, E. O’Riorden, and G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific, Singapore, 1996.

  • [10] R. B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32 (1978), 1025–1039.

  • [11] D. Duffy, Uniformly convergent difference schemes for problems with a small parameter in the leading derivative, Ph.D. thesis, School of Mathematics, Trinity College, Dublin, 1980.

  • [12] L. Liu and Y. Chen, Maximum norm a posteriori error estimates for a singularly perturbed differential difference equation with small delay, Appl. Math. Comp. 227 (2014), 801–810.

  • [13] C. V. Pao, Nonlinear parabolic and elliptic equations, Plenus Press, New York, 1992.

  • [14] C. V. Pao, Monotone iterations for numerical solutions of reaction-diffusion -convection equations with time delay, Nume. Methods for PDEs 14 (1998), 339–351.

  • [15] X. Lu, Combined methods for numerical solutions of parabolic problems with time delays, Appl. Math. Comput. 89 (1994), 213–254.

  • [16] X. Lu, Monotone method and convergence acceleration for finite difference solution of parabolic problems with delays, Nume. Methods for PDEs, 11 (1995), 591–602.

  • [17] I. Boglaev, Monotone Schwarz iterates for a semilinear parabolic convection-diffusion problem, J. Comput. and Appl. Math. 183 (2005), 191–209.

  • [18] V. Sangwan and B. V. R. Kumar, Finite element analysis for mass-lumped three-step Taylor Galerkin method for time dependent singularly perturbed problems with exponentially fitted splines, Num. Func. Anal. and Opt. 33 (2012), 638–660.

  • [19] I. Boglaev, On a domain decompostion algorithm for a singularly perturbed reaction-diffusion problem, J. Comput. and Appl. Math. 98 (1998), 213–232.

  • [20] V. Sangwan, Finite element analysis and parallel computation of singulary perturbed problems using three-step Taylor Galerkin method, Ph.D. thesis,IIT Kanpur, India, 2011.

  • [21] D. Y. Tzou, Micro-to-macroscale heat transfer, Taylor and Francis, Washington, DC, 1997.

  • [22] D. D. Joseph, L. Preziosi, Addendum to the paper heat waves, Rev. Mod. Phys. 62 (1989), 375–391.

  • [23] Q. Liu, X. Wang, D. De Kee, Mass transport through swelling membranes, Int. J. Eng. Sci. 43 (2005), 1464–1470.

  • [24] M. Bestehorn, E. V. Grigorieva, Formation and propagation of localized status in extended systems, Ann. Phys. (Liepzig), 13 (2004), 423–431.

  • [25] M A. Ezzat, M. I. Othman, A. M. S. El-Karamany, State space approach to two-dimensional generalized thermo-viscoelasticity with relaxation times, Int. J. Eng. Sci., 40 (2002), 1251–1274.

  • [26] L. Bobisud, Second-order linear parabolic equations with small parameter, Arch. Rational Mech. Anal. 27 (1967), 385–397.

  • [27] P. D. Lax and B. Wendroff, System of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237.

  • [28] P. D. Lax and B. Wendroff, On the stability of difference schemes, Comm. Pure Appl. Math. 15 (1962), 363–371.

  • [29] P. D. Lax and B. Wendroff, Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math. 17 (1964), 381–398.

  • [30] M. Garbey, Yu. A. Kuznetsov and Yu. V. Vassilevski, A parallel Schwarz method for a convection-diffusion problem, SIAM J. Sci. Comput. 22 (2000), 891–916.

About the article

Received: 2015-10-29

Accepted: 2016-12-14

Published Online: 2017-01-20

Published in Print: 2017-02-01


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2015-0156. Export Citation

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