Skip to main content
Top
Published in: Journal of Applied Mathematics and Computing 1-2/2013

01-07-2013 | Original Research

A posteriori error analysis for defect correction method for two parameter singular perturbation problems

Authors: Mohan K. Kadalbajoo, Anuradha Jha

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2013

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Defect correction method is used for two parameter singular perturbation problem on Bakhvalov-Shishkin mesh. Use of defect correction method on Bakhvalov-Shishkin mesh gives a second order convergence. A posteriori error estimate is obtained. The numerical examples are given to establish the second order convergence in practice.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Bakhvalov, N.S.: Towards optimization of methods for solving boundary value problems in the presence of a boundary layer. Zh. Vychisl. Mat. Mat. Fiz. 9, 101–117 (1969) (in Russian) Bakhvalov, N.S.: Towards optimization of methods for solving boundary value problems in the presence of a boundary layer. Zh. Vychisl. Mat. Mat. Fiz. 9, 101–117 (1969) (in Russian)
2.
go back to reference Böhmer, K., Hemker, P., Stetter, H.J.: The defect correction approach. Computing, Suppl. 5, 1–5 (1984) CrossRef Böhmer, K., Hemker, P., Stetter, H.J.: The defect correction approach. Computing, Suppl. 5, 1–5 (1984) CrossRef
3.
go back to reference Ervin, V., Layton, W.: An analysis of defect-correction method for a model convection-diffusion equation. SIAM J. Numer. Anal. 26(1), 169–179 (1989) MathSciNetMATHCrossRef Ervin, V., Layton, W.: An analysis of defect-correction method for a model convection-diffusion equation. SIAM J. Numer. Anal. 26(1), 169–179 (1989) MathSciNetMATHCrossRef
4.
5.
go back to reference Hemker, P.: The use of defect correction for the solution of a singularly perturbed O.D.E. Seminarber. Humboldt-Univ. Berlin Sekt. Math. 46, 91–103 (1982) MATH Hemker, P.: The use of defect correction for the solution of a singularly perturbed O.D.E. Seminarber. Humboldt-Univ. Berlin Sekt. Math. 46, 91–103 (1982) MATH
6.
go back to reference Linß, T.: An upwind difference scheme on a novel Shishkin-type mesh for a linear convection-diffusion problem. J. Comput. Appl. Math. 110, 93–104 (1999) MathSciNetMATHCrossRef Linß, T.: An upwind difference scheme on a novel Shishkin-type mesh for a linear convection-diffusion problem. J. Comput. Appl. Math. 110, 93–104 (1999) MathSciNetMATHCrossRef
7.
go back to reference Linß, T.: Layer-adapted meshes for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 192, 1061–1105 (2003) MATHCrossRef Linß, T.: Layer-adapted meshes for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 192, 1061–1105 (2003) MATHCrossRef
8.
go back to reference Linß, T.: Error expansion for a first order upwind scheme applied to a model convection-diffusion problem. IMA J. Numer. Anal. 24(2), 239–253 (2004) MathSciNetMATHCrossRef Linß, T.: Error expansion for a first order upwind scheme applied to a model convection-diffusion problem. IMA J. Numer. Anal. 24(2), 239–253 (2004) MathSciNetMATHCrossRef
9.
go back to reference Linß, T.: Layer adapted meshes for convection diffusion problems. Habilitation thesis, Technische Universität Dreston (2006) Linß, T.: Layer adapted meshes for convection diffusion problems. Habilitation thesis, Technische Universität Dreston (2006)
10.
go back to reference Linß, T.: A posteriori error estimation for a singularly perturbed problem with two small parameters. Int. J. Numer. Anal. Model. 7(3), 491–506 (2010) MathSciNetMATH Linß, T.: A posteriori error estimation for a singularly perturbed problem with two small parameters. Int. J. Numer. Anal. Model. 7(3), 491–506 (2010) MathSciNetMATH
11.
go back to reference Linß, T.: Layer Adapted Meshes for Reaction-Convection Diffusion Problems. Lecture Notes in Mathematics, vol. 1985. Springer, Berlin (2010) MATHCrossRef Linß, T.: Layer Adapted Meshes for Reaction-Convection Diffusion Problems. Lecture Notes in Mathematics, vol. 1985. Springer, Berlin (2010) MATHCrossRef
12.
go back to reference Linß, T., Kopteva, N.: A posteriori error estimation for defect correction method applied to convection-diffusion problems. Int. J. Numer. Anal. Model. 7(4), 718–733 (2010) MathSciNetMATH Linß, T., Kopteva, N.: A posteriori error estimation for defect correction method applied to convection-diffusion problems. Int. J. Numer. Anal. Model. 7(4), 718–733 (2010) MathSciNetMATH
13.
go back to reference Linß, T., Roos, H.G.: Analysis of a finite difference scheme for a singularly perturbed problem with two small parameters. J. Math. Anal. Appl. 289(2), 355–366 (2004) MathSciNetMATHCrossRef Linß, T., Roos, H.G.: Analysis of a finite difference scheme for a singularly perturbed problem with two small parameters. J. Math. Anal. Appl. 289(2), 355–366 (2004) MathSciNetMATHCrossRef
14.
go back to reference Natesan, S., Gracia, J.L., Clavero, C.: Singularly perturbed boundary value problem with two small parameters—a defect correction approach. In: Proceedings of the International Conference on Boundary and Interior Layers—Computational and Asymptotic Methods, BAIL 2004, ONERA—Centre de Toulouse, France, pp. 1–6 Natesan, S., Gracia, J.L., Clavero, C.: Singularly perturbed boundary value problem with two small parameters—a defect correction approach. In: Proceedings of the International Conference on Boundary and Interior Layers—Computational and Asymptotic Methods, BAIL 2004, ONERA—Centre de Toulouse, France, pp. 1–6
15.
go back to reference O’Riordan, E., Shishkin, G.I., Picket, M.L.: Singularly perturbed problems modelling reaction-convection-diffusion processes. Comput. Methods Appl. Math. 3(3), 424–442 (2003) MathSciNetMATH O’Riordan, E., Shishkin, G.I., Picket, M.L.: Singularly perturbed problems modelling reaction-convection-diffusion processes. Comput. Methods Appl. Math. 3(3), 424–442 (2003) MathSciNetMATH
16.
go back to reference Gracia, J.L., O’Riordan, E., Picket, M.L.: A parameter robust second order numerical method for a singularly perturbed two-parameter problem. Appl. Numer. Math. 56(7), 962–980 (2006) MathSciNetMATHCrossRef Gracia, J.L., O’Riordan, E., Picket, M.L.: A parameter robust second order numerical method for a singularly perturbed two-parameter problem. Appl. Numer. Math. 56(7), 962–980 (2006) MathSciNetMATHCrossRef
17.
go back to reference Roos, H.G., Uzelac, Z.: The SDFEM for a convection diffusion problem with two small parameters. Comput. Methods Appl. Math. 3(3), 443–458 (2003) MathSciNetMATH Roos, H.G., Uzelac, Z.: The SDFEM for a convection diffusion problem with two small parameters. Comput. Methods Appl. Math. 3(3), 443–458 (2003) MathSciNetMATH
19.
go back to reference Surla, K., Uzelac, Z., Teofanov, L.: The discrete minimum principle for quadratic spline discretization of a singularly perturbed problem. Math. Comput. Simul. 79, 2490–2505 (2009) MathSciNetMATHCrossRef Surla, K., Uzelac, Z., Teofanov, L.: The discrete minimum principle for quadratic spline discretization of a singularly perturbed problem. Math. Comput. Simul. 79, 2490–2505 (2009) MathSciNetMATHCrossRef
Metadata
Title
A posteriori error analysis for defect correction method for two parameter singular perturbation problems
Authors
Mohan K. Kadalbajoo
Anuradha Jha
Publication date
01-07-2013
Publisher
Springer-Verlag
Published in
Journal of Applied Mathematics and Computing / Issue 1-2/2013
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-012-0628-y

Other articles of this Issue 1-2/2013

Journal of Applied Mathematics and Computing 1-2/2013 Go to the issue

Premium Partner