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Convergence Analysis of Two Numerical Schemes Applied to a Nonlinear Elliptic Problem

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Abstract

For a given nonlinear problem discretized by standard finite elements, we propose two iterative schemes to solve the discrete problem. We prove the well-posedness of the corresponding problems and their convergence. Next, we construct error indicators and prove optimal a posteriori estimates where we treat separately the discretization and linearization errors. Some numerical experiments confirm the validity of the schemes and allow us to compare them.

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References

  1. Adams, R.A.: Sobolev Spaces. Acadamic Press, INC (1978)

    MATH  Google Scholar 

  2. Bernardi, C., Dakroub, J., Mansour, G., Sayah, T.: A posteriori analysis of iterative algorithms for a nonlinear problem. J. Sci. Comput. 65(2), 672–697 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chaillou, A.-L., Suri, M.: Computable error estimators for the approximation of nonlinear problems by linearized models. Comput. Methods Appl. Mech. Eng. 196, 210–224 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chaillou, A.-L., Suri, M.: A posteriori estimation of the linearization error for strongly monotone nonlinear operators. Comput. Methods Appl. Mech. Eng. 205, 72–87 (2007)

    MathSciNet  MATH  Google Scholar 

  5. El Alaoui, L., Ern, A., Vohralík, M.: Guaranteed and robust a posteriori error estimate and balancing discretization and linearization errore for monotone nonlinear problems. Comput. Methods Appl. Mech. Eng. 200, 2782–2795 (2011)

    Article  MATH  Google Scholar 

  6. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–266 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Verfürth, R.: A Posteriori Error Estimation Techniques For Finite Element Methods. Numerical Mathematics and Scientific Computation, Oxford (2013)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions - C.N.R.S., Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 05, France

    Christine Bernardi & Jad Dakroub

  2. Unité de recherche EGFEM, Faculté des sciences, Université Saint-Joseph, Beirut, Lebanon

    Jad Dakroub, Gihane Mansour, Farah Rafei & Toni Sayah

Authors
  1. Christine Bernardi
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  2. Jad Dakroub
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  3. Gihane Mansour
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  4. Farah Rafei
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  5. Toni Sayah
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Corresponding author

Correspondence to Toni Sayah.

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Bernardi, C., Dakroub, J., Mansour, G. et al. Convergence Analysis of Two Numerical Schemes Applied to a Nonlinear Elliptic Problem. J Sci Comput 71, 329–347 (2017). https://doi.org/10.1007/s10915-016-0301-y

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  • DOI: https://doi.org/10.1007/s10915-016-0301-y

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