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On the Numerical Solution of the Dirichlet Problem for the Elliptic \((\sigma _2)\) Equation

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Modeling, Simulation and Optimization for Science and Technology

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 34))

Abstract

The numerical solution of the Dirichlet problem for the real elliptic \((\sigma _2)\) equation for arbitrary domains in three dimensions is addressed with a least-squares method and a relaxation algorithm. This iterative approach allows to solve a sequence of linear variational problems and of algebraic eigenvalue problems independently. Mixed finite element approximations with a Tychonoff regularization are used for the discretization. Efficient algebraic solvers for the eigenvalue problems are coupled with a conjugate gradient algorithm for the solution of linear variational problems. Numerical results show the convergence of the iterative sequence to the exact solution, when such a solution exists. When a smooth solution does not exist, the proposed method allows to obtain an approximate solution in a least-squares sense.

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Acknowledgments

The author acknowledges the partial support of the National Science Foundation Grants NSF DMS-0412267 and NSF DMS-0913982. The author thanks Prof. E. Dean, Prof. R. Glowinski (Univ. of Houston), Prof. D. Sorensen (Rice University), Prof. M. Picasso (EPFL) for helpful comments and discussions. The author thanks Prof. M. Picasso and Prof. J. Rappaz, as part of this work has been done during an academic stay at MATHICSE, EPFL, Switzerland.

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Authors and Affiliations

  1. Geneva School of Business Administration (Haute Ecole de Gestion de Genève), Route de Drize 7, 1227, Carouge, Switzerland

    Alexandre Caboussat

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  1. Alexandre Caboussat
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Correspondence to Alexandre Caboussat .

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Editors and Affiliations

  1. College of Technology, University of Houston, Houston, Texas, USA

    William Fitzgibbon

  2. Department of Mathematics, University of Houston, Houston, Texas, USA

    Yuri A. Kuznetsov

  3. Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Pekka Neittaanmäki

  4. Laboratoire Jacques-Louis Lions, University of Paris VI, Paris, France

    Olivier Pironneau

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Caboussat, A. (2014). On the Numerical Solution of the Dirichlet Problem for the Elliptic \((\sigma _2)\) Equation. In: Fitzgibbon, W., Kuznetsov, Y., Neittaanmäki, P., Pironneau, O. (eds) Modeling, Simulation and Optimization for Science and Technology. Computational Methods in Applied Sciences, vol 34. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9054-3_2

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  • DOI: https://doi.org/10.1007/978-94-017-9054-3_2

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  • Online ISBN: 978-94-017-9054-3

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