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Numerical Solution of a Class of Moving Boundary Problems with a Nonlinear Complementarity Approach

Author

Listed:
  • Grigori Chapiro

    (Federal University of Juiz de Fora)

  • Angel E. R. Gutierrez

    (Instituto de Matemática y Ciencias Afines (IMCA))

  • José Herskovits

    (Military Institute of Engineering
    Federal University of Rio de Janeiro)

  • Sandro R. Mazorche

    (Federal University of Juiz de Fora)

  • Weslley S. Pereira

    (Federal University of Juiz de Fora)

Abstract

Parabolic-type problems, involving a variational complementarity formulation, arise in mathematical models of several applications in Engineering, Economy, Biology and different branches of Physics. These kinds of problems present several analytical and numerical difficulties related, for example, to time evolution and a moving boundary. We present a numerical method that employs a global convergent nonlinear complementarity algorithm for solving a discretized problem at each time step. Space discretization was implemented using both the finite difference implicit scheme and the finite element method. This method is robust and efficient. Although the present method is general, at this stage we only apply it to two one-dimensional examples. One of them involves a parabolic partial differential equation that describes oxygen diffusion problem inside one cell. The second one corresponds to a system of nonlinear differential equations describing an in situ combustion model. Both models are rewritten in the quasi-variational form involving moving boundaries. The numerical results show good agreement when compared to direct numerical simulations.

Suggested Citation

  • Grigori Chapiro & Angel E. R. Gutierrez & José Herskovits & Sandro R. Mazorche & Weslley S. Pereira, 2016. "Numerical Solution of a Class of Moving Boundary Problems with a Nonlinear Complementarity Approach," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 534-550, February.
    • Handle: RePEc:spr:joptap:v:168:y:2016:i:2:d:10.1007_s10957-015-0816-7
    DOI: 10.1007/s10957-015-0816-7
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    References listed on IDEAS

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    1. J. Herskovits, 1998. "Feasible Direction Interior-Point Technique for Nonlinear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 121-146, October.
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    Cited by:

    1. Canelas, Alfredo & Pereira, Antonio & Roche, Jean R. & Brancher, Jean P., 2019. "Solution of the equilibrium problem in electromagnetic casting considering a solid inclusion in the melt," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 160(C), pages 126-137.
    2. Angel E. R. Gutierrez & Sandro R. Mazorche & José Herskovits & Grigori Chapiro, 2017. "An Interior Point Algorithm for Mixed Complementarity Nonlinear Problems," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 432-449, November.

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