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

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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.

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Acknowledgments

The authors thank Prof. Jean R. Roche and anonymous referees for helping to improve the text. This work was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ).

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Authors and Affiliations

  1. Department of Mathematics, Federal University of Juiz de Fora, Juiz de Fora, Brazil

    Grigori Chapiro, Sandro R. Mazorche & Weslley S. Pereira

  2. Instituto de Matemática y Ciencias Afines (IMCA), Lima, Peru

    Angel E. R. Gutierrez

  3. Department of Mechanical and Materials Engineering, Military Institute of Engineering, Rio de Janeiro, Brazil

    José Herskovits

  4. Mechanical Engineering Program, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    José Herskovits

Authors
  1. Grigori Chapiro
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  2. Angel E. R. Gutierrez
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  3. José Herskovits
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  4. Sandro R. Mazorche
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  5. Weslley S. Pereira
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Correspondence to Grigori Chapiro.

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Communicated by Ilio Galligani.

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Chapiro, G., Gutierrez, A.E.R., Herskovits, J. et al. Numerical Solution of a Class of Moving Boundary Problems with a Nonlinear Complementarity Approach. J Optim Theory Appl 168, 534–550 (2016). https://doi.org/10.1007/s10957-015-0816-7

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