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Weak formulation of Hadamard variation applied to the filtration problem

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Abstract

Quantities defined using a solution of an elliptic boundary value problem may vary when the boundary of the domain is perturbed. Such a variation with respect to domain perturbation is called Hadamard variation. We present a weak formulation of Hadamard variation and apply it to the filtration (or dam) problem. We obtain first Hadamard variations of quantities arising in the variational principle of the filtration problem. The correctness of the obtained first variations is confirmed by numerical experiments. Using the first variations, we propose a numerical iterative scheme for the filtration problem. Numerical examples show good performance of the presented scheme.

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References

  1. Alt H.W.: A free boundary problem associated with the flow of ground water. Arch. Ration. Mech. Anal. 64, 111–126 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alt H.W.: Numerical solution of steady-state porous flow free boundary problems. Numer. Math. 36, 73–98 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alt H.W., Gilardi G.: The behavior of the free boundary for the dam problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9, 571–626 (1982)

    MathSciNet  MATH  Google Scholar 

  4. Azegami H., Takeuchi K.: A smoothing method for shape optimization: traction method using the robin condition. Int. J. Comput. Methods 3, 21–33 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baiocchi C.: Su un problema di frontiera libera connesso a questioni di idraulica. Ann. Mat. Pura Appl. (4) 92, 107–127 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baiocchi C., Capelo A.: Variational and Quasivariational Inequalities. Applications to Free-Boundary Problems. Wiley, New York (1984)

    MATH  Google Scholar 

  7. Baiocchi C., Comincioli V., Magenes E., Pozzi G.A.: Free boundary problems in the theory of fluid flow through porous media: existence and uniqueness theorems. Ann. Mat. Pura Appl. (4) 97, 1–82 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brezis H., Kinderlehrer D., Stampacchia G.: Sur une nouvelle formulation du problème de l’écoulement à travers une digue. C. R. Acad. Sci. Paris 287, 711–714 (1978)

    MathSciNet  MATH  Google Scholar 

  9. Carrillo-Menendez J., Chipot M.: Sur l’unicité de la solution du problème de l’écoulement à travers une digue. C. R. Acad. Sci. Paris 292, 191–194 (1981)

    MathSciNet  Google Scholar 

  10. Friedman A.: Vaiational Principles and Free-Boundary Problems. Wiley-Interscience, New York (1982)

    Google Scholar 

  11. Haslinger J., Hoffmann K.-H., Mäkinen R.A.E.: Optimal control/dual approach for the numerical solution of a dam problem. Adv. Math. Sci. Appl. 2, 189–213 (1993)

    MathSciNet  MATH  Google Scholar 

  12. Kaizu S., Azegami H.: Optimal shape problems and traction method. Trans. Jpn. Soc. Ind. Appl. Math. 16, 277–290 (2006) (in Japanese)

    Google Scholar 

  13. Palmerio B., Dervieux A.: Hadamard’s variational formula for a mixed problem and an application to a problem related to a Signorini-like variational inequality. Numer. Funct. Anal. Optim. 1, 113–144 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sokolowski J., Zolesio J-P.: Introduction to Shape Optimization. Springer, Berlin (1992)

    MATH  Google Scholar 

  15. Suzuki T., Tsuchiya T.: Convergence analysis of trial free boundary methods for the two-dimensional filtration problem. Numer. Math. 100, 537–564 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka, 560-0043, Japan

    Takashi Suzuki

  2. Graduate School of Science and Engineering, Ehime University, Matsuyama, 790-8577, Japan

    Takuya Tsuchiya

Authors
  1. Takashi Suzuki
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  2. Takuya Tsuchiya
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Corresponding author

Correspondence to Takuya Tsuchiya.

Additional information

The authors are partially supported by Grant-in-Aid for Scientific Research (C), Project number 22540139, Japan Society for the Promotion of Science.

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Suzuki, T., Tsuchiya, T. Weak formulation of Hadamard variation applied to the filtration problem. Japan J. Indust. Appl. Math. 28, 327–350 (2011). https://doi.org/10.1007/s13160-011-0044-y

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  • DOI: https://doi.org/10.1007/s13160-011-0044-y

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