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A Newton method for Bernoulli’s free boundary problem in three dimensions

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The present paper is dedicated to the numerical solution of Bernoulli’s free boundary problem in three dimensions. We reformulate the given free boundary problem as a shape optimization problem and compute the shape gradient and Hessian of the given shape functional. To approximate the shape problem we apply a Ritz–Galerkin discretization. The necessary optimality condition is resolved by Newton’s method. All information of the state equation, required for the optimization algorithm, are derived by boundary integral equations which we solve numerically by a fast wavelet Galerkin scheme. Numerical results confirm that the proposed Newton method yields an efficient algorithm to treat the considered class of problems.

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References

  1. Acker A. (1988). On the geometric form of Bernoulli configurations. Math Meth Appl Sci 10: 1–14

    Article  MATH  MathSciNet  Google Scholar 

  2. Acker, A.: Theorems and counterexamples on the geometry of solutions to Bernoulli free boundary problems. In: (Concus, P., et al., eds.) Advances in Geometric Analysis and Continuum Mechanics (Stanford, CA, 1993), pp. 8–21. Int. Press, Cambridge, MA (1995)

  3. Alt H. W. and Caffarelli L. A. (1981). Existence and regularity for a minimum problem with free boundary. J Reine Angew Math 325: 105–144

    MATH  MathSciNet  Google Scholar 

  4. Colton, D., Kress, R.: Integral equation methods in scattering theory. In: Pure and Applied Mathematics. Wiley, Chichester (1983)

  5. Dennis J. E. and Schnabel R. B. (1983). Numerical methods for nonlinear equations and unconstrained optimization techniques. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  6. Delfour M. and Zolesio J.-P. (2001). Shapes and geometries. SIAM, Philadelphia

    MATH  Google Scholar 

  7. Dahmen W., Harbrecht H. and Schneider R. (2006). Compression techniques for boundary integral equations – optimal complexity estimates. SIAM J Numer Anal 43: 2251–2271

    Article  MATH  MathSciNet  Google Scholar 

  8. Dahmen W. and Kunoth A. (1992). Multilevel preconditioning. Numer Math 63: 315–344

    Article  MATH  MathSciNet  Google Scholar 

  9. Eppler K. (2000). Boundary integral representations of second derivatives in shape optimization. Discuss Math Differ Incl Control Optim 20: 63–78

    MATH  MathSciNet  Google Scholar 

  10. Eppler K. (2000). Optimal shape design for elliptic equations via BIE-methods. Int J Appl Math Comput Sci 10: 487–516

    MATH  MathSciNet  Google Scholar 

  11. Eppler, K., Harbrecht, H.: Shape optimization for 3D electrical impedance tomography. In: (Glowin, R., et al., eds.) Free and Moving Boundaries: Analysis, Simulation and Control, Lecture Notes in Pure and Applied Mathematics, vol. 252, pp. 165–184. Marcel Dekker, Boca Raton (2007)

  12. Eppler K. and Harbrecht H. (2006). Efficient treatment of stationary free boundary problems. Appl Numer Math 56: 1326–1339

    Article  MATH  MathSciNet  Google Scholar 

  13. Eppler, K., Harbrecht, H.: Tracking Neumann data for stationary free boundary problems. Preprint No. 313, Berichtsreihe des SFB 611, Universität Bonn (2006, submitted)

  14. Eppler K., Harbrecht H. and Schneider R. (2007). On convergence in elliptic shape optimization. SIAM J Control Optim 45: 61–83

    Article  MathSciNet  Google Scholar 

  15. Flucher M. and Rumpf M. (1997). Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J Reine Angew Math 486: 165–204

    MATH  MathSciNet  Google Scholar 

  16. Grossmann C. and Terno J. (1993). Numerik der Optimierung. Teubner, Stuttgart

    MATH  Google Scholar 

  17. Harbrecht H. and Schneider R. (2006). Wavelet Galerkin schemes for boundary integral equations – implementation and quadrature. SIAM J Sci Comput 27: 1347–1370

    Article  MATH  MathSciNet  Google Scholar 

  18. Haslinger J., Kozubek T., Kunisch K. and Peichl G. (2003). Shape optimization and fictitious domain approach for solving free boundary value problems of Bernoulli type. Comput Optim Appl 26: 231–251

    Article  MATH  MathSciNet  Google Scholar 

  19. Murat, F., Simon, J.: Étude de problèmes d’optimal design. In: (Céa, J., eds.) Optimization Techniques, Modeling and Optimization in the Service of Man. Lect. Notes Comput. Sci., vol. 41, pp. 54–62. Springer, Berlin (1976)

  20. Pironneau O. (1983). Optimal shape design for elliptic systems. Springer, New York

    Google Scholar 

  21. Potthast, R.: Fréchet-Differenzierbarkeit von Randintegraloperatoren und Randwertproblemen zur Helmholtzgleichung und den zeitharmonischen Maxwellgleichungen. Ph.D Thesis, Universität Göttingen (1994)

  22. Simon J. (1980). Differentiation with respect to the domain in boundary value problems. Numer Funct Anal Optim 2: 649–687

    Article  MATH  MathSciNet  Google Scholar 

  23. Sokolowski J. and Zolesio J.-P. (1992). Introduction to shape optimization. Springer, Berlin

    MATH  Google Scholar 

  24. Tiihonen T. (1997). Shape optimization and trial methods for free-boundary problems. RAIRO Modél Math Anal Numér 31: 805–825

    MATH  MathSciNet  Google Scholar 

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Correspondence to Helmut Harbrecht.

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Harbrecht, H. A Newton method for Bernoulli’s free boundary problem in three dimensions. Computing 82, 11–30 (2008). https://doi.org/10.1007/s00607-008-0260-8

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  • DOI: https://doi.org/10.1007/s00607-008-0260-8

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