Exterior Dirichlet and Neumann Problems in Domains with Random Boundaries

Abstract

An approximation of statistical moments of solutions to exterior Dirichlet and Neumann problems with random boundary surfaces is investigated. A rigorous shape calculus approach has been used to approximate these statistical moments by those of the corresponding shape derivatives, which are computed by boundary integral equation methods. Examples illustrate our theoretical results.

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References

  1. 1.

    Alfeld, P., Neamtu, M., Schumaker, L.L.: Bernstein–Bézier polynomials on spheres and sphere-like surfaces. Comput. Aided Geom. Des. 13, 333–349 (1996)

    Article  Google Scholar 

  2. 2.

    Alfeld, P., Neamtu, M., Schumaker, L.L.: Dimension and local bases of homogeneous spline spaces. SIAM J. Math. Anal. 27, 1482–1501 (1996)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Alfeld, P., Neamtu, M., Schumaker, L.L.: Fitting scattered data on sphere-like surfaces using spherical splines. J. Comput. Appl. Math. 73, 5–43 (1996)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Antoine Henrot, M.P.: Variation et optimisation de formes. Mathématiques et Applications, vol. 48. Springer, Berlin (2005)

    Google Scholar 

  5. 5.

    Bejan, A.: Shape and Structure, from Engineering to Nature. Cambridge University Press, New York (2000)

    Google Scholar 

  6. 6.

    Canuto, C., Kozubek, T.: A fictitious domain approach to the numerical solution of PDEs in stochastic domains. Numer. Math. 107, 257–293 (2007)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Amrouche, J .S.Cherif, Necasova, Sarka: Shape sensitivity analysis of the dirichlet laplacian in a half-space. Bull. Pol. Acad. Sci. Math. 52, 365–380 (2004)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chernov, A.: Abstract sensitivity analysis for nonlinear equations and applications. In: Kunisch, G.O.K., Steinbach, O. (eds.) Numerical Mathematics and Advanced Applications, pp. 407–414. American Mathematical Society, Graz (2008)

    Google Scholar 

  9. 9.

    Chernov, A., Pham, T.D., Tran, T.: A shape calculus based method for a transmission problem with random interface. Comput. Math. Appl. (2015). https://doi.org/10.1016/j.camwa.2015.06.021

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chernov, A., Schwab, C.: First order \(k\)-th moment finite element analysis of nonlinear operator equations with stochastic data. Math. Comput. 82, 1859–1888 (2013)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dambrine, M., Harbrecht, H., Puig, B.: Computing quantities of interest for random domains with second order shape sensitivity analysis. ESAIM: M2AN 49, 1285–1302 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Giga, Y.: Surface Evolution Equations: A Level Set Approach. Monographs in Mathematics, vol. 99. Birkhäuser Verlag, Basel (2006)

    Google Scholar 

  13. 13.

    Harbrecht, H.: On output functionals of boundary value problems on stochastic domains. Math. Methods Appl. Sci. 33, 91–102 (2010)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Harbrecht, H., Li, J.: First order second moment analysis for stochastic interface problems based on low-rank approximation. ESAIM Math. Model. Numer. Anal. 47, 1533–1552 (2013)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Harbrecht, H., Schneider, R., Schwab, C.: Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109, 385–414 (2008)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008)

    Google Scholar 

  17. 17.

    Pham, T.D., Tran, T., Chernov, A.: Pseudodifferential equations on the sphere with spherical splines. Math. Models Methods Appl. Sci. 21, 1933–1959 (2011)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Sauter, S.A., Schwab, C.: Boundary Element Methods. Springer Series in Computational Mathematics, vol. 39. Springer, Berlin (2011). (Translated and expanded from the 2004 German original)

    Google Scholar 

  19. 19.

    Sokołowski, J., Zolésio, J.-P.: Introduction to Shape Optimization. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992). (Shape sensitivity analysis)

    Google Scholar 

  20. 20.

    Xiu, D., Tartakovsky, D.M.: Numerical methods for differential equations in random domains. SIAM J. Sci. Comput. 28, 167–1185 (2006). (electronic)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

Duong Thanh Pham and Dũng Dinh’s research was funded by the Department of Science and Technology–Ho Chi Minh City (HCMC-DOST), and the Institute for Computational Science and Technology (ICST) at Ho Chi Minh city, Vietnam, under Contract 21/2017/HD-KHCNTT on 21/09/2017. A part of this paper was done when Duong Pham and Dũng Dinh were working at and Thanh Tran was visiting Vietnam Institute for Advanced Study in Mathematics (VIASM). These authors thank VIASM for providing a fruitful research environment and working condition. Thanh Tran was partially supported by the Australian Research Council under the Grant DP160101755.

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Communicated by Ahmad Izani Md. Ismail.

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Pham, D.T., Tran, T., Dinh, D. et al. Exterior Dirichlet and Neumann Problems in Domains with Random Boundaries. Bull. Malays. Math. Sci. Soc. 43, 1311–1342 (2020). https://doi.org/10.1007/s40840-019-00741-9

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Keywords

  • Dirichlet and Neumann problems
  • Random surfaces
  • Statistical moments
  • Shape derivative

Mathematics Subject Classification

  • 65N30
  • 65N38
  • 65N15
  • 65N50