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A Meshless Solution to the p-Laplace Equation

  • Francisco Manuel Bernal Martinez
  • Manuel Segura Kindelan
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 11)

The p-Laplace equation is a non-linear elliptic PDE which plays an important role in the modeling of many phenomena in areas such as glaciology, non-Newtonian rheology or edge-preserving image deblurring. We have linearized it and applied a scheme introduced by G. Fasshauer which allows to solve it in the framework of Kansa's method. In order to confirm the validity of the approach, a 2D example (the pressure distribution in Hele-Shaw flow) has been numerically solved. The convergence and accuracy of the method are discussed, and an improvement based on smoothing up the linearized PDE is suggested.

Keywords

p-Laplace Non-linear methods Kansa Method 

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References

  1. 1.
    G. Aronsson and U. Janfalk, On Hele-Shaw flow of power-law fluids, Eur. J. Appl. Math. 3, 343–336 (1992)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    R. K. Beatson and H.-Q. Bui, Mollification Formulas and Implicit Smoothing, Research report UCDMS 2003/19 (2003)Google Scholar
  3. 3.
    F. Bernal and M. Kindelan, An RBF meshless method for injection molding modeling, Lecture Notes in Computational Science and Engineering, Springer (2006)Google Scholar
  4. 4.
    F. Bernal and M. Kindelan, RBF meshless modeling of non-Newtonian Hele-Shaw flow, Eng. Anal. Boun. Elem. 31, 863–874 (2007)CrossRefGoogle Scholar
  5. 5.
    F. Bernal and M. Kindelan, Meshless Simulation of Hele-Shaw Flow, 14th European Conference on Mathematics for Industry ECMI-2006, to appear in Progress in Industrial Mathematics at ECMI 2006, Mathematics in Industry (2006)Google Scholar
  6. 6.
    J. C. Carr, R. K. Beatson, B. C. McCallum, W. R. Fright, T. J. McLennan, and T. J. Mitchell, Smooth Surface Reconstruction from Noisy Range Data. In GRAPHITE 2003, ACM Press, New York, 119–126 (2003)Google Scholar
  7. 7.
    A. H. D. Cheng, M. A. Golberg, E. J. Kansa, and G. Zammito, Exponential convergence and h-c multiquadric collocation method for partial differential equations. Numer. Meth. Part. D. Eq. 19(5), 571–594 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    G. E. Fasshauer, Newton iteration with multiquadrics for the solution of nonlinear PDEs, Comput. Math. Appl. 43, 423–438 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    G. E. Fasshauer, Solving differential equations with radial basis functions: Multilevel methods and smoothing, Adv. Comput. Math. 11, 139–159 (1999)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    G. E. Fasshauer, On Smoothing for Multilevel Approximation with Radial Basis Functions, Approximation Theory XI, Vol. II: Computational Aspects, C. K. Chui, and L. L. Schumaker (eds.), Vanderbilt University Press, Nashville, TN 55–62 (1999)Google Scholar
  11. 11.
    G. E. Fasshauer, E. C. Gartland, and J. W. Jerome, Algorithms defined by Nash iteration: Some implementations via multilevel collocation and smoothing, J. Comp. Appl. Math. 119, 161–183 (2000)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    A. I. Fedoseyev, M. J. Friedman, and E. J. Kansa, Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary, Comput. Math. Appl. 43, 439–455 (2002)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    C. A. Hieber and S. F. Shen, A finite-element/finite difference simulation of the injection-molding filling process, J. Non-Newton. Fluid Mech. 7, 1–32 (1979)CrossRefGoogle Scholar
  14. 14.
    E. J. Kansa, Multiquadrics — a scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates, Comput. Math. Appl. 19, 127–145 (1990)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    E. J. Kansa, Multiquadrics— a scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl. 19, 147–161 (1990)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    L. Ling, R. Opfer, and R. Schaback, Results on meshless collocation techniques, Eng. Anal. Bound. Elems. 30(4), 247–253 (2006)CrossRefGoogle Scholar
  17. 17.
    C. T. Mouat and R. K. Beatson, RBF Collocation, Research report UCDMS 2002/3 (2002)Google Scholar

Copyright information

© Springer Science + Business Media B.V 2009

Authors and Affiliations

  • Francisco Manuel Bernal Martinez
    • 1
  • Manuel Segura Kindelan
    • 1
  1. 1.G. Millán Institute for Modeling, Simulation and Industrial MathematicsUniversidad Carlos III de MadridLeganés

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