Skip to main content
SpringerLink
Log in

A Meshless Solution to the p-Laplace Equation

  • Chapter
Progress on Meshless Methods

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 11))

  • 1389 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Aronsson and U. Janfalk, On Hele-Shaw flow of power-law fluids, Eur. J. Appl. Math. 3, 343–336 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. R. K. Beatson and H.-Q. Bui, Mollification Formulas and Implicit Smoothing, Research report UCDMS 2003/19 (2003)

    Google Scholar 

  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. F. Bernal and M. Kindelan, RBF meshless modeling of non-Newtonian Hele-Shaw flow, Eng. Anal. Boun. Elem. 31, 863–874 (2007)

    Article  Google Scholar 

  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. 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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  8. G. E. Fasshauer, Newton iteration with multiquadrics for the solution of nonlinear PDEs, Comput. Math. Appl. 43, 423–438 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. G. E. Fasshauer, Solving differential equations with radial basis functions: Multilevel methods and smoothing, Adv. Comput. Math. 11, 139–159 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  16. L. Ling, R. Opfer, and R. Schaback, Results on meshless collocation techniques, Eng. Anal. Bound. Elems. 30(4), 247–253 (2006)

    Article  Google Scholar 

  17. C. T. Mouat and R. K. Beatson, RBF Collocation, Research report UCDMS 2002/3 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. G. Millán Institute for Modeling, Simulation and Industrial Mathematics, Universidad Carlos III de Madrid, 28911, Leganés

    Francisco Manuel Bernal Martinez & Manuel Segura Kindelan

Authors
  1. Francisco Manuel Bernal Martinez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuel Segura Kindelan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Manuel Bernal Martinez .

Editor information

Editors and Affiliations

  1. Departamento de Engenharia Mecânica e Gestão Industrial, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, Porto, 4200-465, Portugal

    A. J. M. Ferreira

  2. University of California Davis, California, USA

    E. J. Kansa

  3. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA

    G. E. Fasshauer

  4. Instituto Superior Técnico, DECivil/ICIST, TU Lisbon, Portugal

    V. M. A. Leitão

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science + Business Media B.V

About this chapter

Cite this chapter

Martinez, F.M.B., Kindelan, M.S. (2009). A Meshless Solution to the p-Laplace Equation. In: Ferreira, A.J.M., Kansa, E.J., Fasshauer, G.E., Leitão, V.M.A. (eds) Progress on Meshless Methods. Computational Methods in Applied Sciences, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8821-6_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4020-8821-6_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-8820-9

  • Online ISBN: 978-1-4020-8821-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation