Skip to main content
SpringerLink
Log in

Numerical Solving a Boundary Value Problem for the Eikonal Equation

  • Conference paper
  • First Online:
Finite Difference Methods. Theory and Applications (FDM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11386))

Included in the following conference series:

  • 1331 Accesses

Abstract

In the present work, a Dirichlet problem is studied for the eikonal equation. A nonlinear boundary value problem formulated here can be treated as the limit of the diffusion–reaction problem with a diffusion parameter tending to zero. For numerical solving the singularly perturbed diffusion–reaction problem, monotone approximations are used. Predictions for a 3D model problem are presented to demonstrate possibilities of the developed numerical algorithm. The standard piecewise-linear finite-element approximation is employed to constructed discretization in space.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

References

  1. Belyaev, A.G., Fayolle, P.A.: On variational and PDE-based distance function approximations. In: Computer Graphics Forum, vol. 34, pp. 104–118. Wiley Online Library (2015)

    Google Scholar 

  2. Bhattacharya, T., DiBenedetto, E., Manfredi, J.: Limits as \(p \rightarrow \infty \) of \(\triangle _p u_p = f\) and related extremal problems. Rend. Sem. Mat. Univ. Polytec. Torino 47, 15–68 (1989)

    Google Scholar 

  3. Bitzadze, A.V.: Some Classes of Partial Differential Equations. Nauka, Moscow (1981). (in Russian)

    Google Scholar 

  4. Born, M., Wolf, E.: Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  5. Cai, Q., Kollmannsberger, S., Sala-Lardies, E., Huerta, A., Rank, E.: On the natural stabilization of convection dominated problems using high order Bubnov-Galerkin finite elements. Comput. Math. Appl. 66(12), 2545–2558 (2014)

    Article  MathSciNet  Google Scholar 

  6. Ciarlet, P.G., Raviart, P.A.: Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2(1), 17–31 (1973)

    Article  MathSciNet  Google Scholar 

  7. Crane, K., Weischedel, C., Wardetzky, M.: Geodesics in heat: a new approach to computing distance based on heat flow. ACM Trans. Graph. (TOG) 32(5), 152 (2013)

    Article  Google Scholar 

  8. Fu, Z., Jeong, W.K., Pan, Y., Kirby, R.M., Whitaker, R.T.: A fast iterative method for solving the eikonal equation on triangulated surfaces. SIAM J. Sci. Comput. 33(5), 2468–2488 (2011)

    Article  MathSciNet  Google Scholar 

  9. Fu, Z., Kirby, R.M., Whitaker, R.T.: A fast iterative method for solving the eikonal equation on tetrahedral domains. SIAM J. Sci. Comput. 35(5), C473–C494 (2013)

    Article  MathSciNet  Google Scholar 

  10. Gilles, A., Pierre, K.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, 2nd edn. Springer, New York (2006). https://doi.org/10.1007/978-0-387-44588-5

    Book  MATH  Google Scholar 

  11. Gómez, J.V., Alvarez, D., Garrido, S., Moreno, L.: Fast methods for eikonal equations: an experimental survey. arXiv preprint arXiv:1506.03771 (2015)

  12. Gurumoorthy, K.S., Rangarajan, A.: A Schrödinger equation for the fast computation of approximate euclidean distance functions. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 100–111. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02256-2_9

    Chapter  Google Scholar 

  13. Jeong, W.K., Whitaker, R.T.: A fast iterative method for eikonal equations. SIAM J. Sci. Comput. 30(5), 2512–2534 (2008)

    Article  MathSciNet  Google Scholar 

  14. Kružkov, S.N.: Generalized solutions of the Hamilton-Jacobi equations of eikonal type. I. Formulation of the problems; existence, uniqueness and stability theorems; some properties of the solutions. Sb.: Math. 27(3), 406–446 (1975)

    MATH  Google Scholar 

  15. Letniowski, F.W.: Three-dimensional Delaunay triangulations for finite element approximations to a second-order diffusion operator. SIAM J. Sci. Stat. Comput. 13(3), 765–770 (1992)

    Article  MathSciNet  Google Scholar 

  16. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, New York (2012). https://doi.org/10.1007/978-1-4612-5282-5

    Book  MATH  Google Scholar 

  17. Roos, H., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems. Springer Series in Computational Mathematics, vol. 24. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-34467-4

    Book  MATH  Google Scholar 

  18. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics. Springer, Berlin (2006). https://doi.org/10.1007/978-3-662-03359-3

    Book  MATH  Google Scholar 

  19. Tsai, Y.H.R., Cheng, L.T., Osher, S., Zhao, H.K.: Fast sweeping algorithms for a class of Hamilton-Jacobi equations. SIAM J. Numer. Anal. 41(2), 673–694 (2003)

    Article  MathSciNet  Google Scholar 

  20. Tsitsiklis, J.: Fast marching methods. IEEE Trans. Autom. Control 40, 1528–1538 (1995)

    Article  Google Scholar 

Download references

Acknowledgements

Petr Vabishchevich gratefully acknowledges support from the the Russian Federation Government (# 14.Y26.31.0013).

Author information

Authors and Affiliations

  1. Nuclear Safety Institute, Russian Academy of Sciences, 52, B. Tulskaya, Moscow, Russia

    Alexander G. Churbanov & Petr N. Vabishchevich

  2. North–Eastern Federal University, 58, Belinskogo, Yakutsk, Russia

    Petr N. Vabishchevich

Authors
  1. Alexander G. Churbanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petr N. Vabishchevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Petr N. Vabishchevich .

Editor information

Editors and Affiliations

  1. IICT, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. University of Rousse, Rousse, Bulgaria

    Lubin Vulkov

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Churbanov, A.G., Vabishchevich, P.N. (2019). Numerical Solving a Boundary Value Problem for the Eikonal Equation. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-11539-5_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-11538-8

  • Online ISBN: 978-3-030-11539-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation