An Adaptive Finite Element Method for the Infinity Laplacian

  • Omar LakkisEmail author
  • Tristan Pryer
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)


We construct a finite element method (FEM) for the infinity Laplacian. Solutions of this problem are well known to be singular in nature so we have taken the opportunity to conduct an a posteriori analysis of the method deriving residual based estimators to drive an adaptive algorithm. It is numerically shown that optimal convergence rates are regained using the adaptive procedure.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Aronsson, Construction of singular solutions to the p-harmonic equation and its limit equation for p = . Manuscr. Math. 56(2), 135–158 (1986)Google Scholar
  2. 2.
    G. Awanou, Pseudo time continuation and time marching methods for Monge–Ampère type equations (2012). In revision – tech report available on
  3. 3.
    E.N. Barron, L.C. Evans, R. Jensen, The infinity Laplacian, Aronsson’s equation and their generalizations. Trans. Am. Math. Soc. 360(1), 77–101 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    L.A. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43 (American Mathematical Society, Providence, 1995)Google Scholar
  5. 5.
    M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)Google Scholar
  6. 6.
    M.G. Crandall, L.C. Evans, R.F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differ. Equ. 13(2), 123–139 (2001)zbMATHMathSciNetGoogle Scholar
  7. 7.
    A. Dedner, T. Pryer, Discontinuous Galerkin methods for nonvariational problems (2013). Submitted – tech report available on ArXiV
  8. 8.
    S. Esedoglu, A.M. Oberman, Fast semi-implicit solvers for the infinity laplace and p-laplace equations, Arxiv, 2011Google Scholar
  9. 9.
    L.C. Evans, O. Savin, C 1, α regularity for infinity harmonic functions in two dimensions, Calc. Var. Partial Differ. Equ. 32(3), 325–347 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    X. Feng, M. Neilan, Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. J. Sci. Comput. 38(1), 74–98 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Ration. Mech. Anal. 123(1), 51–74 (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    O. Lakkis, T. Pryer, A finite element method for second order nonvariational elliptic problems. SIAM J. Sci. Comput. 33(2), 786–801 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    G. Lu, P. Wang, Inhomogeneous infinity Laplace equation. Adv. Math. 217(4), 1838–1868 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    A.M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions. Math. Comput. 74(251), 1217–1230 (2005). (Electronic)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SussexBrightonEngland, UK
  2. 2.Department of Mathematics and StatisticsUniversity of ReadingReadingEngland, UK

Personalised recommendations