Skip to main content

An Accelerated Method for Nonlinear Elliptic PDE

Abstract

We propose two numerical methods for accelerating the convergence of the standard fixed point method associated with a nonlinear and/or degenerate elliptic partial differential equation. The first method is linearly stable, while the second is provably convergent in the viscosity solution sense. In practice, the methods converge at a nearly linear complexity in terms of the number of iterations required for convergence. The methods are easy to implement and do not require the construction or approximation of the Jacobian. Numerical examples are shown for Bellman’s equation, Isaacs’ equation, Pucci’s equations, the Monge–Ampère equation, a variant of the infinity Laplacian, and a system of nonlinear equations.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

References

  1. Falcone, M., Finzi Vita, S., Giorgi, T., Smits, R.G.: A semi-Lagrangian scheme for the game p-Laplacian via p-averaging. Appl. Numer. Math. 73, 63–80 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  2. Feng, X., Glowinski, R., Neilan, M.: Recent developments in numerical methods for fully nonlinear second order partial differential equations. SIAM Rev. 55, 205–267 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  3. Oberman, A.: A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions. Math. Comput. 74(251), 1217–1230 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  4. Oberman, A.: Finite difference methods for the infinity Laplace and p-Laplace equations. J. Comput. Appl. Math. 254, 65–80 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  5. Feng, X., Neilan, M.: Mixed finite element methods for the fully nonlinear Monge–Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal. 47(2), 1226–1250 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  6. Feng, X., Neilan, M.: Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. J. Sci. Comput. 38(1), 74–98 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  7. Feng, X., Neilan, M.: A modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations. SIAM J. Numer. Anal. 47(4), 2952–2981 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  8. Feng, X., Neilan, M.: Analysis of Galerkin methods for the fully nonlinear Monge–Ampère equation. J. Sci. Comput. 47(3), 303–327 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  9. Feng, X., Neilan, M.: Galerkin methods for the fully nonlinear Monge–Ampére equation. arXiv:0712.1240 (2007)

  10. Neilan, M.: A nonconforming Morley finite element method for the fully nonlinear Monge–Ampère equation. Numer. Math. 115(3), 371–394 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  11. Lakkis, O., Pryer, T.: An adaptive finite element method for the infinity Laplacian. arXiv:1311.3930 (2013)

  12. Oberman, A.M.: Wide stencil finite difference schemes for the elliptic Monge–Ampère equation and functions of the eigenvalues of the Hessian. Discrete Continuous Dyn. Syst. Ser. B 10(1), 221–238 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  13. Benamou, J.-D., Froese, B.D., Oberman, A.M.: Two numerical methods for the elliptic Monge–Ampère equation. ESAIM Math. Model. Numer. Anal. 44(4), 737–758 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  14. Froese, B.D., Oberman, A.M.: Convergent finite difference solvers for viscosity solutions of the elliptic Monge–Ampère equation in dimensions two and higher. SIAM J. Numer. Anal. 49(4), 1692–1714 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  15. Froese, B.D., Oberman, A.M.: Fast finite difference solvers for singular solutions of the elliptic Monge–Ampère equation. J. Comput. Phys. 230(3), 818–834 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  16. Froese, B.D., Oberman, A.M.: Convergent filtered schemes for the Monge–Ampère partial differential equation. SIAM J. Numer. Anal. 51(1), 423–444 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  17. Dean, E.J., Glowinski, R.: Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach. Comptes Rendus Math. 336(9), 779–784 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  18. Dean, E.J., Glowinski, R.: Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: a least-squares approach. Comptes Rendus Math. 339(12), 887–892 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  19. Dean, E.J., Glowinski, R.: An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge–Ampére equation in two dimensions. Electron. Trans. Numer. Anal. 22, 71–96 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Dean, E.J., Glowinski, R.: Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type. Comput. Methods Appl. Mech. Eng. 195(13), 1344–1386 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  21. Dean, E.J., Glowinski, R.: On the numerical solution of a two-dimensional Pucci’s equation with Dirichlet boundary conditions: a least-squares approach. Comptes Rendus Math. 341(6), 375–380 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  22. Böhmer, K.: On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46(3), 1212–1249 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  23. Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asympotot. Anal. 4, 271–283 (1991)

    MathSciNet  MATH  Google Scholar 

  24. Oberman, A.: Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44(2), 879–895 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  25. Caffarelli, L.A., Souganidis, P.E.: A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs. Commun. Pure Appl. Math. 61, 1–17 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  26. Loeper, G., Rapetti, F.: Numerical solution of the Monge–Ampè equation by a Newton’s algorithm. Comptes Rendus Math. 340(4), 319–324 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  27. Awanou, G.: Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: classical solutions. IMA J. Numer. Anal. http://imajna.oxfordjournals.org/content/early/2014/05/30/imanum.dru028 (2014)

  28. Nesterov, Y.: A method of solving a convex programming problem with convergence rate O\((1/k^2)\). Sov. Math. Dokl. 27(2), 372–376 (1983)

  29. Beck, A., Taboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

  30. Hundsdorfer, W., Ruuth, S.J., Spiteri, R.J.: Monotonicity-preserving linear multistep methods. SIAM J. Numer. Anal. 41(2), 605–623 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  31. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  32. Gottlieb, S.: On high order strong stability preserving Runge–Kutta and multi step time discretizations. J. Sci. Comput. 25(1), 105–128 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  33. Ruuth, S.: Global optimization of explicit strong-stability-preserving Runge–Kutta methods. Math. Comput. 75(253), 183–207 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  34. Bresten, C., Gottlieb, S., Grant, Z., Higgs, D., Ketcheson, D. I., Németh, A.: Strong stability preserving multistep Runge–Kutta methods. arXiv:1307.8058 (2013)

  35. Kuo, H.J., Trudinger, N.S.: Positive difference operators on general meshes. Duke Math. J. 83(2), 415–433 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  36. Isaacs, R.: Differential Games. Wiley, New York (1965)

    MATH  Google Scholar 

  37. Cabré, X., Caffarelli, L.A.: Interior \(C^{2,\alpha }\) regularity theory for a class of non convex fully nonlinear elliptic equation. J. Math. Pures Appl. 82(5), 573–612 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  38. Krylov, N.V.: On the rate of convergence of finite-difference approximations for elliptic Isaacs’ equation in smooth domains. arXiv:1402.0252 (2014)

  39. Barron, E., Evans, L.C., Jensen, R.: The infinity Laplacian. Aronsson’s equation and their generalizations. Trans. Am. Math. Soc. 360(1), 77–101 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  40. Evans, L.C., Yu, Y.: Various properties of solutions of the infinity-Laplacian equation. Commun. Partial Differ. Equ. 30(9), 1401–1428 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  41. Crandall, M.G., Evans, L.C., Gariepy, R.F.: Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differ. Equ. 13(2), 123–139 (2001)

    MathSciNet  MATH  Google Scholar 

  42. Evans, L.C., Smart, C.K.: Everywhere differentiability of infinity harmonic functions. Calc. Var. Partial Differ. Equ. 42(1–2), 289–299 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  43. Aronsson, G.: On the partial differential equation \(u_x^2 u_{xx}+ 2 u_{x} u_{y} u_{xy}+ u_y^2 u_{yy}= 0\). Ark. Mat. 7(5), 395–425 (1968)

    MathSciNet  Article  Google Scholar 

  44. Aronsson, G.: On certain singular solutions of the partial differential equation \(u_x^2 u_{xx}+ 2 u_{x} u_{y} u_{xy}+ u_y^2 u_{yy}= 0\). Manuscr. Math. 47, 133–151 (1984)

    MathSciNet  Article  Google Scholar 

  45. Caffarelli, L.A., Glowinski, R.: Numerical solution of the Dirichlet problem for a Pucci equation in dimension two. Application to homogenization. J. Numer. Math. 16(3), 185–216 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  46. Caboussat, A., Glowinski, R., Sorensen, D.C.: A least-squares method for the numerical solution of the Dirichlet problem for the elliptic Monge–Ampère equation in dimension two. ESAIM Control Optim. Calc. Var. 19(03), 780–810 (2013)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgments

H. Schaeffer was supported by NSF 1303892 and University of California President’s Postdoctoral Fellowship Program. T. Y. Hou was supported by NSF DMS 1318377, NSF DMS 0908546, and DOE DE FG02 06ER25727. The authors would like to thank the editor and reviewers for their helpful feedback.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hayden Schaeffer.

Appendix

Appendix

We provide a proof of Proposition 4.2 below. The arguments follows from a direct computation of the lower bound.

Proof

We can expand \(\alpha _n\) in terms of the sequences \(\{\gamma _j\}^n_{j=1}\) as follows:

$$\begin{aligned} \alpha _n&=(1+\alpha _{n-1})\gamma _n=(1+(1+\alpha _{n-2})\gamma _{n-1})\gamma _n \\&=(1+(1+(1+\alpha _{n-2})\gamma _{n-2})\gamma _{n-1})\gamma _n\ldots \\&=\sum ^{n}_{j=1} \prod ^{n}_{k=j} \gamma _k \end{aligned}$$

Next, define \(\xi _k{:}{=}1-\gamma _k\), by the assumptions of Proposition 4.2, \(1-\xi _k \ge 1- \frac{1}{k+1}\), so

$$\begin{aligned} \prod ^{n}_{k=j} (1-\xi _k) \ge \prod ^{n}_{k=j} (1 - \frac{1}{k+1}) =\frac{j}{n+1}. \end{aligned}$$

Therefore the sequence \(\alpha _n\) can be bound below by:

$$\begin{aligned} \alpha _n&=\sum ^{n}_{j=1} \prod ^{n}_{k=j} \gamma _k \ge \sum ^{n}_{j=1} \frac{j}{n+1} = \frac{n}{2}. \end{aligned}$$

Also, the partial sums of \(\alpha _n\) are given by:

$$\begin{aligned} \sum ^{n-1}_{j=1} \alpha _j \ge \sum ^{n-1}_{j=1} \frac{j}{2} = \frac{n^2-n}{4}. \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Schaeffer, H., Hou, T.Y. An Accelerated Method for Nonlinear Elliptic PDE. J Sci Comput 69, 556–580 (2016). https://doi.org/10.1007/s10915-016-0215-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-016-0215-8

Keywords

  • Nonlinear elliptic PDE
  • Degenerate elliptic PDE
  • Accelerated convergence
  • Elliptic systems
  • Finite difference methods
  • Viscosity solutions
  • Fixed point methods

Mathematics Subject Classification

  • 65B05
  • 65N06
  • 65M22
  • 49L25
  • 35J60
  • 35B51
  • 35J70