Skip to main content

Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations

Abstract

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods. The main idea of the proposed vanishing moment method is to approximate a fully nonlinear second order PDE by a higher order, in particular, a quasilinear fourth order PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.

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

References

  1. 1.

    Aleksandrov, A.D.: Certain estimates for the Dirichlet problem. Sov. Math. Dokl. 1, 1151–1154 (1961)

    Google Scholar 

  2. 2.

    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/02) (electronic)

    Article  MathSciNet  Google Scholar 

  3. 3.

    Aronsson, G., Crandall, M.G., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bull. Am. Math. Soc. 41(4), 439–505 (2004) (electronic)

    MATH  Article  MathSciNet  Google Scholar 

  4. 4.

    Aronsson, G., Evans, L.C., Wu, Y.: Fast/slow diffusion and growing sandpiles. J. Differ. Equ. 131(2), 304–335 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    Baginski, F.E., Whitaker, N.: Numerical solutions of boundary value problems for \(\mathcal{K}\) -surfaces in R 3. Numer. Methods Partial Differ. Equ. 12(4), 525–546 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    Barles, G., Jakobsen, E.R.: Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal. 43(2), 540–558 (2005) (electronic)

    MATH  Article  MathSciNet  Google Scholar 

  7. 7.

    Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4(3), 271–283 (1991)

    MATH  MathSciNet  Google Scholar 

  8. 8.

    Bernardi, C., Maday, Y.: Spectral methods. In: Handbook of Numerical Analysis. Handb. Numer. Anal., vol. V, pp. 209–485. North-Holland, Amsterdam (1997)

    Google Scholar 

  9. 9.

    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)

    MATH  Google Scholar 

  10. 10.

    Bryson, S., Levy, D.: High-order central WENO schemes for multidimensional Hamilton-Jacobi equations. SIAM J. Numer. Anal. 41(4), 1339–1369 (2003) (electronic)

    MATH  Article  MathSciNet  Google Scholar 

  11. 11.

    Caffarelli, L.: The Monge-Ampère equation and optimal transportation, an elementary review. In: Optimal Transportation and Applications, Martina Franca, 2001. Lecture Notes in Math., vol. 1813, pp. 1–10. Springer, Berlin (2003)

    Google Scholar 

  12. 12.

    Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Commun. Pure Appl. Math. 37(3), 369–402 (1984)

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43. Am. Math. Soc., Providence (1995)

    MATH  Google Scholar 

  14. 14.

    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer, New York (1988)

    MATH  Google Scholar 

  15. 15.

    Caselles, V., Morel, J.M., Sbert, C.: An axiomatic approach to image interpolation. IEEE Trans. Image Process. 7, 376–386 (1999)

    Article  MathSciNet  Google Scholar 

  16. 16.

    Chang, S.-Y.A., Gursky, M.J., Yang, P.C.: An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. Math. (2) 155(3), 709–787 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  17. 17.

    Cheng, S.Y., Yau, S.T.: On the regularity of the Monge-Ampère equation det ( 2 u/ x i sx j )=F(x,u). Commun. Pure Appl. Math. 30(1), 41–68 (1977)

    MATH  Article  MathSciNet  Google Scholar 

  18. 18.

    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40. SIAM, Philadelphia (2002). Reprint of the 1978 original (North-Holland, Amsterdam; MR0520174 (58 #25001))

    Google Scholar 

  19. 19.

    Ciarlet, P.G., Raviart, P.-A.: A mixed finite element method for the biharmonic equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations. Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., pp. 125–145. Academic Press, San Diego (1974). Publication No. 33

    Google Scholar 

  20. 20.

    Cockburn, B.: Continuous dependence and error estimation for viscosity methods. Acta Numer. 12, 127–180 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  21. 21.

    Cockburn, B., Kanschat, G., Schötzau, D.: The local discontinuous Galerkin method for linearized incompressible fluid flow: a review. Comput. Fluids 34(4–5), 491–506 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  22. 22.

    Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.): Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000). Theory, computation and applications, Papers from the 1st International Symposium held in Newport, RI, May 24–26, 1999

    MATH  Google Scholar 

  23. 23.

    Courant, R., Hilbert, D.: Methods of Mathematical Physics. Wiley Classics Library, vol. II. Wiley, New York (1989). Partial differential equations, Reprint of the 1962 original, A Wiley-Interscience Publication

    Google Scholar 

  24. 24.

    Crandall, M.G., Evans, L.C., Lions, P.-L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282(2), 487–502 (1984)

    MATH  Article  MathSciNet  Google Scholar 

  25. 25.

    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)

    MATH  Article  MathSciNet  Google Scholar 

  26. 26.

    Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)

    MATH  Article  MathSciNet  Google Scholar 

  27. 27.

    Crandall, M.G., Lions, P.-L.: Convergent difference schemes for nonlinear parabolic equations and mean curvature motion. Numer. Math. 75(1), 17–41 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  28. 28.

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

    MATH  MathSciNet  Google Scholar 

  29. 29.

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

    MATH  MathSciNet  Google Scholar 

  30. 30.

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

    MATH  MathSciNet  Google Scholar 

  31. 31.

    Dean, E.J., Glowinski, R.: Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type. Comput. Methods Appl. Mech. Eng. 195(13–16), 1344–1386 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  32. 32.

    Elliott, C.M., French, D.A., Milner, F.A.: A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54(5), 575–590 (1989)

    MATH  Article  MathSciNet  Google Scholar 

  33. 33.

    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. Am. Math. Soc., Providence (1998)

    MATH  Google Scholar 

  34. 34.

    Falk, R.S., Osborn, J.E.: Error estimates for mixed methods. RAIRO Anal. Numér. 14(3), 249–277 (1980)

    MATH  MathSciNet  Google Scholar 

  35. 35.

    Feng, X.: Convergence of the vanishing moment method for the Monge-Ampère equation. Trans. Am. Math. Soc. (2008, submitted)

  36. 36.

    Feng, X., Karakashian, O.A.: Two-level non-overlapping Schwarz preconditioners for a discontinuous Galerkin approximation of the biharmonic equation. J. Sci. Comput. 22/23, 289–314 (2005)

    Article  MathSciNet  Google Scholar 

  37. 37.

    Feng, X., Karakashian, O.A.: Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition. Math. Comput. 76, 1093–1117 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  38. 38.

    Feng, X., Prohl, A.: Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comput. 73, 541–567 (2003)

    Article  MathSciNet  Google Scholar 

  39. 39.

    Feng, X., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation of the mean curvature flows. Numer. Math. 94, 33–65 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  40. 40.

    Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99, 47–84 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  41. 41.

    Feng, X., Prohl, A.: Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem. Interfaces Free Bound. 7, 1–28 (2005)

    MATH  MathSciNet  Article  Google Scholar 

  42. 42.

    Feng, X., Wu, H.-j.: A posteriori error estimates and adaptive finite element methods for the Cahn-Hilliard equation and the Hele-Shaw flow. J. Comput. Math. (2008, in press)

  43. 43.

    Feng, X., Wu, H.-j.: A posteriori error estimates and an adaptive finite element method for the Allen-Cahn equation and the mean curvature flow. J. Sci. Comput. 24(2), 121–146 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  44. 44.

    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Stochastic Modelling and Applied Probability, vol. 25. Springer, New York (2006)

    MATH  Google Scholar 

  45. 45.

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition

    MATH  Google Scholar 

  46. 46.

    Girault, V., Rivière, B., Wheeler, M.F.: A splitting method using discontinuous Galerkin for the transient incompressible Navier-Stokes equations. M2AN Math. Model. Numer. Anal. 39(6), 1115–1147 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  47. 47.

    Guan, B.: On the existence and regularity of hypersurfaces of prescribed Gauss curvature with boundary. Indiana Univ. Math. J. 44(1), 221–241 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  48. 48.

    Guan, B., Guan, P.: Convex hypersurfaces of prescribed curvatures. Ann. Math. (2) 156(2), 655–673 (2002)

    MATH  Article  Google Scholar 

  49. 49.

    Guan, B., Spruck, J.: Locally convex hypersurfaces of constant curvature with boundary. Commun. Pure Appl. Math. 57(10), 1311–1331 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  50. 50.

    Gutiérrez, C.E.: The Monge-Ampère Equation. Progress in Nonlinear Differential Equations and Their Applications, vol. 44. Birkhäuser, Boston (2001)

    MATH  Google Scholar 

  51. 51.

    Gutiérrez, C.E., Huang, Q.: W 2,p estimates for the parabolic Monge-Ampère equation. Arch. Ration. Mech. Anal. 159(2), 137–177 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  52. 52.

    Hermann, L.: Finite element bending analysis for plates. J. Eng. Mech. Div. 93, 49–83 (1967)

    Google Scholar 

  53. 53.

    Jakobsen, E.R.: On the rate of convergence of approximation schemes for Bellman equations associated with optimal stopping time problems. Math. Models Methods Appl. Sci. 13(5), 613–644 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  54. 54.

    Johnson, C.: On the convergence of a mixed finite-element method for plate bending problems. Numer. Math. 21, 43–62 (1973)

    MATH  Article  MathSciNet  Google Scholar 

  55. 55.

    Krylov, N.V.: The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients. Appl. Math. Optim. 52(3), 365–399 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  56. 56.

    Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996)

    MATH  Google Scholar 

  57. 57.

    Lin, C.-T., Tadmor, E.: High-resolution nonoscillatory central schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21(6), 2163–2186 (2000) (electronic)

    MATH  Article  MathSciNet  Google Scholar 

  58. 58.

    McCann, R.J., Oberman, A.M.: Exact semi-geostrophic flows in an elliptical ocean basin. Nonlinearity 17(5), 1891–1922 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  59. 59.

    Miyoshi, T.: A finite element method for the solutions of fourth order partial differential equations. Kumamoto J. Sci. Math. 9, 87–116 (1972/73)

    MathSciNet  Google Scholar 

  60. 60.

    Miyoshi, T.: A mixed finite element method for the solution of the von Kármán equations. Numer. Math. 26(3), 255–269 (1976)

    Article  MathSciNet  Google Scholar 

  61. 61.

    Mozolevski, I., Süli, E.: A priori error analysis for the hp-version of the discontinuous Galerkin finite element method for the biharmonic equation. Comput. Methods Appl. Math. 3(4), 596–607 (2003) (electronic)

    MATH  MathSciNet  Google Scholar 

  62. 62.

    Neilan, M.: Numerical methods for second order fully nonlinear partial differential equations. Ph.D. Thesis, The University of Tennessee, Knoxville (2008, in preparation)

  63. 63.

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

    MATH  MathSciNet  Google Scholar 

  64. 64.

    Oberman, A.M.: Wide stencil finite difference schemes for elliptic Monge-Ampére equation and functions of the eigenvalues of the Hessian. Preprint (2007)

  65. 65.

    Oliker, V.I., Prussner, L.D.: On the numerical solution of the equation ( 2 z/ x 2)( 2 z/ y 2)−(( 2 z/ x y))2=f and its discretizations. I. Numer. Math. 54(3), 271–293 (1988)

    MATH  Article  MathSciNet  Google Scholar 

  66. 66.

    Osher, S., Shu, C.-W.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907–922 (1991)

    MATH  Article  MathSciNet  Google Scholar 

  67. 67.

    Oukit, A., Pierre, R.: Mixed finite element for the linear plate problem: the Hermann-Miyoshi model revisited. Numer. Math. 74(4), 453–477 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  68. 68.

    Shen, J.: Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput. 15(6), 1489–1505 (1994)

    MATH  Article  MathSciNet  Google Scholar 

  69. 69.

    Wang, L.: On the regularity theory of fully nonlinear parabolic equations. I. Commun. Pure Appl. Math. 45(1), 27–76 (1992)

    MATH  Article  Google Scholar 

  70. 70.

    Zhang, Y.-T., Shu, C.-W.: High-order WENO schemes for Hamilton-Jacobi equations on triangular meshes. SIAM J. Sci. Comput. 24(3), 1005–1030 (2002) (electronic)

    Article  MathSciNet  Google Scholar 

  71. 71.

    Zhao, H.: A fast sweeping method for Eikonal equations. Math. Comput. 74(250), 603–627 (2005) (electronic)

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xiaobing Feng.

Additional information

This work was partially supported by the NSF grants DMS-0410266 and DMS-0710831.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Feng, X., Neilan, M. Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations. J Sci Comput 38, 74–98 (2009). https://doi.org/10.1007/s10915-008-9221-9

Download citation

Keywords

  • Fully nonlinear PDEs
  • Monge-Ampère type equations
  • Moment solutions
  • Vanishing moment method
  • Viscosity solutions
  • Finite element method
  • Mixed finite element method
  • Spectral and discontinuous Galerkin methods