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Journal of Scientific Computing

, Volume 77, Issue 3, pp 1534–1565 | Cite as

Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

  • Xiaobing Feng
  • Thomas Lewis
Article
  • 28 Downloads

Abstract

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs. The proposed LDG methods are natural extensions of a narrow-stencil finite difference framework recently proposed by the authors for approximating viscosity solutions. The idea of the methodology is to use multiple approximations of first and second order derivatives as a way to resolve the potential low regularity of the underlying viscosity solution. Consistency and generalized monotonicity properties are proposed that ensure the numerical operator approximates the differential operator. The resulting algebraic system has several linear equations coupled with only one nonlinear equation that is monotone in many of its arguments. The structure can be explored to design nonlinear solvers. This paper also presents and analyzes numerical results for several numerical test problems in two dimensions which are used to gauge the accuracy and efficiency of the proposed LDG methods.

Keywords

Fully nonlinear PDEs Viscosity solutions Discontinuous Galerkin methods 

Mathematics Subject Classification

65N30 65M60 35J60 35K55 

References

  1. 1.
    Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4(3), 271–283 (1991)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations, Vol. 43 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (1995)Google Scholar
  3. 3.
    Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Debrabant, K., Jakobsen, E.: Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Math. Comput. 82, 1433–1462 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Feng, X., Glowinski, R., Neilan, M.: Recent developments in numerical methods for second order fully nonlinear partial differential equations. SIAM Rev. 55(2), 205–267 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Feng, X., Jensen, M.: Convergent semi-Lagrangian methods for the Monge–Ampère equation on unstructured grids. SIAM J. Numer. Anal. 55, 691–712 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Feng, X., Kao, C., Lewis, T.: Convergent finite difference methods for one-dimensional fully nonlinear second order partial differential equations. J. Comput. Appl. Math. 254, 81–98 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Feng, X., Lewis, T.: A narrow-stencil finite difference method for approximating viscosity solutions of fully nonlinear elliptic partial differential equations with applications to Hamilton–Jacobi–Bellman equations (2018)Google Scholar
  9. 9.
    Feng, X., Lewis, T.: Mixed interior penalty discontinuous Galerkin methods for one-dimensional fully nonlinear second order elliptic and parabolic equations. J. Comput. Math. 32(2), 107–135 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Feng, X., Lewis, T.: Local discontinuous Galerkin methods for one-dimensional second order fully nonlinear elliptic and parabolic equations. J. Sci. Comput. 59, 129–157 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Feng, X., Lewis, T.: Mixed interior penalty discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic equations in high dimensions. Numer. Methods Partial Differ.Equ. 30(5), 1538–1557 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Feng, X., Lewis, T., Neilan, M.: Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations. J. Comput. Appl. Math. 299, 68–91 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Feng, X., Neilan, M.: The vanishing moment method for fully nonlinear second order partial differential equations: formulation, theory, and numerical analysis (2011). arXiv:1109.1183v2
  15. 15.
    Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Applications of Mathematics, No. 1. Springer, Berlin (1975)CrossRefGoogle Scholar
  16. 16.
    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, Volume 25 of Applications of Mathematics. Springer, New York (1993)Google Scholar
  17. 17.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer, Berlin (2001) (reprint of the 1998 edition)zbMATHGoogle Scholar
  18. 18.
    Jensen, M., Smears, I.: On the convergence of finite element methods for Hamilton–Jacobi–Bellman equations. SIAM J. Numer. Anal. 51, 137–162 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lewis, T.L.: Finite difference and discontinuous Galerkin finite element methods for fully nonlinear second order partial differential equations. Ph.D. thesis, University of Tennessee (2013). http://trace.tennessee.edu/utk_graddiss/2446
  20. 20.
    Lewis, T., Neilan, M.: Convergence analysis of a symmetric dual-wind discontinuous Galerkin method. J. Sci. Comput. 59(3), 602–625 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co., Inc., River Edge (1996)CrossRefGoogle Scholar
  22. 22.
    Neilan, M., Salgado, A.J., Zhang, W.: Numerical analysis of strongly nonlinear PDEs. Acta Numer. arXiv:1610.07992 [math.NA] (2017) (to appear)
  23. 23.
    Nitsche, J.A.: Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendung von Teilraumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36, 9–15 (1970/71)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nochetto, R.H., Ntogakas, D., Zhang, W.: Two-scale method for the Monge–Ampére equation: convergence rates (2017). arXiv:1706.06193 [math.NA]
  25. 25.
    Pogorelov, A.V.: Monge–Ampère Equations of Elliptic Type. P. Noordhoff Ltd., Groningen (1964)zbMATHGoogle Scholar
  26. 26.
    Salgado, A.J., Zhang, W.: Finite element approximation of the Isaacs equation (2016). arXiv:1512.09091v1 [math.NA]
  27. 27.
    Shu, C.-W.: High order numerical methods for time dependent Hamilton–Jacobi equations. In: Mathematics and computation in imaging science and information processing, Vol. 11 of Lecture Notes Series Institute Mathematics Science Natural University Singapore, pp. 47–91. World Sci. Publ., Hackensack (2007)Google Scholar
  28. 28.
    Smears, I., Süli, E.: Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman equations with Cordes coefficients. SIAM J. Numer. Anal. 52, 993–1016 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Yan, J., Osher, S.: A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations. J. Comput. Phys. 230, 232–244 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of TennesseeKnoxvilleUSA
  2. 2.Department of Mathematics and StatisticsThe University of North Carolina at GreensboroGreensboroUSA

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