Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form



We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form \(A(x) : D^2 u(x) = f(x)\) in a bounded but not necessarily convex domain \(\Omega \) and study it in the max norm. The fine scale is given by the meshsize h, whereas the coarse scale \(\epsilon \) is dictated by an integro-differential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle for any uniformly positive definite matrix A provided that the mesh is face weakly acute. We establish a discrete Alexandroff–Bakelman–Pucci (ABP) estimate which is suitable for finite element analysis. Its proof relies on a discrete Alexandroff estimate which expresses the min of a convex piecewise linear function in terms of the measure of its sub-differential, and thus of jumps of its gradient. The discrete ABP estimate leads, under suitable regularity assumptions on A and u, to pointwise error estimates of the form
$$\begin{aligned} \Vert \,u - u^{\epsilon }_h\,\Vert _{L^{\infty }(\Omega )} \le \, C(A,u) \, h^{2\alpha /(2 + \alpha )} \big | \ln h \big | \qquad 0< \alpha \le 2, \end{aligned}$$
provided \(\epsilon \approx h^{2/(2+\alpha )}\). Such a convergence rate is at best of order \( h \big | \ln h \big |\), which turns out to be quasi-optimal.


Piecewise linear finite elements Discrete maximum principle Discrete Alexandroff estimate Discrete Alexandroff–Bakelman–Pucci estimate Elliptic PDEs in non-divergence form 2-scale approximation Maximum-norm error estimates 

Mathematics Subject Classification

65N30 65N15 35B50 35D35 35J57 


  1. 1.
    G. Barles and E. R. Jakobsen. Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal., 43(2):540–558 (electronic), 2005.Google Scholar
  2. 2.
    G. Barles and P. E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal., 4:271–283, 1991.MathSciNetMATHGoogle Scholar
  3. 3.
    S. Bartels. Stability and convergence of finite-element approximation schemes for harmonic maps. SIAM J. Numer. Anal., 43(1):220–238 (electronic), 2005.Google Scholar
  4. 4.
    J.-D. Benamou, B. D. Froese and A. M. Oberman. Numerical solution of the optimal transportation problem using the Monge-Ampère equation. J. Comput. Phys., 260:107–126, 2014.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    S. N. Bernstein. Sur la généralisation du probléme de Dirichlet. Math. Ann., 62:253–271, 1906; 69:82–136. 1910.Google Scholar
  6. 6.
    J. F. Bonnans and H. Zidani. Consistency of generalized finite difference schemes for the stochastic HJB equation. SIAM J. Numer. Anal., 41(3):1008–1021, 2003.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    S. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Berlin: Springer 2014.Google Scholar
  8. 8.
    L. Caffarelli. Elliptic second order equations. Rend. Sem. Mat. Fis. Milano, 58:253–284 (1990), 1988.MathSciNetGoogle Scholar
  9. 9.
    L. Caffarelli and X. Cabré. Fully nonlinear elliptic equations, volume 43 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1995.Google Scholar
  10. 10.
    L. Caffarelli, M.G. Crandall, M. Kocan, and A. Świech, On viscosity solutions of fully nonlinear equations with measurable ingredients. Commun. Pure Appl. Math., 1996.Google Scholar
  11. 11.
    L. Caffarelli and L. Silvestre. Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations. In Nonlinear partial differential equations and related topics, vol. 229, pp. 67–85. Am. Math. Soc., Providence, RI, 2010.Google Scholar
  12. 12.
    L. Caffarelli and P. E. Souganidis. A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs. Comm. Pure Appl. Math., 61:1–17, 2008.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    F. Camilli and M. Falcone. An approximation scheme for the optimal control of diffusion processes. RAIRO Modél. Math. Anal. Numér., 29(1):97–122, 1995.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ph. Ciartet and P.A. Raviart. Maximum principle and uniform convergence for the finite element method. Comp. Meths. Appl. Mech. Eng., 2(1):17-31. 1973.MathSciNetCrossRefGoogle Scholar
  15. 15.
    F. Chiarenza, M. Frasca, and P. Longo. Interior \(W^2_p\) estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche Mat., 40(1):149–168, 1991.MathSciNetMATHGoogle Scholar
  16. 16.
    F. Chiarenza, M. Frasca, and P. Longo. \(W^2_p\)-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Amer. Math. Soc., 336(2):841–853, 1993.MATHGoogle Scholar
  17. 17.
    P. G. Ciarlet. Basic error estimates for elliptic problems. In Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, pp. 17–351. North-Holland, Amsterdam, 1991.Google Scholar
  18. 18.
    E. J. Dean and R. Glowinski. 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.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    K. Debrabant and E. R. Jakobsen. Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Math. Comp., 82(283):1433–1462, 2013.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    X. Feng, L. Hennings, and M. Neilan. Finite element methods for second order linear elliptic partial differential equations in non-divergence form. Math. Comput., (to appear).Google Scholar
  21. 21.
    W. H. Fleming and H. M. Soner. Controlled Markov processes and viscosity solutions, volume 25 of Stochastic Modelling and Applied Probability. Springer, New York, Second Edition, 2006.Google Scholar
  22. 22.
    D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, Second Edition, 1983.Google Scholar
  23. 23.
    B. Grünbaum. Convex polytopes, volume 221 of Graduate Texts in Mathematics. Springer, New York, Second Edition, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler.Google Scholar
  24. 24.
    Q. Han and F. Lin. Elliptic partial differential equations, volume 1 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997.Google Scholar
  25. 25.
    R. R. Jensen. Uniformly elliptic PDEs with bounded, measurable coefficients. J. Fourier Anal. Appl., 2(3):237–259, 1995.MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    M. Jensen and I. Smears. On the convergence of finite element methods for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal., 51(1):137–162, 2013.MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    D. Kim. Second order elliptic equations in \({\mathbb{R}}^d\) with piecewise continuous coefficients. Potential Anal., 26:189–212, 2007.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    M. Kocan. Approximation of viscosity solutions of elliptic partial differential equations on minimal grids Numer. Math., 72:73–92, 1995.MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    S. Korotov, M. Křížek, and P. Neittaanmäki. Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle. Math. Comput., 70(233):107–119 (electronic), 2001.Google Scholar
  30. 30.
    N. V. Krylov. On the rate of convergence of finite-difference approximations for Bellman’s equations. Algebra i Analiz, 9:245–256, 1997.MathSciNetGoogle Scholar
  31. 31.
    N. V. Krylov. On the rate of convergence of finite-difference approximations for Bellman’s equations with variable coefficients. Probab. Theory Related Fields, 117:1–16, 2000.MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    A. Kufner, O. John, and S. Fučík. Function spaces. Noordhoff International Publishing, Leyden; Academia, Prague, 1977. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis.Google Scholar
  33. 33.
    H. J. Kuo and N. S. Trudinger. Discrete methods for fully nonlinear elliptic equations. SIAM J. Numer. Anal., 29:123–135, 1992.MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    H-J. Kuo and N. S. Trudinger. A note on the discrete Aleksandrov-Bakelman maximum principle. In Proceedings of 1999 International Conference on Nonlinear Analysis (Taipei), vol. 4, pp. 55–64, 2000.Google Scholar
  35. 35.
    H. J. Kushner and P. Dupuis. Numerical methods for stochastic control problems in continuous time, volume 24 of Applications of Mathematics (New York). Springer, New York, Second Edition, 2001. Stochastic Modelling and Applied Probability.Google Scholar
  36. 36.
    O. A. Ladyzhenskaya and N. N. Ural’tseva. Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York-London, 1968.Google Scholar
  37. 37.
    O. Lakkis and T. Pryer. A finite element method for second order nonvariational elliptic problems. SIAM J. Sci. Comput., 33(2):786–801, 2011.MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    A. Lorenzi. On elliptic equations with piecewise constant coefficients, II. Ann. Scuola Norm. Sup. Pisa, 26(3), 839-870, 1972.MathSciNetMATHGoogle Scholar
  39. 39.
    A. Maugeri, D. K. Palagachev, and L. G. Softova. Elliptic and parabolic equations with discontinuous coefficients, volume 109 of Mathematical Research. Wiley-VCH Verlag Berlin GmbH, Berlin, 2000.Google Scholar
  40. 40.
    T. S. Motzkin and W. Wasow. On the approximation of linear elliptic differential equations by difference equations with positive coefficients. J. Math. Physics, 31:253–59, 1953.MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    N. Nadirashvili. Nonuniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24(3):537–549, 1997.MathSciNetMATHGoogle Scholar
  42. 42.
    R. H. Nochetto, M. Paolini, and C. Verdi. An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates. Math. Comput., 57(195):73–108, S1–S11, 1991.Google Scholar
  43. 43.
    R. H. Nochetto and W. Zhang. Pointwise rates of convergence for the Oliker-Prussner method for the Monge-Ampère equation. (submitted).Google Scholar
  44. 44.
    R. H. Nochetto and W. Zhang. Two-scale FEM for equations in non-divergence form: Calderón-Zygmund theory. (in preparation).Google Scholar
  45. 45.
    M. Safonov. Nonuniqueness for second-order elliptic equations with measurable coefficients. SIAM J. Math. Anal., 30(4):879–895 (electronic), 1999.Google Scholar
  46. 46.
    A. H. Schatz and L. B. Wahlbin. On the quasi-optimality in \(L_{\infty }\) of the \(\dot{H}^{1}\)-projection into finite element spaces. Math. Comp., 38(157):1–22, 1982.MATHGoogle Scholar
  47. 47.
    I. Smears and E. Süli. Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordès coefficients. SIAM J. Numer. Anal., 51(4):2088–2106, 2013.MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    I. Smears and E. Süli. Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equations with Cordès coefficients. SIAM J. Numer. Anal., 52(2):993–1016, 2014.MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    A. H. Stroud. Approximate calculation of multiple integrals. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. Prentice-Hall Series in Automatic Computation.Google Scholar
  50. 50.
    S. Walker. FELICITY: Finite Element Implementation and Computational Interface Tool for you, Tutorial, 2013.Google Scholar
  51. 51.
    C. Wang and J. Wang. A primal-dual weak galerkin finite element method for second order elliptic equations in non-divergence form. http://arxiv.org/abs/1510.03499.

Copyright information

© SFoCM 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA

Personalised recommendations