Free Access
Issue
ESAIM: M2AN
Volume 55, Number 2, March-April 2021
Page(s) 449 - 478
DOI https://doi.org/10.1051/m2an/2020081
Published online 15 March 2021
  1. L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000). [Google Scholar]
  2. G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991) 271–283. [MathSciNet] [Google Scholar]
  3. J. Blechschmidt, R. Herzog and M. Winkler, Error estimation for second-order PDEs in non-variational form. Preprint arXiv:1909.12676 (2019). [Google Scholar]
  4. J. Blechta, J. Málek and M. Vohralk, Localization of the W−1, q norm for local a posteriori efficiency. IMA J. Numer. Anal. 40 (2020) 914–950. [Google Scholar]
  5. O. Bokanowski, S. Maroso and H. Zidani, Some convergence results for Howard’s algorithm. SIAM J. Numer. Anal. 47 (2009) 3001–3026. [Google Scholar]
  6. J.F. Bonnans and H. Zidani, Consistency of generalized finite difference schemes for the stochastic HJB equation. SIAM J. Numer. Anal. 41 (2003) 1008–1021. [Google Scholar]
  7. S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal. 41 (2003) 306–324. [Google Scholar]
  8. S.C. Brenner, T. Gudi, M. Neilan and L.Y. Sung, C0 penalty methods for the fully nonlinear Monge–Ampère equation. Math. Comput. 80 (2011) 1979–1995. [Google Scholar]
  9. S.C. Brenner and E.L. Kawecki, Adaptive C0 interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients. J. Comput. Appl. Math. 113241 (2020). [Google Scholar]
  10. S.C. Brenner and L.Y. Sung, Virtual enriching operators. Calcolo 56 (2019) 25. [Google Scholar]
  11. L.A. Caffarelli and X. Cabré, Fully nonlinear elliptic equations. In: Vol. 43 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (1995). [Google Scholar]
  12. C. Carstensen, T. Gudi and M. Jensen, A unifying theory of a posteriori error control for discontinuous Galerkin FEM. Numer. Math. 112 (2009) 363–379. [Google Scholar]
  13. F. Chiarenza, M. Frasca and P. Longo, W2,p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 336 (1993) 841–853. [Google Scholar]
  14. P.G. Ciarlet, The finite element method for elliptic problems. In: Vol. 40 of Classics in Applied Mathematics. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). [Google Scholar]
  15. P.G. Ciarlet, Linear and Nonlinear Functional Analysis With Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (2013). [Google Scholar]
  16. H.O. Cordes, Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen. Math. Ann. 131 (1956) 278–312. [Google Scholar]
  17. M.G. Crandall and P.L. Lions, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion. Numer. Math. 75 (1996) 17–41. [Google Scholar]
  18. M.G. Crandall, H. Ishii and P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. [Google Scholar]
  19. K. Debrabant and E.R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Math. Comput. 82 (2013) 1433–1462. [Google Scholar]
  20. D.A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods. In: Vol. 69. of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg (2012). [Google Scholar]
  21. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, revised edition. In: Textbooks in Mathematics. CRC Press, Boca Raton, FL (2015). [Google Scholar]
  22. X. Feng, R. Glowinski and M. Neilan, Recent developments in numerical methods for fully nonlinear second order partial differential equations. SIAM Rev. 55 (2013) 205–267. [Google Scholar]
  23. X. Feng and M. Jensen, Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids. SIAM J. Numer. Anal. 55 (2017) 691–712. [Google Scholar]
  24. W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions, 2nd edition. In: Vol. 25 of Stochastic Modelling and Applied Probability. Springer, New York (2006). [Google Scholar]
  25. I. Fonseca, G. Leoni and R. Paroni, On Hessian matrices in the space BH. Commun. Contemp. Math. 7 (2005) 401–420. [Google Scholar]
  26. D. Gallistl, Variational formulation and numerical analysis of linear elliptic equations in nondivergence form with Cordes coefficients. SIAM J. Numer. Anal. 55 (2017) 737–757. [Google Scholar]
  27. D. Gallistl, Numerical approximation of planar oblique derivative problems in nondivergence form. Math. Comput. 88 (2019) 1091–1119. [Google Scholar]
  28. D. Gallistl and E. Süli, Mixed finite element approximation of the Hamilton–Jacobi–Bellman equation with Cordes coefficients. SIAM J. Numer. Anal. 57 (2019) 592–614. [Google Scholar]
  29. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. . In: Classics in Mathematics. Reprint of the 1998 edition. Springer-Verlag, Berlin (2001). [Google Scholar]
  30. T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79 (2010) 2169–2189. [Google Scholar]
  31. P. Houston, D. Schötzau and T.P. Wihler, Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math. Models Methods Appl. Sci. 17 (2007) 33–62. [Google Scholar]
  32. M. Jensen, L2 (Hγ1) finite element convergence for degenerate isotropic Hamilton–Jacobi–Bellman equations. IMA J. Numer. Anal. 37 (2017) 1300–1316. [Google Scholar]
  33. M. Jensen and I. Smears, On the convergence of finite element methods for Hamilton–Jacobi–Bellman equations. SIAM J. Numer. Anal. 51 (2013) 137–162. [Google Scholar]
  34. O.A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374–2399. [Google Scholar]
  35. E.L. Kawecki, Finite element methods for Monge-Ampère type equations. Ph.D. thesis, University of Oxford (2018). [Google Scholar]
  36. E.L. Kawecki, A DGFEM for nondivergence form elliptic equations with Cordes coefficients on curved domains. Numer. Methods Part. Differ. Equ. 35 (2019) 1717–1744. [Google Scholar]
  37. E.L. Kawecki, A discontinuous Galerkin finite element method for uniformly elliptic two dimensional oblique boundary-value problems. SIAM J. Numer. Anal. 57 (2019) 751–778. [Google Scholar]
  38. E.L. Kawecki and I. Smears, Convergence of adaptive discontinuous Galerkin and C0-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations. Preprint arXiv 2006.07215 (2020). [Google Scholar]
  39. M. Kocan, Approximation of viscosity solutions of elliptic partial differential equations on minimal grids. Numer. Math. 72 (1995) 73–92. [Google Scholar]
  40. N.V. Krylov, Nonlinear elliptic and parabolic equations of the second order. Translated from the Russian by P. L. Buzytsky [P. L. Buzytski]. In: Vol. 7 of Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht (1987). [Google Scholar]
  41. H.J. Kuo and N.S. Trudinger, Linear elliptic difference inequalities with random coefficients. Math. Comput. 55 (1990) 37–53. [Google Scholar]
  42. H.J. Kushner and P. Dupuis, Numerical methods for stochastic control problems in continuous time, 2nd edition. In: Vol. 24 of Applications of Mathematics (New York). Springer-Verlag, New York (2001). [Google Scholar]
  43. O. Lakkis and T. Pryer, A finite element method for second order nonvariational elliptic problems. SIAM J. Sci. Comput. 33 (2011) 786–801. [Google Scholar]
  44. O. Lakkis and T. Pryer, A finite element method for nonlinear elliptic problems. SIAM J. Sci. Comput. 35 (2013) A2025–A2045. [Google Scholar]
  45. A. Maugeri, D.K. Palagachev and L.G. Softova, Elliptic and parabolic equations with discontinuous coefficients. In: Vol. 109 of Mathematical Research. Wiley-VCH Verlag Berlin GmbH, Berlin (2000). [Google Scholar]
  46. T.S. Motzkin and W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients. J. Math. Phys. 31 (1953) 253–259. [Google Scholar]
  47. M. Neilan, A.J. Salgado and W. Zhang, Numerical analysis of strongly nonlinear PDEs. Acta Numer. 26 (2017) 137–303. [Google Scholar]
  48. M. Neilan and M. Wu. Discrete Miranda-Talenti estimates and applications to linear and nonlinear PDEs. J. Comput. Appl. Math. 356 (2019) 358–376. [Google Scholar]
  49. R.H. Nochetto and W. Zhang, Discrete ABP estimate and convergence rates for linear elliptic equations in non-divergence form. Found. Comput. Math. 18 (2018) 537–593. [Google Scholar]
  50. M.V. Safonov, Nonuniqueness for second-order elliptic equations with measurable coefficients. SIAM J. Math. Anal. 30 (1999) 879–895. [Google Scholar]
  51. A.J. Salgado and W. Salgado, Finite element approximation of the Isaacs equation. ESAIM: M2AN 53 (2019) 351–374. [EDP Sciences] [Google Scholar]
  52. J. Schöberl, C++11 implementation of finite elements in NGSolve. Tech. Rep. ASC Report 30/2014, Institute for Analysis and Scientific Computing, Vienna University of Technology (2014). [Google Scholar]
  53. I. Smears, Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton–Jacobi–Bellman equations. J. Sci. Comput. 74 (2018) 145–174. [Google Scholar]
  54. I. Smears and E. Süli, Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordès coefficients. SIAM J. Numer. Anal. 51 (2013) 2088–2106. [Google Scholar]
  55. I. Smears and E. Süli, Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman equations with Cordes coefficients. SIAM J. Numer. Anal. 52 (2014) 993–1016. [Google Scholar]
  56. I. Smears and E. Süli, Discontinuous Galerkin finite element methods for time-dependent Hamilton–Jacobi–Bellman equations with Cordes coefficients. Numer. Math. 133 (2016) 141–176. [Google Scholar]
  57. A. Veeser and P. Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. I—Abstract theory. SIAM J. Numer. Anal. 56 (2018) 1621–1642. [Google Scholar]
  58. R. Verfürth, A posteriori error estimation techniques for finite element methods. In: Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013). [Google Scholar]
  59. E. Zeidler, Nonlinear functional analysis and its applications. II/B. Nonlinear monotone operators, Translated from the German by the author and Leo F. Boron. Springer-Verlag, New York (1990). [Google Scholar]

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