Free Access
Volume 55, Number 2, March-April 2021
Page(s) 449 - 478
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]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you