EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 56 / No 2 (March-April 2022) ESAIM: M2AN, 56 2 (2022) 679-704 References
Open Access
Issue
ESAIM: M2AN
Volume 56, Number 2, March-April 2022
Page(s) 679 - 704
DOI https://doi.org/10.1051/m2an/2022017
Published online 15 March 2022
  1. Y. Achdou, F. Camilli and I. Capuzzo Dolcetta, Homogenization of Hamilton-Jacobi equations: numerical methods. Math. Models Methods Appl. Sci. 18 (2008) 1115–1143. [CrossRef] [MathSciNet] [Google Scholar]
  2. O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40 (2001) 1159–1188. [Google Scholar]
  3. O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170 (2003) 17–61. [Google Scholar]
  4. O. Alvarez and M. Bardi, Ergodic problems in differential games. In: Advances in Dynamic Game Theory. Vol. 9 of Ann. Internat. Soc. Dynam. Games. Birkhäuser Boston, Boston, MA (2007) 131–152. [CrossRef] [Google Scholar]
  5. O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations. Mem. Amer. Math. Soc. 204 (2010) vi+77. [Google Scholar]
  6. O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations. J. Differ. Equ. 243 (2007) 349–387. [CrossRef] [Google Scholar]
  7. M. Arisawa and P.-L. Lions, On ergodic stochastic control. Comm. Part. Differ. Equ. 23 (1998) 2187–2217. [CrossRef] [Google Scholar]
  8. O. Bokanowski, S. Maroso and H. Zidani, Some convergence results for Howard’s algorithm. SIAM J. Numer. Anal. 47 (2009) 3001–3026. [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. 388 (2021) 113241. [CrossRef] [Google Scholar]
  10. L. Caffarelli, M.G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math. 49 (1996) 365–397. [CrossRef] [MathSciNet] [Google Scholar]
  11. F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes. ESAIM: M2AN 29 (1995) 97–122. [CrossRef] [EDP Sciences] [Google Scholar]
  12. F. Camilli and E.R. Jakobsen, A finite element like scheme for integro-partial differential Hamilton–Jacobi–Bellman equations. SIAM J. Numer. Anal. 47 (2009) 2407–2431. [CrossRef] [MathSciNet] [Google Scholar]
  13. F. Camilli and C. Marchi, Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs. Nonlinearity 22 (2009) 1481–1498. [CrossRef] [MathSciNet] [Google Scholar]
  14. Y. Capdeboscq, T. Sprekeler and E. Süli, Finite element approximation of elliptic homogenization problems in nondivergence-form. ESAIM: M2AN 54 (2020) 1221–1257. [CrossRef] [EDP Sciences] [Google Scholar]
  15. 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. [CrossRef] [MathSciNet] [Google Scholar]
  16. L.C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. R. Soc. Edinburgh Sect. A 111 (1989) 359–375. [CrossRef] [Google Scholar]
  17. L.C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. R. Soc. Edinburgh Sect. A 120 (1992) 245–265. [CrossRef] [Google Scholar]
  18. M. Falcone and M. Rorro, On a variational approximation of the effective Hamiltonian. In: Numerical Mathematics and Advanced Applications. Springer, Berlin (2008) 719–726. [CrossRef] [Google Scholar]
  19. 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]
  20. 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]
  21. C. Finlay and A.M. Oberman, Approximate homogenization of convex nonlinear elliptic PDEs. Commun. Math. Sci. 16 (2018) 1895–1906. [Google Scholar]
  22. C. Finlay and A.M. Oberman, Approximate homogenization of fully nonlinear elliptic PDEs: estimates and numerical results for Pucci type equations. J. Sci. Comput. 77 (2018) 936–949. [CrossRef] [MathSciNet] [Google Scholar]
  23. W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edition. Vol. 25 of Stochastic Modelling and Applied Probability. Springer, New York (2006). [Google Scholar]
  24. D. Gallistl, Numerical approximation of planar oblique derivative problems in nondivergence form. Math. Comp. 88 (2019) 1091–1119. [Google Scholar]
  25. 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]
  26. D. Gallistl, T. Sprekeler and E. Süli, Mixed finite element approximation of periodic Hamilton–Jacobi–Bellman problems with application to numerical homogenization. Multiscale Model. Simul. 19 (2021) 1041–1065. [CrossRef] [MathSciNet] [Google Scholar]
  27. R. Glowinski, S. Leung and J. Qian, A simple explicit operator-splitting method for effective Hamiltonians. SIAM J. Sci. Comput. 40 (2018) A484–A503. [CrossRef] [Google Scholar]
  28. D.A. Gomes and A.M. Oberman, Computing the effective Hamiltonian using a variational approach. SIAM J. Control Optim. 43 (2004) 792–812. [CrossRef] [MathSciNet] [Google Scholar]
  29. H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm. Pure Appl. Math. 42 (1989) 15–45. [CrossRef] [MathSciNet] [Google Scholar]
  30. M. Jensen, Formula finite element convergence for degenerate isotropic Hamilton–Jacobi–Bellman equations. IMA J. Numer. Anal. 37 (2017) 1300–1316. [MathSciNet] [Google Scholar]
  31. 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]
  32. 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]
  33. 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]
  34. 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]
  35. E.L. Kawecki and T. Pryer, Virtual element methods for non-divergence form equations. To appear. [Google Scholar]
  36. E.L. Kawecki and I. Smears, Unified analysis of discontinuous Galerkin and C0-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations. ESAIM: M2AN 55 (2021) 449–478. [CrossRef] [EDP Sciences] [Google Scholar]
  37. 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, Found. Comput. Math. (2021) DOI: 10.1007/s10208-021-09493-0. [Google Scholar]
  38. E.L. Kawecki, O. Lakkis and T. Pryer, A finite element method for the Monge-Ampère equation with transport boundary conditions. Preprint arXiv:1807.03535 (2018). [Google Scholar]
  39. P.-L. Lions, Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations. II. Viscosity solutions and uniqueness. Comm. Part. Differ. Equ. 8 (1983) 1229–1276. [CrossRef] [Google Scholar]
  40. P.-L. Lions, A remark on Bony maximum principle. Proc. Amer. Math. Soc. 88 (1983) 503–508. [CrossRef] [MathSciNet] [Google Scholar]
  41. S. Luo, Y. Yu and H. Zhao, A new approximation for effective Hamiltonians for homogenization of a class of Hamilton-Jacobi equations. Multiscale Model. Simul. 9 (2011) 711–734. [Google Scholar]
  42. N. Nadirashvili and S. Vlăduț, Nonclassical solutions of fully nonlinear elliptic equations. Geom. Funct. Anal. 17 (2007) 1283–1296. [CrossRef] [MathSciNet] [Google Scholar]
  43. N. Nadirashvili and S. Vlăduț, Singular solutions of Hessian fully nonlinear elliptic equations. Adv. Math. 228 (2011) 1718–1741. [CrossRef] [MathSciNet] [Google Scholar]
  44. 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]
  45. M. Neilan, A.J. Salgado and W. Zhang, Numerical analysis of strongly nonlinear PDEs. Acta Numer. 26 (2017) 137–303. [Google Scholar]
  46. A.M. Oberman, R. Takei and A. Vladimirsky, Homogenization of metric Hamilton-Jacobi equations. Multiscale Model. Simul. 8 (2009) 269–295. [Google Scholar]
  47. J. Qian, Two approximations for effective Hamiltonians arising from homogenization of Hamilton-Jacobi equations. UCLA CAM report 03–39 (2003). [Google Scholar]
  48. J. Qian, H.V. Tran and Y. Yu, Min-max formulas and other properties of certain classes of nonconvex effective Hamiltonians. Math. Ann. 372 (2018) 91–123. [CrossRef] [MathSciNet] [Google Scholar]
  49. A.J. Salgado and W. Zhang, Finite element approximation of the Isaacs equation. ESAIM: M2AN 53 (2019) 351–374. [Google Scholar]
  50. 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]
  51. 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]
  52. 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]
  53. 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]
  54. T. Sprekeler and H.V. Tran, Optimal convergence rates for elliptic homogenization problems in nondivergence-form: analysis and numerical illustrations. Multiscale Model. Simul. 19 (2021) 1453–1473. [CrossRef] [MathSciNet] [Google Scholar]
  55. K. Vemaganti, Discontinuous Galerkin methods for periodic boundary value problems. Numer. Methods Part. Differ. Equ. 23 (2007) 587–596. [CrossRef] [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