Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 
 

 

Local a posteriori error estimates for time-dependent Hamilton-Jacobi equations


Authors: Bernardo Cockburn, Ivan Merev and Jianliang Qian
Journal: Math. Comp. 82 (2013), 187-212
MSC (2010): Primary 65M15, 65M12; Secondary 49L25
DOI: https://doi.org/10.1090/S0025-5718-2012-02610-X
Published electronically: June 5, 2012
MathSciNet review: 2983021
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we obtain the first local a posteriori error estimate for time-dependent Hamilton-Jacobi equations. Given an arbitrary domain $ \Omega $ and a time $ T$, the estimate gives an upper bound for the $ L^\infty $-norm in $ \Omega $ at time $ T$ of the difference between the viscosity solution $ u$ and any continuous function $ v$ in terms of the initial error in the domain of dependence and in terms of the (shifted) residual of $ v$ in the union of all the cones of dependence with vertices in $ \Omega $. The estimate holds for general Hamiltonians and any space dimension. It is thus an ideal tool for devising adaptive algorithms with rigorous error control for time-dependent Hamilton-Jacobi equations. This result is an extension to the global a posteriori error estimate obtained by S. Albert, B. Cockburn, D. French, and T. Peterson in A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part II: The time-dependent case, Finite Volumes for Complex Applications, vol. III, June 2002, pp. 17-24. Numerical experiments investigating the sharpness of the a posteriori error estimates are given.


References [Enhancements On Off] (What's this?)

  • 1. R. Abgrall, Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes, Comm. Pure Appl. Math. 49 (1996), 1339-1373. MR 1414589 (98d:65121)
  • 2. S. Albert, B. Cockburn, D. French, and T. Peterson, A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part I: The steady state case, Math. Comp. 71 (2002), 49-76. MR 1862988 (2002m:65107)
  • 3. -, A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part II: The time-dependent case, Finite Volumes for Complex Applications, (R. Herbin and D. Kröner, eds.), vol. III, Hermes Penton Science, June 2002, 17-24. MR 2007401
  • 4. T. Barth and J. Sethian, Numerical schemes for Hamilton-Jacobi and level set equations on triangular domains, J. Comput. Phys. 145 (1998), 1-40. MR 1640142 (99d:65277)
  • 5. Y. Chen and B. Cockburn, An adaptive high order discontinuous Galerkin method with error control for time Hamilton-Jacobi equations. Part I: The one-dimensional steady state case, J. Comput. Phys. 226 (2007), 1027-1058. MR 2356867 (2008k:65239)
  • 6. B. Cockburn and B. Yenikaya, An adaptive method with rigorous error control for the Hamilton-Jacobi equations. Part I: The one-dimensional steady state case, Appl. Numer. Math. 52 (2005), 175-195. MR 2116910 (2005i:65200)
  • 7. -, An adaptive method with rigorous error control for the Hamilton-Jacobi equations. Part II: The two-dimensional steady state case, J. Comput. Phys. 209 (2005), 391-405. MR 2151991 (2006b:65185)
  • 8. M. G. Crandall, L. C. Evans, and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 478-502. MR 732102 (86a:35031)
  • 9. M. G. Crandall and P. L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp. 43 (1984), 1-19. MR 744921 (86j:65121)
  • 10. M. Falcone and R. Ferretti, Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations, Numer. Math. 67 (1994), 315-344. MR 1269500 (95d:49045)
  • 11. C.  Hu and C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 21 (1999), 666-690. MR 1718679 (2000g:65095)
  • 12. H. Ishii, Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations, Indiana Univ. Math. Journal 33 (1984), 721-748. MR 756156 (85h:35057)
  • 13. S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal. 28 (1991), 907-922. MR 1111446 (92e:65118)
  • 14. J. Qian, Approximations for viscosity solutions of Hamilton-Jacobi equations with locally varying time and space grids, SIAM J. Numer. Anal. 2006, 2391-2401. MR 2206440 (2007b:49055)
  • 15. P. E. Souganidis, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, J. Diff. Eqns. 59 (1985), 1-43. MR 803085 (86k:35028)
  • 16. G. B. Whitham, Linear and nonlinear waves, Wiley, New York, NY, 1974. MR 0483954 (58:3905)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65M15, 65M12, 49L25

Retrieve articles in all journals with MSC (2010): 65M15, 65M12, 49L25


Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Ivan Merev
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
Email: merev001@math.umn.edu

Jianliang Qian
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: qian@math.msu.edu

DOI: https://doi.org/10.1090/S0025-5718-2012-02610-X
Keywords: A posteriori error estimates, Hamilton-Jacobi equations
Received by editor(s): May 7, 2010
Received by editor(s) in revised form: May 27, 2011
Published electronically: June 5, 2012
Additional Notes: The first author was partially supported by the National Science Foundation (Grant DMS-0712955) and by the Minnesota Supercomputing Institute.
The third author was partially supported by the National Science Foundation (NSF 0810104 and NSF 0830161).
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society