Abstract
We present an a posteriori estimate for a first order semi-Lagrangian method for Hamilton–Jacobi equations. The result requires piecewise C 1,1 regularity of the viscosity solution and is stated for the Bellman equation related to the infinite horizon problem, although it can be applied to more general Hamilton–Jacobi equations with convex Hamiltonians. This estimate suggests different numerical indicators that can be used to construct an adaptive algorithm for the approximation of the viscosity solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Abgrall, Numerical discretization of the first-order Hamilton–Jacobi equation on triangular meshes, Comm. Pure Appl. Math. 49 (1996) 1339–1373.
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, Preprint (2001).
M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations (Birkhäuser, Boston, 1997).
M.W. Bern, J.E. Flaherty and M. Luskin, Grid Generation and Adaptive Algorithms (Springer, New York, 1999).
F. Camilli and L. Grüne, Numerical Approximation of the Maximal solutions for a class of degenerate Hamilton–Jacobi equations, SIAM J. Numer. Anal. 38 (2000) 1540–1560.
M.G. Crandall and P.L. Lions, Two approximations of solutions of Hamilton–Jacobi equations, Math. Comp. 43 (1984) 1–19.
K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, in: Acta Numerica (Cambridge Univ. Press, Cambridge, 1995) pp. 105–158.
M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim. 15 (1987) 1–13 and 23 (1991) 213–214.
M. Falcone and R. Ferretti, Discrete-time high-order schemes for viscosity solutions of Hamilton–Jacobi equations, Numer. Math. 67 (1994) 315–344.
M. Falcone and R. Ferretti, Semi-Lagrangian schemes for Hamilton–Jacobi equations, discrete representation formulae and Godunov methods, J. Comput. Phys. (2001) to appear.
M. Falcone and C. Makridakis, eds., Numerical Methods for Viscosity Solutions and Applications (World Scientific, Singapore, 2001).
L. Grüne, An adaptive grid scheme for the discrete Hamilton–Jacobi–Bellman equation, Numer. Math. 75 (1997) 319–337.
L. Grüne, Adaptive grid generation for evolutive Hamilton–Jacobi–Bellman equations, in [11], pp. 319–337.
C. Hu and C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM J. Sci. Comput. 21 (1999) 666–690.
G. Kossioris, C. Makridakis and P.E. Souganidis, Finite volume schemes for Hamilton–Jacobi equations, Numer. Math. 83 (1999) 427–442.
C.T. Lin and E. Tadmor, High-resolution nonoscillatory central schemes for Hamilton–Jacobi equations, SIAM J. Sci. Comput. 21 (2000) 2163–2186.
P.L. Lions, Generalized Solutions of Hamilton–Jacobi Equations (Pitman, London, 1982).
S. Osher and C.W. Shu, High-order essentially non oscillatory schemes for Hamilton–Jacobi equations, SIAM J. Numer. Anal. 28 (1991) 907–922.
M. Sagona, Numerical methods for degenerate Eikonal type equations and applications, Tesi di Dottorato, Dipartimento di Matematica, Università di Napoli “Federico II” (November 2001).
M. Sagona and A. Seghini, An adaptive scheme on unstructured grids for the shape-from-shading problem, in [11].
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sagona, M., Seghini, A. An a Posteriori Error Estimate for a Semi-Lagrangian Scheme for Hamilton–Jacobi Equations. Numerical Algorithms 33, 453–460 (2003). https://doi.org/10.1023/A:1025509315400
Issue Date:
DOI: https://doi.org/10.1023/A:1025509315400