A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part I: The steady state case

Albert, Samuel; Cockburn, Bernardo; French, Donald; Peterson, Todd

doi:10.1090/S0025-5718-01-01346-1

A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part I: The steady state case
HTML articles powered by AMS MathViewer

by Samuel Albert, Bernardo Cockburn, Donald A. French and Todd E. Peterson;

Math. Comp. 71 (2002), 49-76

DOI: https://doi.org/10.1090/S0025-5718-01-01346-1

Published electronically: October 4, 2001

PDF | Request permission

Abstract:

A new upper bound is provided for the L$^\infty$-norm of the difference between the viscosity solution of a model steady state Hamilton-Jacobi equation, $u$, and any given approximation, $v$. This upper bound is independent of the method used to compute the approximation $v$; it depends solely on the values that the residual takes on a subset of the domain which can be easily computed in terms of $v$. Numerical experiments investigating the sharpness of the a posteriori error estimate are given.

References

R. Abgrall, Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes, Comm. Pure Appl. Math. 49 (1996), no. 12, 1339–1373. MR 1414589, DOI 10.1002/(SICI)1097-0312(199612)49:12<1339::AID-CPA5>3.0.CO;2-B
Timothy J. Barth and James A. Sethian, Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, J. Comput. Phys. 145 (1998), no. 1, 1–40. MR 1640142, DOI 10.1006/jcph.1998.6007
Bernardo Cockburn, Discontinuous Galerkin methods for convection-dominated problems, High-order methods for computational physics, Lect. Notes Comput. Sci. Eng., vol. 9, Springer, Berlin, 1999, pp. 69–224. MR 1712278, DOI 10.1007/978-3-662-03882-6_{2}
B. Cockburn and H. Gau, A posteriori error estimates for general numerical methods for scalar conservation laws, Mat. Aplic. Comp. 14 (1995), 37–45.
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), no. 2, 487–502. MR 732102, DOI 10.1090/S0002-9947-1984-0732102-X
Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp. 43 (1984), no. 167, 1–19. MR 744921, DOI 10.1090/S0025-5718-1984-0744921-8
K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational differential equations, Cambridge University Press, Cambridge, 1996. MR 1414897
Marizio Falcone and Roberto Ferretti, Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations, Numer. Math. 67 (1994), no. 3, 315–344. MR 1269500, DOI 10.1007/s002110050031
Changqing Hu and Chi-Wang Shu, A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 21 (1999), no. 2, 666–690. MR 1718679, DOI 10.1137/S1064827598337282
G. Kossioris, Ch. Makridakis, and P. E. Souganidis, Finite volume schemes for Hamilton-Jacobi equations, Numer. Math. 83 (1999), no. 3, 427–442. MR 1715569, DOI 10.1007/s002110050457
C.-T. Lin and E. Tadmor, L$^1$-stability and error estimates for approximate Hamiliton-Jacobi solutions, Numer. Math. 87 (2001), 701–735.
Zh. L. Korobitsyna, The method of fictitious domains for the Laplace equation with nonhomogeneous boundary conditions, Zh. Vychisl. Mat. i Mat. Fiz. 33 (1993), no. 7, 1095–1103 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 33 (1993), no. 7, 963–969. MR 1231792
Stanley Osher and Chi-Wang Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal. 28 (1991), no. 4, 907–922. MR 1111446, DOI 10.1137/0728049
Benoît Perthame and Richard Sanders, The Neumann problem for nonlinear second order singular perturbation problems, SIAM J. Math. Anal. 19 (1988), no. 2, 295–311. MR 930028, DOI 10.1137/0519022
J. A. Sethian, Level set methods, Cambridge Monographs on Applied and Computational Mathematics, vol. 3, Cambridge University Press, Cambridge, 1996. Evolving interfaces in geometry, fluid mechanics, computer vision, and materials science. MR 1409367
V. A. Dobrushkin, Analytic solution of a static boundary value problem of the plane theory of elasticity, Differentsial′nye Uravneniya 21 (1985), no. 4, 717–720, 736 (Russian). MR 791123
Lars B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR 1439050, DOI 10.1007/BFb0096835