Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
Published electronically: June 5, 2012
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?)


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: http://dx.doi.org/10.1090/S0025-5718-2012-02610-X
PII: S 0025-5718(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