We present a local a posteriori error estimate for general numerical methods for
time-dependent Hamilton-Jacobi equations. Since Hamilton-Jacobi equations find
applications in many areas there is an interest in constructing efficient algorithms
that produce numerical approximations with a guaranteed precision set beforehand
by the practitioner. To develop such algorithms, it is important to be able
to estimate the quality of any given approximation in terms of computable quantities
only, and this is what a posteriori error estimates provide. Given an arbitrary
2 Rd and a time T > 0, the a posteriori error estimate gives an upper
bound for the L1-norm of the difference between the viscosity solution u and any
continuous function v in
at time T. The estimate holds for general Hamiltonians
and any space dimensions d. The case
= Rd reduces 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 both the global and local a posteriori error estimates are provided.
The results confirm that the a posteriori error estimates are very efficient and are
thus an ideal tool for devising adaptive algorithms with rigorous error control for time-dependent Hamilton-Jacobi equations.
University of Minnesota Ph.D. dissertation. May 2010. Major: Mathematics. Advisor: Prof. Bernardo Cockburn. 1 computer file (PDF); viii, 67 pages. Ill. (some col.)
Merev, Ivan Georgiev.
A posteriori error estimates for time-dependent Hamilton-Jacobi equations..
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