TY - JOUR

T1 - Local a posteriori error estimates for time-dependent Hamilton-Jacobi equations

AU - Cockburn, Bernardo

AU - Merev, Ivan

AU - Qian, Jianliang

PY - 2013

Y1 - 2013

N2 - In this paper, we obtain the first local a posteriori error estimate for time-dependent Hamilton-Jacobi equations. Given an arbitrary domain Ω and a time T, the estimate gives an upper bound for the L∞-norm in Ω 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 Ω. 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.

AB - In this paper, we obtain the first local a posteriori error estimate for time-dependent Hamilton-Jacobi equations. Given an arbitrary domain Ω and a time T, the estimate gives an upper bound for the L∞-norm in Ω 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 Ω. 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.

KW - A posteriori error estimates

KW - Hamilton-Jacobi equations

UR - http://www.scopus.com/inward/record.url?scp=84872171762&partnerID=8YFLogxK

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U2 - 10.1090/S0025-5718-2012-02610-X

DO - 10.1090/S0025-5718-2012-02610-X

M3 - Article

AN - SCOPUS:84872171762

VL - 82

SP - 187

EP - 212

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 281

ER -