An adaptive method with rigorous error control for the Hamilton-Jacobi equations. Part II: The two-dimensional steady-state case

Authors:
Bernardo Cockburn;Bayram Yenikaya
Affiliations:
School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN 55455, USA;School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN 55455, USA
Venue:
Journal of Computational Physics
Year:
2005

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Abstract

In this paper, we devise and study an adaptive method for finding approximations to the viscosity solution of Hamilton-Jacobi equations. The method, which is an extension to two space dimensions of a similar method previously proposed for one space dimension, is studied in the framework of steady-state Hamilton-Jacobi equations with periodic boundary conditions. It seeks numerical approximations whose L^~-distance to the viscosity solution is no bigger than a prescribed tolerance. A thorough numerical study is carried out which shows that a strict error control is achieved and that the method exhibits an optimal computational complexity which does not depend on the value of the tolerance or on the type of Hamiltonian.