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

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.