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.