| We devise and study an adaptive method for finding approximations to the viscosity solution of Hamilton-Jacobi equations. The method is studied in the framework of steady state Hamilton-Jacobi equations with periodic boundary conditions. It seeks numerical approximations whose L infinity-distance to the viscosity solution is no bigger than a prescribed tolerance. The method proceeds as follows. On any given grid, the approximate solution is computed by using a well-known monotone scheme; then, the quality of the approximation is tested by using an approximate a posteriori error estimate. If the error is bigger than the prescribed tolerance, a new grid is generated and the same process is repeated until the desired quality of the approximation is achieved. Adaptive meshes are used to reduce the amount of computations, and hence to devise an efficient method. A thorough numerical study is carried out on one-dimensional and two-dimensional problems, 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. This is the first adaptive method for Hamilton-Jacobi equations with these characteristics. |
