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

Abstract

In this paper, we introduce a new adaptive method for finding approximations for Hamilton–Jacobi equations whose L ∞ -distance to the viscosity solution is no bigger than a prescribed tolerance. This is done on the simple setting of a one-dimensional model problem with periodic boundary conditions. We consider this to be a stepping stone towards the more challenging goal of constructing such methods for general Hamilton–Jacobi equations. 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 computed by solving a differential equation whose devising is the main contribution of the paper. A thorough numerical study of the method is performed which shows that rigorous error control is achieved, even though only an approximate a posteriori error estimate is used; the method is thus reliable. Furthermore, the numerical study also shows that the method is efficient and that it has an optimal computational complexity. These properties are independent of the value of the tolerance. Finally, we provide extensive numerical evidence indicating that the adaptive method converges to an approximate solution that can be characterized solely in terms of the tolerance, the artificial viscosity of the monotone scheme and the exact solution.