An adaptive high-order discontinuous Galerkin method with error control for the Hamilton-Jacobi equations. I: The one-dimensional steady state case (Q2456732)

The authors introduce and study a numerical scheme in order to approximate the viscosity solution of a one-dimensional steady-state Hamilton-Jacobi equation. The scheme contains three basic steps, namely: a) an iterative solver used to solve the nonlinear system resulting from the discontinuous Galerkin method with polynomials of a given order; b) a new a posteriori error estimate; and c) a way of computing a new grid by the ratio of the a posteriori error estimate to the tolerance. With this scheme the authors carry out numerical experiments on six, qualitatively different, test problems. They show that the new a posteriori error estimate is really sharp and the adaptive method is effective and robust.