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

Abstract

We propose and study an adaptive version of the discontinuous Galerkin method for Hamilton–Jacobi equations. It works as follows. Given the tolerance and the degree of the polynomial of the approximate solution, the adaptive algorithm finds a mesh on which the approximate solution has an L ∞-distance to the viscosity solution no bigger than the prescribed tolerance. The algorithm uses three main tools. The first is an iterative solver combining the explicit Runge–Kutta discontinuous Galerkin method and the implicit Newton’s method that enables us to solve the Hamilton–Jacobi equations efficiently. The second is a new a posteriori error estimate based on the approximate resolution of an approximate problem for the actual error. The third is a method that allows us to find a new mesh as a function of the old mesh and the ratio of the a posteriori error estimate to the tolerance. We display extensive numerical evidence that indicates that, for any given polynomial degree, the method achieves its goal with optimal complexity independently of the tolerance. This is done in the framework of one-dimensional steady-state model problems with periodic boundary conditions.