An adaptive high order discontinuous Galerkin method with error control for the Hamilton-Jacobi equations

In the first part of this thesis, we propose and study an adaptive version of the discontinuous Galerkin method for the one-dimensional 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 Linfinity-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. The second part of this thesis is devoted to a study of the discontinuous Galerkin method for the two-dimensional steady-state Hamilton-Jacobi equations. The algorithm is also based on an iterative solver combining the explicit Runge-Kutta discontinuous Galerkin finite element method and the implicit Newton's method. There are two main contributions of this algorithm. The first is an appropriate incorporation of a slope limiter into the Newton's method. The second is the recovery of the solution to the Hamilton-Jacobi equations from that to the corresponding conservation laws. As a result, at least (k + 1)-th and k-th order of accuracy, for even and odd k respectively, is observed for smooth problems when k-th degree polynomials are used. We also study a simple adaptive version of this method applied to problems with nonsmooth solutions. Given the degree of the polynomial of the approximate solution, k ≥ 2, the algorithm starts with a uniform mesh and refines the subsequent meshes nonuniformly depending on whether the slope limiter is applied or not. We obtain faster convergence to the viscosity solution when the adaptive method is applied. This is done in the framework of two-dimensional steady-state model problems with periodic boundary conditions.