Finite-element Discretization of Static Hamilton-Jacobi Equations based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton–Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss–Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.