L1-Approximation of Stationary Hamilton-Jacobi Equations

We describe a nonlinear finite element technique to approximate the solutions of stationary Hamilton-Jacobi equations in two space dimensions using continuous finite elements of arbitrary degree. The method consists of minimizing a functional containing the L-norm of the Hamiltonian plus a discrete entropy. It is shown that the approximate sequence converges to the unique viscosity solution under appropriate hypotheses on the Hamiltonian and the mesh family.