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 und...

We prove optimality principles for continuous bounded nonnegative viscosity solutions of Hamilton-Jacobi-Bellman equations. In particular we provide a representation formula for viscosity supersolutions as value functions of suitable obstacle control problems. This representation formula is applied to extend the Lyapunov direct method for stability to controlled Ito stochastic differential equa...

The spectral viscosity approximate solution of convex Hamilton–Jacobi equations with periodic boundary conditions is studied. It is proved in this paper that the approximation and its gradient remain uniformly bounded, formally spectral accurate, and converge to the unique viscosity solution. The L1-convergence rate of the order 1− ε∀ε > 0 is obtained.

The viscosity solution of static Hamilton-Jacobi equations with a pointsource condition has an upwind singularity at the source, which makes all formally high-order finite-difference scheme exhibit first-order convergence and relatively large errors. To obtain designed high-order accuracy, one needs to treat this source singularity during computation. In this paper, we apply the factorization i...

In this paper, we obtain the first local a posteriori error estimate for time-dependent Hamilton-Jacobi equations. Given an arbitrary domain Ω and a time T , the estimate gives an upper bound for the L∞-norm in Ω at time T of the difference between the viscosity solution u and any continuous function v in terms of the initial error in the domain of dependence and in terms of the (shifted) resid...

We study an optimal control problem for viscosity solutions of a Hamilton-Jacobi equation describing the propagation of a one dimensional graph with the control being the speed function. The existence of an optimal control is proved together with an approximate controllability result in the H−1 norm. We prove convergence of a discrete optimal control problem based on a monotone finite differenc...

A new upper bound is provided for the L∞-norm of the difference between the viscosity solution of a model steady state Hamilton-Jacobi equation, u, and any given approximation, v. This upper bound is independent of the method used to compute the approximation v; it depends solely on the values that the residual takes on a subset of the domain which can be easily computed in terms of v. Numerica...