Approximation schemes for viscosity solutions of Hamilton-Jacobi equations

Equations of Hamilton-Jacobi type arise in many areas of applications, including the calculus of variations, control theory and differential games. Recently M. G. Crandall and P.-L. Lions established the correct notion of generalized solutions for these equations. This article discusses the convergence of general approximation schemes to this solution and gives, under certain hypotheses, explicit error estimates. These results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes, with error estimates.