Fully nonlinear elliptic and parabolic equations appear in numerous applications of partial differential equations, ranging from control theory of random processes to differential geometry.
An important class of techniques to solve these equations numerically are given by various finite-difference schemes. We will give a brief introduction to these methods, outline some of their strengths and limitations, and then show that very strong error bounds hold for certain types of schemes. These bounds are new even for linear elliptic equations, but in fact also apply to parabolic Bellman equations with Lipschitz coefficients.
Part of the results is obtained in collaboration with Hongjie Dong from Brown University.