Nonlinear elliptic and parabolic partial differential equations (PDEs) appear in problems from science, engineering, atmospheric/ocean studies, image processesing, and mathematical finance.
The theory of viscosity solutions has been enormously successful in addressing the problems of existence, uniqueness, and stability for a wide class of such equations.
A problem which has not been addressed with as much success is the construction of solutions. In some cases, exact solutions formulas exist, but for the most part, solutions must be found numerically.
In the sprit of the classical 1928 paper of Courant, Freidrichs, and Lewy which used the finite difference method to construct solutions of linear PDEs, we construct solutions to nonlinear degenerate elliptic and parabolic PDEs.
We will present example schemes and computational results, for: valuation in math finance, motion by mean curvature, and the infinity Laplacian.