The author treats the construction of convergent finite difference methods for a class of nonlinear partial differential equations, the class of degenerate elliptic equations with unique viscosity solutions, already defined by M. G. Crandall, H. Ishii and P.-L. Lions [Bull. Am. Math. Soc., New Ser. 27, No. 1, 1–67 (1992; Zbl 0755.35015)].
To get convergence results, theorems about monotonicity and nonexpansivity in the maximum norm are provided and proved for the class of the developed schemes, the so-called degenerate elliptic schemes. The techniques involve schemes for Hamilton-Jacobi equations, free boundary problems and obstacle problems using building blocks of schemes for simpler equations. For time-dependent equations explicit schemes with a nonlinear Courant-Friedrichs-Lewy condition are presented.