Discrete Methods for Fully Nonlinear Elliptic Equations

Abstract

This paper exhibits and proves the stability of discrete approximations for the solution of the Dirichlet problem for fully nonlinear, uniformly elliptic partial differential equations in situations where classical solutions are not known to exist. The resulting solutions are characterized in the viscosity sense of Crandall and Lions [Traps. Amer. Math. Soc., 177 (1983) pp. 1–42] and our stability analysis depends on estimates for nonlinear difference equations of positive type which are consequences of earlier work on linear difference equations with random coefficients.