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Discrete methods for fully nonlinear elliptic equations. (English) Zbl 0745.65058
This work considers fully nonlinear elliptic equations satisfying structure conditions modelled on linear uniformly elliptic equations. Thus, the stability analysis depends on estimates for nonlinear difference equations of positive type which are consequences of earlier work of the authors [Math. Comp. 55, 37-53 (1990; Zbl 0716.39005)] concerned with linear difference equations with random coefficients. Further, when classical solutions do not exist, the concept of a weak solution becomes important for both existence and approximation considerations.
Here the resulting solutions are characterized in the viscosity sense of M. G. Crandall and P.-L. Lions [Trans. Amer. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)]. Developing elliptic difference schemes which are consistent with the nonlinear equation, treating the relationship between consistent difference schemes and viscosity solutions, thereby proving the stability of direct approximations, finally providing iterative schemes for solving the approximating difference equations, the authors emphasize that these works are intended as preliminary theoretical thrusts at the numerical approximation of nonclassical solutions of fully nonlinear equations.

MSC:
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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