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Local estimates for parabolic difference operators. (English) Zbl 0846.39007
The authors prove a local maximum principle and a weak Harnack inequality for parabolic difference inequalities. As applications, they derive discrete analogues of the Hölder and Harnack estimates of N. V. Krylov and M. V. Safonov [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 161-175 (1980; Zbl 0439.35023)] and results pertaining to the stability of discrete schemes for nonlinear parabolic problems.
For related results see the papers of M. Gruber [Math. Z. 185, 23-43 (1984; Zbl 0517.35042)] and the authors [SIAM J. Numer. Anal. 29, No. 1, 123-135 (1992; Zbl 0745.65058)].

MSC:
39A70 Difference operators
35K55 Nonlinear parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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