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On the discrete maximum principle for parabolic difference operators. (English) Zbl 0787.65059
Authors’ summary: We derive a discrete analogue for parabolic difference inequalities of the Krylov maximum principle for parabolic differential inequalities. The result embraces both explicit and implicit difference schemes and extends to the parabolic case our previous work on linear elliptic difference inequalities with random coefficients [Math. Comput. 55, No. 191, 37-53 (1990; Zbl 0716.39005)].

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
39A10 Additive difference equations
35K15 Initial value problems for second-order parabolic equations
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[1] A. D. ALEKSANDROV, Uniqueness conditions and estimates for the solution of the Dirichlet problem, Vestnik Leningrad. Univ. 18, 1963, no. 3, pp. 5-29, English transl., Amer. Math. Soc. Transl. 1968, 2, 68, pp. 89-119. Zbl0177.36802 MR164135 · Zbl 0177.36802
[2] I. YA BAKEL’MAN, Geometric methods for solving elliptic equations, Nauka, Moscow, 1965 (In Russian).
[3] D. GILBARG and N. S. TRUDINGER, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, Heidelberg, New York and Tokyo, 1983. Zbl0361.35003 MR737190 · Zbl 0361.35003
[4] N. V. KRYLOV, Sequence of convex functions and estimates of maximum of the solution of a parabolic equation, Sibirsk Mat. Ž 1976, 17, pp. 290-303 : English translation in Siberian Math. J. 1976, 17, pp. 226-237. Zbl0362.35038 MR420016 · Zbl 0362.35038 · doi:10.1007/BF00967569
[5] N. V. KRYLOV, Nonlinear elliptic and parabolic equations of the second order, Nauka, Moscow, 1985 (In Russian). English translation by D. Reidel Publishing Company, Dordrecht, Holland, 1987. Zbl0619.35004 MR901759 · Zbl 0619.35004
[6] H. J. KUO and N. S. TRUDINGER, Linear elliptic difference inequalities with random coefficients, Math. Comp. 1990, 55, pp. 37-53. Zbl0716.39005 MR1023049 · Zbl 0716.39005 · doi:10.2307/2008791
[7] H. J. KUO and N. S. TRUDINGER, Discrete methods for fully nonlinear elliptic equations, SIAM J. on Numer. Anal. 1992, 29, pp. 123-135. Zbl0745.65058 MR1149088 · Zbl 0745.65058 · doi:10.1137/0729008
[8] T. MOTZKIN and W. WASOW, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Phys. 1952, 31, pp. 253-259. Zbl0050.12501 MR52895 · Zbl 0050.12501
[9] A. I. NAZAROV and N. N. URAL TSEVA, Convex monotone hulls and estimaties of the maximum of the solution of parabolic equations, Zap. Nauchn Sem. LOMI, 1985, 147, pp. 71-86 (In Russian). Zbl0596.35008 MR821477 · Zbl 0596.35008 · eudml:188346
[10] S. J. REYE, Harnack inequalities for parabolic equations in general form with bounded measurable coefficients, Research Report R44-84, Centre for Math. Anal. Aust. Nat. Univ. (1984) (see also Doctoral dissertation : Fully non-linear parabolic differential equations of second order, Aust. Nat. Univ. 1985).
[11] K. TSO, On an Aleksandrov-Bakel’man type maximum principle for second order parabolic equations, Comm. Partial Differential Equations 1985, 10, pp. 543-553. Zbl0581.35027 MR790223 · Zbl 0581.35027 · doi:10.1080/03605308508820388
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