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Mathematics of Computation

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Generalized local maximum principles for finite-difference operators


Author: Achi Brandt
Journal: Math. Comp. 27 (1973), 685-718
MSC: Primary 65Q05
DOI: https://doi.org/10.1090/S0025-5718-1973-0329289-6
MathSciNet review: 0329289
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Abstract: The generalized local maximum principle for a difference operator $ {L_h}$ asserts that if $ {L_h}u(x) > 0$ then $ \Gamma u$ cannot attain its positive maximum at the net-point x. Here $ \Gamma $ is a local net-operator such that $ \Gamma u = u + O(h)$ for any smooth function u. This principle, with simple forms of $ \Gamma $, is proved for some quite general classes of second-order elliptic operators $ {L_h}$, whose associated global matrices are not necessarily monotone. It is shown that these generalized principles can be used for easy derivation of global a priori estimates to the solutions of elliptic difference equations and to their difference-quotients. Some examples of parabolic difference equations are also treated.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1973-0329289-6
Keywords: Elliptic difference operators, parabolic difference operators, boundary value problems, maximum principles, monotone matrices, a priori estimates, estimates for difference-quotients
Article copyright: © Copyright 1973 American Mathematical Society