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 asserts that if then cannot attain its positive maximum at the net-point *x*. Here is a local net-operator such that for any smooth function *u*. This principle, with simple forms of , is proved for some quite general classes of second-order elliptic operators , 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.

**[1]**James H. Bramble,*Fourth-order finite difference analogues of the Dirichlet problem for Poisson’s equation in three and four dimensions*, Math. Comp.**17**(1963), 217–222. MR**0160338**, https://doi.org/10.1090/S0025-5718-1963-0160338-8**[2]**James H. Bramble,*On the convergence of difference approximations for second order uniformly elliptic operators*, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R. I., 1970, pp. 201–209. MR**0260200****[3]**J. H. Bramble and B. E. Hubbard,*On the formulation of finite difference analogues of the Dirichlet problem for Poisson’s equation*, Numer. Math.**4**(1962), 313–327. MR**0149672**, https://doi.org/10.1007/BF01386325**[4]**J. H. Bramble and B. E. Hubbard,*A theorem on error estimation for finite difference analogues of the Dirichlet problem for elliptic equations*, Contributions to Differential Equations**2**(1963), 319–340. MR**0152134****[5]**James H. Bramble and Bert E. Hubbard,*New monotone type approximations for elliptic problems*, Math. Comp.**18**(1964), 349–367. MR**0165702**, https://doi.org/10.1090/S0025-5718-1964-0165702-X**[6]**J. H. Bramble, B. E. Hubbard, and Vidar Thomée,*Convergence estimates for essentially positive type discrete Dirichlet problems*, Math. Comp.**23**(1969), 695–709. MR**0266444**, https://doi.org/10.1090/S0025-5718-1969-0266444-7**[7]**Achi Brandt,*Estimates for difference quotients of solutions of Poisson type difference equations*, Math. Comp.**20**(1966), 473–499. MR**0204897**, https://doi.org/10.1090/S0025-5718-1966-0204897-8**[8]**A. Brandt,*Interior estimates for second-order elliptic differential (or finite-difference) equations via the maximum principle*, Israel J. Math.**7**(1969), 95–121. MR**0252836**, https://doi.org/10.1007/BF02771657**[9]**A. Brandt,*Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle*, Israel J. Math.**7**(1969), 254–262. MR**0249803**, https://doi.org/10.1007/BF02787619**[10]**Philippe G. Ciarlet,*Discrete maximum principle for finite-difference operators*, Aequationes Math.**4**(1970), 338–352. MR**0292317**, https://doi.org/10.1007/BF01844166**[11]**L. Collatz, "Bemerkungen zur Fehlerabschätzung für das Differenzenverfahren bei partiellen Differentialgleichungen,"*Z. Angew. Math. Mech.*, v. 13, 1933, pp. 56-57.**[12]**R. Courant and D. Hilbert,*Methods of mathematical physics. Vol. II: Partial differential equations*, (Vol. II by R. Courant.), Interscience Publishers (a division of John Wiley & Sons), New York-Lon don, 1962. MR**0140802****[13]**E. C. Du Fort and S. P. Frankel,*Stability conditions in the numerical treatment of parabolic differential equations*, Math. Tables and Other Aids to Computation**7**(1953), 135–152. MR**0059077**, https://doi.org/10.1090/S0025-5718-1953-0059077-7**[14]**George E. Forsythe and Wolfgang R. Wasow,*Finite-difference methods for partial differential equations*, Applied Mathematics Series, John Wiley & Sons, Inc., New York-London, 1960. MR**0130124****[15]**S. Gerschgorin, "Fehlerabschätzung für das Differenzenverfahren zur Lösung partieller Differentialgleichungen,"*Z. Angew. Math. Mech.*, v. 10, 1930, pp. 373-382.**[16]**Pentti Laasonen,*On the discretization error of the Dirichlet problem in a plane region with corners*, Ann. Acad. Sci. Fenn. Ser. A I No.**408**(1967), 16. MR**0232009****[17]**T. S. Motzkin and W. Wasow,*On the approximation of linear elliptic differential equations by difference equations with positive coefficients*, J. Math. Physics**31**(1953), 253–259. MR**0052895****[18]**Harvey S. Price,*Monotone and oscillation matrices applied to finite difference approximations*, Math. Comp.**22**(1968), 489–516. MR**0232550**, https://doi.org/10.1090/S0025-5718-1968-0232550-5**[19]**Vidar Thomée,*Elliptic difference operators and Dirichlet’s problem*, Contributions to Differential Equations**3**(1964), 301–324. MR**0163444**

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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