Discrete Green's functions
Authors:
G. T. McAllister and E. F. Sabotka
Journal:
Math. Comp. 27 (1973), 5980
MSC:
Primary 65P05
MathSciNet review:
0341909
Fulltext PDF Free Access
Abstract 
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Abstract: Let be the discrete Green's function over a discrete hconvex region of the plane; i.e., for for . Assume that and are Hölder continuous over and positive. We show that and , where is an mth order difference quotient with respect to the components of P or Q, and denotes an mth order difference quotient only with respect to the components of P.
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 [1]
 J. H. Bramble & V. Thomée, "Pointwise bounds for discrete Green's functions," SIAM J. Numer. Anal., v. 6, 1969, pp. 583590. MR 41 #7870. MR 0263265 (41:7870)
 [2]
 D. F. DeSanto & H. B. Keller, "Numerical studies of transition from laminar to turbulent flow over a flat plate," J. Soc. Indust. Appl. Math., v. 10, 1962, pp. 569595. MR 28 #800. MR 0157568 (28:800)
 [3]
 C. R. Deeter & G. Springer, "Discrete harmonic kernels," J. Math. Mech., v. 14, 1965, pp. 413438. MR 34 #970. MR 0201085 (34:970)
 [4]
 R. Sherman Lehman, "Developments at an analytic corner of solutions of elliptic partial differential equations," J. Math. Mech., v. 8, 1959, pp. 727760. MR 21 #4291. MR 0105552 (21:4291)
 [5]
 G. T. McAllister, "A priori bounds on difference quotients of solutions to some linear uniformly elliptic difference equations," Numer. Math., v. 11, 1968, pp. 1337. MR 37 #2465. MR 0226879 (37:2465)
 [6]
 G. T. McAllister, "An application of a priori bounds on difference quotients to a constructive solution of mildly quasilinear Dirichlet problems," J. Math. Anal. Appl., v. 24, 1968, pp. 582607. MR 38 #2963. MR 0234647 (38:2963)
 [7]
 W. H. McCrea & F. J. W. Whipple, "Random paths in two and three dimensions," Proc. Roy. Soc. Edinburgh, v. 60, 1940, pp. 281298. MR 2, 107. MR 0002733 (2:107f)
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 Moshe Mangad, "Bounds for the twodimensional discrete harmonic Green's function," Math. Comp., v. 20, 1966, pp. 6067. MR 33 #6856. MR 0198701 (33:6856)
 [9]
 J. Nitsche & J. C. C. Nitsche, "Error estimates for the numerical solution of elliptic differential equations," Arch. Rational Mech. Anal., v. 5, 1960, pp. 293306. MR 22 #8664. MR 0117890 (22:8664)
 [10]
 R. B. Simpson, "A fundamental solution for a biharmonic finitedifference operator," Math. Comp., v. 21, 1967, pp. 321339. MR 37 #2466. MR 0226880 (37:2466)
 [11]
 F. Stummel, "Elliptische Differenzenoperatoren unter Dirichlet Randbedingungen," Math. Z., v. 97, 1967, pp. 169211. MR 36 #7346. MR 0224302 (36:7346)
 [12]
 KjellOve Widman, "Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations," Math. Scand., v. 21, 1967, pp. 1737. MR 39 #621. MR 0239264 (39:621)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197303419099
PII:
S 00255718(1973)03419099
Keywords:
Elliptic difference equations,
finite differences
Article copyright:
© Copyright 1973
American Mathematical Society
