Discrete Green's functions

Authors:
G. T. McAllister and E. F. Sabotka

Journal:
Math. Comp. **27** (1973), 59-80

MSC:
Primary 65P05

DOI:
https://doi.org/10.1090/S0025-5718-1973-0341909-9

MathSciNet review:
0341909

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the discrete Green's function over a discrete *h*-convex region of the plane; i.e., for for . Assume that and are Hölder continuous over and positive. We show that and , where is an *m*th order difference quotient with respect to the components of *P* or *Q*, and denotes an *m*th order difference quotient only with respect to the components of *P*.

**[1]**J. H. Bramble and V. Thomée,*Pointwise bounds for discrete Green’s functions*, SIAM J. Numer. Anal.**6**(1969), 583–590. MR**0263265**, https://doi.org/10.1137/0706053**[2]**Daniel F. De Santo and Herbert B. Keller,*Numerical studies of transition from laminar to turbulent flow over a flat plate*, J. Soc. Indust. Appl. Math.**10**(1962), 569–595. MR**0157568****[3]**Charles R. Deeter and George Springer,*Discrete harmonic kernels*, J. Math. Mech.**14**(1965), 413–438. MR**0201085****[4]**R. Sherman Lehman,*Developments at an analytic corner of solutions of elliptic partial differential equations*, J. Math. Mech.**8**(1959), 727–760. MR**0105552****[5]**G. T. McAllister,*A priori bounds on difference quotients of solutions to some linear uniformly elliptic difference equations*, Numer. Math.**11**(1968), 13–37. MR**0226879**, https://doi.org/10.1007/BF02165468**[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.**24**(1968), 582–607. MR**0234647**, https://doi.org/10.1016/0022-247X(68)90012-7**[7]**W. H. McCrea and F. J. W. Whipple,*Random paths in two and three dimensions*, Proc. Roy. Soc. Edinburgh**60**(1940), 281–298. MR**0002733****[8]**Moshe Mangad,*Bounds for the two-dimensional discrete harmonic Green’s function*, Math. Comp.**20**(1966), 60–67. MR**0198701**, https://doi.org/10.1090/S0025-5718-1966-0198701-4**[9]**Joachim Nitsche and Johannes C. C. Nitsche,*Error estimates for the numerical solution of elliptic differential equations*, Arch. Rational Mech. Anal.**5**(1960), 293–306 (1960). MR**0117890**, https://doi.org/10.1007/BF00252911**[10]**R. Bruce Simpson,*A fundamental solution for a biharmonic finite-difference operator*, Math. Comp.**21**(1967), 321–339. MR**0226880**, https://doi.org/10.1090/S0025-5718-1967-0226880-X**[11]**Friedrich Stummel,*Elliptische Differenzenoperatoren unter Dirichletranbedingungen*, Math. Z.**97**(1967), 169–211 (German). MR**0224302**, https://doi.org/10.1007/BF01111697**[12]**Kjell-Ove Widman,*Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations*, Math. Scand.**21**(1967), 17–37 (1968). MR**0239264**, https://doi.org/10.7146/math.scand.a-10841

Retrieve articles in *Mathematics of Computation*
with MSC:
65P05

Retrieve articles in all journals with MSC: 65P05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0341909-9

Keywords:
Elliptic difference equations,
finite differences

Article copyright:
© Copyright 1973
American Mathematical Society