## Discrete Green’s functions

HTML articles powered by AMS MathViewer

- by G. T. McAllister and E. F. Sabotka PDF
- Math. Comp.
**27**(1973), 59-80 Request permission

## Abstract:

Let $G(P;Q)$ be the discrete Green’s function over a discrete*h*-convex region $\Omega$ of the plane; i.e., $a(P){G_{x\bar x}}(P;Q) + c(P){G_{y\bar y}}(P;Q) = - \delta (P;Q)/{h^2}$ for $P \in {\Omega _h},G(P;Q) = 0$ for $P \in \partial {\Omega _h}$. Assume that $a(P)$ and $c(P)$ are Hölder continuous over $\Omega$ and positive. We show that $|{D^{(m)}}G(P;Q)| \leqq {A_m}/\rho _{P\;Q}^m$ and $|{\tilde D^{(m)}}G(P;Q)| \leqq {B_m}d(Q)/\rho _{P\;Q}^{m + 1}$, where ${D^{(m)}}$ is an

*m*th order difference quotient with respect to the components of

*P*or

*Q*, and ${\tilde D^{(m)}}$ denotes an

*m*th order difference quotient only with respect to the components of

*P*.

## References

- J. H. Bramble and V. Thomée,
*Pointwise bounds for discrete Green’s functions*, SIAM J. Numer. Anal.**6**(1969), 583–590. MR**263265**, DOI 10.1137/0706053 - 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**157568**, DOI 10.1137/0110044 - Charles R. Deeter and George Springer,
*Discrete harmonic kernels*, J. Math. Mech.**14**(1965), 413–438. MR**0201085** - R. Sherman Lehman,
*Developments at an analytic corner of solutions of elliptic partial differential equations*, J. Math. Mech.**8**(1959), 727–760. MR**0105552**, DOI 10.1512/iumj.1959.8.58047 - 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**226879**, DOI 10.1007/BF02165468 - 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**234647**, DOI 10.1016/0022-247X(68)90012-7 - W. H. McCrea and F. J. W. Whipple,
*Random paths in two and three dimensions*, Proc. Roy. Soc. Edinburgh**60**(1940), 281–298. MR**2733**, DOI 10.1017/S0370164600020265 - Moshe Mangad,
*Bounds for the two-dimensional discrete harmonic Green’s function*, Math. Comp.**20**(1966), 60–67. MR**198701**, DOI 10.1090/S0025-5718-1966-0198701-4 - 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**117890**, DOI 10.1007/BF00252911 - R. Bruce Simpson,
*A fundamental solution for a biharmonic finite-difference operator*, Math. Comp.**21**(1967), 321–339. MR**226880**, DOI 10.1090/S0025-5718-1967-0226880-X - Friedrich Stummel,
*Elliptische Differenzenoperatoren unter Dirichletranbedingungen*, Math. Z.**97**(1967), 169–211 (German). MR**224302**, DOI 10.1007/BF01111697 - 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**239264**, DOI 10.7146/math.scand.a-10841

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Math. Comp.
**27**(1973), 59-80 - MSC: Primary 65P05
- DOI: https://doi.org/10.1090/S0025-5718-1973-0341909-9
- MathSciNet review: 0341909