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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
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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 $ \vert{D^{(m)}}G(P;Q)\vert \leqq {A_m}/\rho _{P\;Q}^m$ and $ \vert{\tilde D^{(m)}}G(P;Q)\vert \leqq {B_m}d(Q)/\rho _{P\;Q}^{m + 1}$, where $ {D^{(m)}}$ is an mth order difference quotient with respect to the components of P or Q, and $ {\tilde D^{(m)}}$ denotes an mth order difference quotient only with respect to the components of P.


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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

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