The self-adjoint 5-point and 7-point difference operators, the associated Dirichlet problems, Darboux transformations and Lelieuvre formulas

2004/02/20 by Maciej Nieszporski, P. Santini, Nieszporski, Maciej +2 · 1 citation
Mathematics · Physics and Astronomy · #Differential Equations and Numerical Methods #Nonlinear Waves and Solitons #Numerical methods for differential equations #nlin.SI

paper · pdf · doi:10.48550/arxiv.nlin/0402038

20 pages, 6 figures, submitted to Glasgow Mathematical Journal Trust for Island II proceedinds

arxiv created 2004/02/20 · arxiv updated 2009/12/01

Abstract

We present some basic properties of two distinguished discretizations of elliptic operators: the self-adjoint 5-point and 7-point schemes on a two dimensional lattice. We first show that they allow to solve Dirichlet boundary value problems; then we present their Darboux transformations. Finally we construct their Lelieuvre formulas and we show that, at the level of the normal vector and in full analogy with their continuous counterparts, the self-adjoint 5-point scheme characterizes a two dimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar), while the self-adjoint 7-point scheme characterizes a generic 2D lattice.

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