Geodesic
THE SELF-ADJOINT 5-POINT AND 7-POINT DIFFERENCE OPERATORS, THE ASSOCIATED DIRICHLET PROBLEMS, DARBOUX TRANSFORMATIONS AND LELIEUVRE FORMULAE
Glasgow mathematical journal, Tome 47 (2005) no. A, pp. 133-147

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DOI

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 us to solve Dirichlet boundary value problems; then we present their Moutard transformations (distinguished examples of transformation of Darboux type in two dimensions). Finally we construct their Lelieuvre formulae 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.
DOI : 10.1017/S0017089505002351
Mots-clés : 35J55 
NIESZPORSKI, M.; SANTINI, P.M. THE SELF-ADJOINT 5-POINT AND 7-POINT DIFFERENCE OPERATORS, THE ASSOCIATED DIRICHLET PROBLEMS, DARBOUX TRANSFORMATIONS AND LELIEUVRE FORMULAE. Glasgow mathematical journal, Tome 47 (2005) no. A, pp. 133-147. doi: 10.1017/S0017089505002351
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     author = {NIESZPORSKI, M. and SANTINI, P.M.},
     title = {THE {SELF-ADJOINT} {5-POINT} {AND} {7-POINT} {DIFFERENCE} {OPERATORS,} {THE} {ASSOCIATED} {DIRICHLET} {PROBLEMS,} {DARBOUX} {TRANSFORMATIONS} {AND} {LELIEUVRE} {FORMULAE}},
     journal = {Glasgow mathematical journal},
     pages = {133--147},
     year = {2005},
     volume = {47},
     number = {A},
     doi = {10.1017/S0017089505002351},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089505002351/}
}
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