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THE SELF-ADJOINT 5-POINT AND 7-POINT DIFFERENCE OPERATORS, THE ASSOCIATED DIRICHLET PROBLEMS, DARBOUX TRANSFORMATIONS AND LELIEUVRE FORMULAE

Published online by Cambridge University Press:  14 July 2005

M. NIESZPORSKI  and
P.M. SANTINI
M. NIESZPORSKI
Affiliation:
Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski ul. Hoża 74, 00-682 Warszawa, Poland, and Instytut Fizyki Teoretycznej, Uniwersytet w Białymstoku, ul. Lipowa 41, 15-424 Białystok, Poland e-mail: maciejun@fuw.edu.pl
P.M. SANTINI
Affiliation:
Dipartimento di Fisica, Università di Roma “La Sapienza” and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Piazz.le Aldo Moro 2, I-00185 Roma, Italy e-mail: paolo.santini@roma1.infn.it
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Abstract

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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.

Keywords

35J55
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 47 , Issue A , June 2005 , pp. 133 - 147
Copyright
2005 Glasgow Mathematical Journal Trust