On the Definition of Surface Potentials for Finite-Difference Operators

Abstract

For a class of linear constant-coefficient finite-difference operators of the second order, we introduce the concepts similar to those of conventional single- and double-layer potentials for differential operators. The discrete potentials are defined completely independently of any notion related to the approximation of the continuous potentials on the grid. We rather use an approach based on differentiating, and then inverting the differentiation of, a function with surface discontinuity of a particular kind, which is the most general way of introducing surface potentials in the theory of distributions. The resulting finite-difference “surface” potentials appear to be solutions of the corresponding system of linear algebraic equations driven by special source terms. The properties of the discrete potentials in many respects resemble those of the corresponding continuous potentials. Primarily, this pertains to the possibility of representing a given solution to the homogeneous equation on the domain as a variety of surface potentials with the density defined on the domain's boundary. At the same time, the discrete surface potentials can be interpreted as one specific realization of the generalized potentials of Calderon's type, and consequently, their approximation properties can be studied independently in the framework of the difference potentials method by Ryaben'kii. The motivation for introducing and analyzing the discrete surface potentials was provided by the problems of active shielding and control of sound, in which the aforementioned source terms that drive the potentials are interpreted as the acoustic control sources that cancel out the unwanted noise on a predetermined region of interest.