Difference Potentials and their Applications

In this lecture, we introduce the concept of difference potentials with the density from the space of discontinuities or jumps, which extends and generalizes the previous constructions of difference potentials; this new concept is sufficiently universal and at the same time simple. The apparatus of difference potentials constitutes the foundation of the difference potentials method (DPM). Before considering the actual constructions of difference potentials, we discuss some new opportunities that the DPM provides for computations. This brief introductory discussion (as well as the main part of the lecture itself) has a goal of drawing the attention of the scientific computing research community to the DPM and its applications. Although the construction of difference potentials with the density from the space of jumps presents an independent mathematical interest, the subject of this lecture will seem too abstract without discussing the possible applications in the beginning. Moreover, in the ..