The paper deals mainly with numerical methods for solving two-dimensional elliptic equations with piecewise constant coefficients. The simple model case of a single interface line is considered in detail on the base of the fairly known and standard approach with the interface conditions being treated in a variational form. The spaces and norms of the type indicated in the paper were used for domain decomposition methods (or composition methods as was suggested by V. Agoshkov and V. Lebedev) by many even in the eighties.
The authors pay special attention to gradient-based iterations which are easy to implement (especially for complicated problems like Navier-Stokes system which is mentioned by the authors), but it should be noted that other effective iterative stratieges are also well known for the reduced problems on interfaces [see, e.g. E. G. D’yakonov, Optimization in solving elliptic problems. Boca Raton: CRC Press (1996; Zbl 0852.65087) and references therein].