The construction of efficient algorithms for problems of mathematical physics in domains with curvilinear boundary meets serious difficulties. The inconsistency of the spatial mesh and the presence of irregular mesh point break the uniformity of difference schemes, aggravate their properties, and hamper the investigation.
An original discretization algorithm for connected \(n\)-dimensional domains was suggested by V. N. Abrashin and the first author [Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 1991, No. 6, 5-11 (1991; Zbl 0749.65070)]. A distinctive feature of this approach is that a uniform mesh is considered on each straight line of the domain’s division and efficient algorithms for the problems considered in such domains are realized on these meshes. On the basis of this idea we construct difference schemes for multidimensional elliptic equations with Dirichlet boundary conditions.
The iterative procedures of the multicomponent method of variable directions for elliptic equations in a rectangular region were studied in a paper of V. N. Abrashin and N. Zhadaeva [Differ. Uravn. Primen. 43, 22-30 (1988; Zbl 0658.65087)] by using spectral analysis. It is hardly possible to apply spectral analysis to the difference algorithms proposed here, and so they are studied (including stability and convergence) by the method of energy estimates.