On an iterative process for the grid conjugation problem with iterations on the boundary of the solution discontinuity

Abstract

An iterative process for the grid problem of conjugation with iterations on the boundary of the discontinuity of the solution is considered. Similar grid problem arises in difference approximation of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions. The study of iterative processes for the states of such problems is of independent interest for theory and practice. The paper shows that the numerical solution of boundary problems of this type can be efficiently implemented using iterations on the inner boundary of the grid solution discontinuity in combination with other iterative methods for nonlinearities separately in each of the grid subregions. It can be noted that problems for states of controlled processes described by equations of mathematical physics with discontinuous coefficients and solutions arise in mathematical modeling and optimization of heat transfer, diffusion, filtration, elasticity theory, etc. The proposed iterative process reduces the solution of the initial grid boundary problem for a state with a discontinuous solution to a solution of two special boundary problems in two grid subdomains at every fixed iteration. The convergence of the iteration process in the Sobolev grid norms to the unique solution of the grid problem for each initial approximation is proved.