Numerical solution of optimization problems for semi-linear elliptic equations with discontinuous coefficients and solutions

Abstract

In this work we consider optimization problems for processes described by semi-linear partial differential equations of elliptic type with discontinuous coefficients and solutions (with imperfect contact matching conditions), with controls involved in the coefficients. Finite difference approximations of optimization problems are constructed. For the numerical implementation of finite optimization problems differentiability and Lipshitz-continuity of the grid functional of the approximating grid problems are proved. An iterative method for solving boundary value problems of contact for PDEs of elliptic type with discontinuous coefficients and solutions is developed and validated. The convergence of the iterative process is investigated. And the convergence rate of iterations (with calculated constants) is estimated.