Iterative methods for operator equations with adjoint factorization
First, the author gives a brief review of iterative methods for solving linear systems \(Au=f\) with a degenerate matrix \(A\) and \(u,f\in \mathbb{R}^n\). Next, he considers the following operator equations: \[ R^*Ru=f,\quad RR^*u=g, \tag{1} \] where \(R\: H\to H^*\) and the adjoint \(R^*\: H^*\to H\) are linear operators in finite-dimensional Euclidean spaces \(H\) and \(H^*\). Systems (1) are also degenerate. The former system in (1) can be reduced to the system \[ R^*\xi=f,\quad Ru=\xi \] which, in this article, plays the role of a basic system for constructing iterative numerical methods. An analysis of these methods is presented. It is shown that the results are applicable to the numerical solution of boundary value problems for second-order elliptic systems in the divergent form.
scientific article; zbMATH DE number 1498468