In this paper, we discuss a new adaptive approach for iterative solution of sparse linear systems arising from partial differential equations (PDEs) with self-adjoint operators. The idea is to use the a posteriori estimated local distribution of the algebraic error in order to steer and guide the solve process in such way that the algebraic error is reduced more efficiently in the consecutive iterations. We first explain the motivation behind the proposed procedure and show that it can be equivalently formulated as constructing a special combination of preconditioner and initial guess for the original system. We present several numerical experiments in order to identify when the adaptive procedure can be of practical use.