I am interested in solving \begin{equation} Ax = b \end{equation} for a large sparse linear symmetric positive definite matrix $A$ by Conjugate Gradients method. (These systems usually come as discretization of PDEs.) I would like to know if there is some convergence estimate in a situation where I know that the spectrum of $A$ lies in a union of intervals, i. e. \begin{equation} \sigma(A) \subset \bigcup_{i\in\mathcal{I}} [l_i, l_{i+1}] \end{equation} where $\mathcal{I}$ is a suitable index set. The intervals can degenerate into one point sets.
I know about the usual worst-case estimate based on Chebyshev polynomials that corresponds to locating the spectrum into one interval. I remember reading a paper by Axelsson (et al.?) that dealt with a situation where there are few isolated eigenvalues and the rest is in one interval. I wonder if there are some other similar more general results.
