We consider a self-adjoint second-order elliptic boundary value problem with variable zeroth order ("reaction") coefficient and its finite element discretization. In this project, we study the mesh-independent superlinear convergence of the preconditioned conjugate gradient method (CGM) for this type of problem. Our goal is to find an eigenvalue-based estimation of the rate of the superlinear convergence when the reaction coefficient of the elliptic boundary value problem belongs to a general Sobolev space. This work extends previous results where the coefficient was assumed to be continuous.