New methods are developed for convergence error estimation and convergence acceleration in iteratively solved problems. The convergence error estimation method is based on the eigenvalue analysis of linear systems, but it can also be used for nonlinear systems. Newton's method is used to estimate the magnitude and the phase angle of eigenvalues. The convergence of iterative method is accelerated by subtracting convergence error from the iteratively calculated solutions. The performances of these methods are demonstrated for the Laplace, Euler and Navier-Stokes equations.