In order to solve a system of the second kind \(y=g+Ky\) in a Banach space setting numerically, a convenient approximation \(\bar K\) can be chosen to replace \((I-K)^{-1}\) by \((I-\bar K)^{-1}\). The acceleration refinement technique employs the iterates of \((K-\bar K)\) to develop \((I- K)^{-1}\) in the form of infinite series. Successive truncations of the series define a sequence of higher-order approximations to the inverse of \((I-K)\). An error estimate for the k-th iterate is given.
Various known techniques are presented both as refinement procedures and as residual correction methods, as e.g. projection methods or, in the case of integral equations, quadrature methods. The technique is applied to integral equations, in particular to an ill-conditioned weakly singular system and to Love’s equation to show the higher convergence rates of the refinement schemes. The author also considers the question of complexity in connection with optimal orders and numerical run time.