On a Principle of Direct Defect Correction Based on A-Posteriori Error Estimates

A combination of an iterative procedure with realistic a-posteriori error estimates allows the approximate solution of functional equations where an error improvement can be achieved which is controlled by a direct defect correction (or: residual improvement). The underlying mathematical theory is presented which is essentially based on the Inverse Function Theorem. As applications, defect corrections via projection methods for linear problems as well as for nonlinear problems are analyzed. For a linear model problem of a singularly perturbed one-dimensional boundary value problem, computational results are presented where one defect correction step is performed using a self-adaptive finite element method.