Defect correction methods, classic and new

Анотація

Defect correction methods are based on the idea of measuring the quality of an approximate solution to an operator equation by forming the defect, or residual, with respect to the given problem. By an appropriate backsolving procedure, an error estimate is obtained. This process can also be continued in an iterative fashion.  One purpose of this overview is the further dissemination of the underlying concepts. Therefore, we first give a general and consistent review on various types defect correction methods, and its application in the context of discretization schemes for differential equations. After describing the general algorithmic templates we discuss some specific techniques used in the solution of ordinary differential equations. Moreover, new results about the application to implicit problems are presented.