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W. Auzinger:
"Defect Correction Methods";
in: "Encyclopedia of Applied and Computational Mathematics", 1; B. Enquist (ed.); Springer Verlag, Wien, Heidelberg, New York, 2015, (invited), ISBN: 978-3-540-70528-4, 323 - 332.

Defect Correction (DeC) methods (also: `deferred correction methods´) are based on a
particular way to estimate local or global errors, especially for differential and integral
equations. The use of simple and stable integration schemes in combination with defect
(residual) evaluation leads to computable error estimates and, in an iterative fashion,
yields improved numerical solutions.
In the first part of this article, the underlying principle is motivated and described
in a general setting, with focus on the main ideas and algorithmic templates.
In the sequel, we consider its application to ordinary differential equations in more
detail. The proper choice of algorithmic components is not always straightforward, and
we discuss some of the relevant issues. There are many versions and application areas
with various pros and cons, for which we give an overview in the final sections. Applications
to partial differential equations (PDEs) in a variational context are briefly
discussed.
In this overview we are not specifying all algorithmic components in detail,
e.g., concerning the required interpolation and quadrature processes. But these are
numerical standard procedures which are easy to understand and to realize. Also, an
exhaustive survey of the available literature on the topic is not provided here.

defect, residual; defect correction, DeC, deferred correction; original problem, OP; neighboring problem, NP; correction scheme, CS; full approximation scheme; FAS; ordinary differential equation, ODE; partial differential equation, PDE; boundary value pr

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