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W. Auzinger, H. Hofstätter, W. Kreuzer, E. Weinmüller:
"Modified defect correction algorithms for ODEs. Part I: General theory";
Report for ANUM Preprint No. 1/03; 2003.

The well-known method of Iterated Defect Correction (IDeC) is based on the following idea: Compute a simple, basic approximation and form its defect w.r.t. the given ODE via a piecewise interpolant. This defect is used to define an auxiliary, neighboring problem whose exact solution is known. Solving the neighboring problem with the basic discretization scheme yields a global error estimate. This can be used to construct an improved approximation, and the procedure can be iterated. The fixed point of such an iterative process corresponds to a certain collocating solution.

We present a variety of modifications to this algorithm. Some of these have been proposed only recently, and together they form a family of iterative techniques, each with its particular advantages. These modifications are based on techniques like defect quadrature (IQDeC), defect interpolation (IPDeC), and combinations thereof. We investigate the convergence on (locally) equidistant and nonequidistant grids and show how superconvergent approximations can be obtained. Numerical examples illustrate our considerations.

The application to stiff initial value problems will be discussed in Part II of this paper.

http://www.math.tuwien.ac.at/~winfried/papers/anumpp0103.pdf