We discuss new variants of the acceleration technique
known as Iterated Defect Correction (IDeC).
In the classical version of the IDeC procedure, the defect is chosen as
the pointwise residual with
respect to the given ODE.
Here, we show that the classical method can be improved if the defect is
suitably modified. As an example we consider the backward Euler method
for boundary value problems in ODEs. For the defect, we choose a locally
integrated form which amounts to the residual with respect to certain
Runge-Kutta schemes associated with a particular set of quadrature rules.
It is shown that this results in a possible improvement in the
maximal attainable convergence order. Moreover, the grids for this variant
do not have to be chosen as piecewise equidistant, as is the case
for the classical version. It is further demonstrated that the
attainable orders are
determined by the fixed points of the procedure. Finally, an alternative
implementation using the box scheme as the basic method is shown to
approach a high-order solution very efficiently.
In particular, the application of these procedures to singular
boundary value problems of the form