Defect Correction is an efficient method to obtain an a-posteriori
error estimate for the discretization error. It was originally
proposed by Zadunaisky to estimate the global error of Runge-Kutta
schemes, cf. [1]. The idea later became a basis for the
acceleration technique called Iterated Defect Correction (IDeC)
used to iteratively improve the numerical solution, see
[2] and [3].
Numerical experiments show that classical variants of the Defect
Correction work well on meshes which are at least locally
equidistant. Otherwise, the convergence of the procedure shows
order reductions. In order to overcome this difficulty, certain
algorithmic modification are necessary.
We discuss a new, carefully designed modification of the error
estimation procedure for the global error of collocation
schemes applied to solve singular boundary value problems with a
singularity of the first kind,
where B0, B1 and β are suitably chosen to yield a
well-posed problem. This global error estimate is the basis for a
grid selection routine in which the grid is modified with the aim
to equidistribute the global error. Most importantly, we observe
that the grid is refined in a way reflecting only the smoothness
of the solution. The above strategies have been implemented in a
MATLAB solver, SBVP 1.0, freely available from
http://www.math.tuwien.ac.at/~ewa,
cf. [4].
Finally, we present the application of another variant of the IDeC
procedure to stiff ODEs.