On integrating vertex singularities using extrapolation

Abstract

Abstract.A new approach to the integration of vertex singularities is described. This approach is based on a non-uniform subdivision of the region of integration and the technique fits well to the subdivision strategy used in many adaptive algorithms. A nice feature with this approach is that it can be used in any dimension and on any region of integration which can be subdivided into subregions of the same form. The strategy can be applied both to vertex singularities and internal point singularities. In the latter case this can be done without an initial subdivision of the region in order to put the singular point in a vertex. It turns out that the technique has excellent numerical stability properties.Subject Classification: AMS(MOS) 65D30, 65B05.

Generalized Duffy transformation for integrating vertex singularities

“…Extrapolation techniques [20][21][22][23] construct more accurate integration formulae based on asymptotic error expansions of standard quadratures, whereas in adaptive subdivision scheme [24,25] the integration domain is subdivided into uniform/nonuniform subdomains and well-known integration rules are used over the subdomains. Klees [26] shows that for weakly singular integrands, extrapolation and adaptive subdivision techniques behave poorly in terms of both accuracy and efficiency.…”

Integrating Singularities using Non-uniform Subdivision and Extrapolation

“…Example 4) is the only one that can be handled by the technique described in [7] directly. However, all of these problems can be dealt with by that technique after an appropriate transformation.…”

“…Just as in [7] we will base this on a non-uniform subdivision of the region of integration combined with extrapolation. The non-uniform aspect resembles the basic idea with the product rules: to treat the problem differently in the different directions of integration.…”

“…This approach can be used on problems having vertex singularities associated with homogeneous functions. The idea presented in [7] is generally applicable to any dimension and any region which can be subdivided into subregions of the same form, e. g. hypercubes, simplices, etc.. Furthermore, the technique can be applied to internal point singularities as well.…”

“…In a recent paper Espelid (1992) [7] describes a new idea applying extrapolation on a sequence of estimates produced through a non-uniform subdivision of the initial region. This approach can be used on problems having vertex singularities associated with homogeneous functions.…”