Hierarchical quadrature for multidimensional singular integrals

Abstract

We introduce a method for the evaluation of singular integrals arising in the discretisation of integral equations. The method is based on the idea of repeated subdivision of domains. The integrals defined on these subdomains are classified. Each class can be expressed as a sum of regular integrals and representatives of other classes. A system of equations describes the relations between the classes. Therefore the approximate value of the singular integral only depends on the accuracy of the calculation of regular integrals.

Hierarchical quadrature for multidimensional singular integrals – Part II

“…The methods presented in [8] and outlined in the Subsections 3. with n = dim(D x ), m = dim(D y ) and d as given before.…”

“…This section gives a short summarization of [8], collecting only the basic facts and ideas of hierarchical quadrature.…”

“…But, as mentioned in Subsection 3.2 and shown in [8], the subdivision of simplices becomes much more difficult than in the case of hypercubes. Therefore, we only show that the ideas given before can be transferred to the case of simplices.…”

“…4. The given integral is not solvable by the method mentioned in Subsection 3.2 and given [8], as α = −(n + m) + d = −2. The solution can be obtained using the method, based on the modified Hadamard partie finie integrals, given above.…”

“…In [8] we adapt the method given in [2] to arbitrary dimensions n for cubical domains and to n 3 for simplicial ones. But it cannot be applied to certain hypersingular integrals.…”