The continuation approach for singular and near-singular integration

Abstract

This work provides an overview of the continuation approach, a unified framework for understanding and directly computing a large class of singular and near-singular integrals. Singular surface integrals are viewed merely as limits, of ‘continuations’, of non-singular (but perhaps near-singular) ones. The analysis presents a clear and intuitive picture of the behaviour of these integrals, which leads to general formulae for singular and near-singular integral evaluations, the former being a special case of the latter. We also obtain necessary and sufficient boundedness conditions for the singular integrals. When these conditions are met, the continuation singular integral encompasses the classical cases of a Cauchy principal value, jump terms, and Hadamard finite part. The analysis exploits the functional homogeneity of many Green's functions, and it covers integrals on smooth (flat and curved) as well as non-smooth surfaces. Some numerical integration examples are presented.