Numerical calculation of regular and singular integrals in boundary integral equations using Clenshaw–Curtis quadrature rules

Abstract

The efficient and accurate calculation of integrals, especially singular and nearly singular integrals, poses a great challenge in boundary integral equation (BIE) methods. In this paper, the Clenshaw–Curtis integration technique is proposed to calculate both regular and singular integrals in the BIE in a unified way. Taking the meshless discretization of two-dimensional Helmholtz BIE by linear integration cell as an example, four types of Clenshaw–Curtis quadrature rules are established for regular, weakly singular, hypersingular, and nearly hypersingular integrals. Explicit expressions of integration weights in all quadrature rules are derived, and the degree of precision of these quadrature rules is analyzed theoretically. Integration points in all quadrature rules are the same, which is very useful for meshless analysis of BIEs. The values of integration points and weights for regular, weakly singular and hypersingular integrals are tabulated in the interval [−1, 1], which can be directly used to facilitate the practical application of BIE methods. Numerical results are provided to verify the effectiveness of these Clenshaw–Curtis quadrature rules.