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Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric three-dimensional problems are considered. This paper focuses on the numerical integration of the functions on the standard domain [-1, 1], with a logarithmic singularity at the centre. The substitution x = tp, where p (≥ 3) is an odd integer is given particular attention, as this returns a regular integral and the domain unchanged. Gauss-Legendre quadrature rules are applied to the transformed integrals for a number of values of p. It is shown that a high value for p typically gives more accurate results.

american journal of computational mathematics

Quadrature Rules for Functions with a Mid-Point Logarithmic Singularity in the Boundary Element Method Based on the &amp;lt;i&amp;gt;x = t&amp;lt;sup&amp;gt;p&amp;lt;/sup&amp;gt;&amp;lt;/i&amp;gt; Substitution