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**A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals.**
*(English)*
Zbl 1194.65143

Summary: A new transformation technique is introduced for evaluating the two-dimensional nearly singular integrals, which arise in the solution of Laplace’s equation in three dimensions, using the boundary element method, when the source point is very close to the element of integration. The integrals are evaluated using (in a product fashion) a transformation which has recently been used to evaluate one-dimensional near singular integrals. This sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element. The method is straightforward to implement and, when it is compared with a number of existing techniques for evaluating two-dimensional near singular integrals, it is found that the sinh method is superior to the existing methods considered, both for potential integrals across the full range of \(b\) values considered \((0<b\leq10)\), and for flux integrals where \(b>0.01\). For smaller values of \(b\), the use of the \(L_1^{-1/5}\) method is recommended for flux integrals.

### MSC:

65N38 | Boundary element methods for boundary value problems involving PDEs |

### Keywords:

nonlinear co-ordinate transformation; boundary element method; nearly singular integrals; numerical integration; sinh function
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\textit{B. M. Johnston} et al., Int. J. Numer. Methods Eng. 69, No. 7, 1460--1479 (2007; Zbl 1194.65143)

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