## Abstract

Although the boundary element method (BEM) has been developed over forty years, the single-layer potential approach is incomplete for solving not only the interior 2D problem in case of a degenerate scale but also the exterior problem with bounded potential at infinity for any scale. The indirect boundary element method (IBEM) is revisited to examine the uniqueness of the solution by using the necessary and sufficient boundary integral equation (BIE). For the necessary and sufficient BIE, a free constant and an extra constraint are simultaneously introduced into the conventional IBEM. The reason why a free constant and an extra constraint are both required is clearly explained by using the degenerate kernel. In order to complete the range of the IBEM lacking a constant term in the case of a degenerate scale, we provide a complete base with a constant. On the other hand, the formulation of the IBEM does not contain a constant field in the degenerate kernel expansion for the exterior problem. To satisfy the bounded potential at infinity, the integration of boundary density is enforced to be zero. Besides, sources can be distributed on either the real boundary or the auxiliary (artificial) boundary in this IBEM. The enriched IBEM is not only free of the degenerate-scale problem for the interior problem but also satisfies the bounded potential at infinity for the exterior problem. Finally, three examples, a circular domain, an infinite domain with two circular holes and an eccentric annulus were demonstrated to illustrate the validity and the effectiveness of the necessary and sufficient BIE.

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## Acknowledgments

The authors wish to thank the support from the Ministry of Science Technology of Taiwan under contract NSC 102-2221-E-019-034 and MOST 103-2221-E-012-MY3. We also appreciate Prof. Houde Han and Dr. Y. T. Lee for useful discussions.

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Chen, JT., Chang, YL., Kao, SK. *et al.* Revisit of the Indirect Boundary Element Method: Necessary and Sufficient Formulation.
*J Sci Comput* **65**, 467–485 (2015). https://doi.org/10.1007/s10915-014-9974-2

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DOI: https://doi.org/10.1007/s10915-014-9974-2