¹National Sun Yat-sen University, Kaohsiung, Taiwan

²I-Shou University, Kaohsiung, Taiwan

³Zhejiang University of Technology, Hangzhou, China

⁴Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, Russian Federation

⁵Chung Hua University, Hsin-Chu, Taiwan, mglee@chu.edu.tw

Dual techniques have been used in many engineering papers to deal with singularity and ill-conditioning of the boundary element method (BEM). In the first part of the two-part article, our efforts were focused on studying the theoretical aspects of this problem, including the analysis of errors and the study of stability. We provided the theoretical analysis for Laplace equation in elliptic domains with elliptic holes. To bypass the degenerate scales of Dirichlet problems, the second and the first kinds of the null field methods (NFM) are used for the exterior and the interior boundaries, simultaneously. This approach is called the dual null field method (DNFM).

This paper is the second part of the study. Numerical results for degenerate models of an elliptic domain with one elliptic hole at 𝑎 + 𝑏 = 2 are carried out to verify the theoretical analysis obtained. The collocation Trefftz method (CTM) is also designed for comparisons. Both the DNFM and the CTM can provide excellent numerical performances. The convergence rates are the same but the stability of CTM is excellent and can achieve the constant condition numbers, Cond = 𝑂(1).

Zi-Cai Li, Prof., National Sun Yat-sen University, Kaohsiung, 80424, Taiwan, zicili1@gmail.com

Hung-Tsai Huang, Prof., I-Shou University, Kaohsiung, 84001, Taiwan, huanght@isu.edu.tw

Li-Ping Zhang, Prof., Zhejiang University of Technology, Hangzhou, 310023, China, zhanglp@zjut.edu.cn

Anna A. Lempert, Cand. Sci. (Phys.Math.), Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664033, Russian Federation, lempert@icc.ru

Ming-Gong Lee, Prof., Chung Hua University, Hsin-Chu, 30012, Taiwan, mglee@chu.edu.tw

1. Chen J.T., Han H., Kuo S.R., Kao S.K. Regularization for ill-conditioned systems of integral equation of first kind with logarithmic kernel. Inverse Problems in Science and Engineering, 2014, vol. 22, no. 7, pp. 1176-1195. https://doi.org/10.1080/17415977.2013.856900
2. Chen J.T., Hong H.K., Review of dual finite element methods with emphasis on hypersingular integrals and divergent series. Appl. Mech. Rev., 1999, vol. 52, no. 1, pp. 17-33. https://doi.org/10.1115/1.3098922
3. Chen J.T., Lee Y.T., Chang Y.L., Jian J. A self-regularized approach for rank-deficient systems in the BEM of 2D Laplace problems. Inverse Problems in Science and Engineering, 2017, vol. 25, no. 1, pp. 89-113. https://doi.org/10.1080/17415977.2016.1138948
4. Kuo S.R., Kao S.K., Huang Y.L., Chen J.T. Revisit of the degenerate scale for an infinite plane problem containing two circular holes using conformal mapping. Applied Mathematics Letters, 2019, vol. 92, pp. 99-107. https://doi.org/10.1016/j.aml.2018.11.023
5. Lee M.G., Li Z.C., Zhang L.P., Huang H.T., Chiang J.Y. Algorithm singularity of the null-field method for Dirichlet problems of Laplace’s equation in annular and circular domains. Eng. Anal. Bound. Elem., 2014, vol. 41, pp. 160-172. https://doi.org/10.1016/j.enganabound.2014.01.013
6. Li Z.C., Chiang J.Y., Huang H.T., Lee M.G. The new interior field method for Laplace’s equation on circular domains with circular holes. Eng. Anal. Bound. Elem., 2016, vol. 67, pp. 173-185. https://doi.org/10.1016/j.enganabound.2016.03.006
7. Li Z.C., Huang H.T., Wei Y., Cheng A.H.-D. Effective Condition Number for Numerical Partial Differential Equations. Beijing, Science Press, 2014, 286 p.
8. Li Z.C., Lu T.T., Hu H.Y., Cheng A.H.-D. Trefftz and Collocation Methods. Southampton, Boston, WIT press, 2008, 432 p.
9. Lee M.G., Zhang L.P., Li Z.C., Kazakov A.L. Dual Null Field Method for Dirichlet Problems of Laplace’s Equation in Circular Domains with Circular Holes. Siberian Electronic Mathematical Reports, 2021, vol. 19(1), pp. 393-422. https://doi.org/10.33048/semi.2021.18.028
10. Li Z.C., Huang H.T., Zhang L.P., Lempert A.A., Lee M.G. Analysis of Dual Null Field Methods for Dirichlet Problems of Laplace’s Equation in Circular Domains with Circular Holes: Bypassing the degenerate scales. The Bulletin of Irkutsk State University. Series Mathematics, submitted, June 2021.
11. Li Z.C., Zhang L.P., Lee M.G., Interior field methods for Neumann problems of Laplace’s equation in elliptic domains, Comparisons with degenerate scales. Eng. Anal. Bound. Elem, 2016, vol. 71, pp. 190-202. https://doi.org/10.1016/j.enganabound.2016.07.003
12. Li Z.C., Zhang L.P., Wei Y., Lee M.G., Chiang J.Y. Boundary methods for Dirichlet problems of Laplace’s equation in elliptic domains with elliptic holes. Eng. Anal. Bound. Elem., 2015, vol. 61, pp. 92-103. https://doi.org/10.1016/j.enganabound.2015.07.001
13. Morse P.M., Feshbach H. Methods of Theoretical Physics. New York, McGraw-Hill, Inc., 1953, 997 p.
14. Portela A., Aliabadi M.H., Rooke D.P. The dual boundary element method: effective implementation for crack problems. Int. J. Numer. Methods Engrg., 1992, vol. 33, pp. 1269–1287. https://doi.org/10.1002/nme.1620330611
15. Krni´c L. Types of Bases in the Algebra of Logic. Glasnik Matematicko-Fizicki i Astronomski, series 2, 1965, vol. 20, pp. 23–32.
16. Lau D., Miyakawa M. Classification and enumerations of bases in 𝑃_𝑘(2). Asian-European Journal of Mathematics, 2008, vol. 1, no. 2, pp. 255-282.
17. Miyakawa M., Rosenberg I., Stojmenovi´c I. Classification of Three-valued logical functions preserving 0. Discrete Applied Mathematics, 1990, vol. 28, pp. 231-249. https://doi.org/10.1016/0166-218X(90)90005-W