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DOI: https://doi.org/10.33048/semi.2021.18.028

Abstract: The dual techniques have been widely used in many engineering papers, to deal with singularity and ill-conditioning of the boundary element method (BEM). In this paper, we consider Laplace's equation with circular domains with one circular hole. The explicit algebraic equations of the first and second kinds of the null field method (NFM) are provided for applications. Traditionally, the first and the second kinds of the NFM are used for the Dirichlet and the Neumann problems, respectively. To bypass the degenerate scales of Dirichlet problems, however, the second and the first kinds of the NFM are used for the exterior and the interior boundaries, simultaneously, called the dual NFM (DNFM) in this paper. The excellent stability and the optimal convergence rates are explored in this paper. By using the simple Gaussian elimination or the iteration methods, numerical solutions can be easily obtained. Recently, the study on degenerate scales is active, many removal techniques are proposed, where the advanced solution methods may be needed, such as the truncated singular value decomposition (TSVD) and the overdetermined systems. In contrast, the solution methods of the DNFM in this paper are much simpler, with a little risk of the algorithm singularity from degenerate scales.

Keywords: Laplace's equations, dual techniques, null field method, boundary element method, dual null field method.

Funding agency	Grant number
Ministry of Science and Technology, Taiwan	109-2923-E-216-001-MY3
Russian Foundation for Basic Research	20-51-S52003
The reported study was funded by the Ministry of Science and Technology (MOST), Grant 109-2923-E-216-001- MY3 and RFBR, research project 20-51-S52003.

Received August 20, 2020, published April 16, 2021

Bibliographic databases:

Citation: M. G. Lee, L. P. Zhang, Z. C. Li, A. L. Kazakov, “Dual null field method for Dirichlet problems of Laplace's equation in circular domains with circular holes”, Sib. Èlektron. Mat. Izv., 18:1 (2021), 393–422

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This publication is cited in the following articles:

1. Z.-C. Li, H.-Ts. Huang, L.-P. Zhang, A. A. Lempert, Lee Ming-Gong, “Analysis of dual null field methods for Dirichlet problems of Laplace's equation in elliptic domains with elliptic holes: bypassing degenerate scale”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 37 (2021), 47–62    
2. Zi-Cai Li, Hung-Tsai Huang, Li-Ping Zhang, Anna A. Lempert, Ming-Gong Lee, “Numerical experiments of the dual null field method for Dirichlet problems of Laplace's equation in elliptic domains with elliptic holes”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 39 (2022), 80–95    
3. O. A. Nefedova, L. F. Spevak, A. L. Kazakov, Li Ming-Gong, “Primenenie metoda nulevogo polya dlya resheniya dvumernogo nelineinogo uravneniya teploprovodnosti”, Kompyuternye issledovaniya i modelirovanie, 15:6 (2023), 1449–1467    

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles