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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 927873, 15 pages
http://dx.doi.org/10.1155/2013/927873
Research Article

Null Field and Interior Field Methods for Laplace’s Equation in Actually Punctured Disks

1Department of Applied Mathematics, I-Shou University, Kaohsiung 84001, Taiwan
2Department of Leisure and Recreation Management, Ph.D. Program in Engineering Science, Chung Hua University, Hsinchu 30012, Taiwan
3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
4Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Received 18 May 2013; Revised 26 August 2013; Accepted 29 August 2013

Academic Editor: Rodrigo Lopez Pouso

Copyright © 2013 Hung-Tsai Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For solving Laplace’s equation in circular domains with circular holes, the null field method (NFM) was developed by Chen and his research group (see Chen and Shen (2009)). In Li et al. (2012) the explicit algebraic equations of the NFM were provided, where some stability analysis was made. For the NFM, the conservative schemes were proposed in Lee et al. (2013), and the algorithm singularity was fully investigated in Lee et al., submitted to Engineering Analysis with Boundary Elements, (2013). To target the same problems, a new interior field method (IFM) is also proposed. Besides the NFM and the IFM, the collocation Trefftz method (CTM) and the boundary integral equation method (BIE) are two effective boundary methods. This paper is devoted to a further study on NFM and IFM for three goals. The first goal is to explore their intrinsic relations. Since there exists no error analysis for the NFM, the second goal is to drive error bounds of the numerical solutions. The third goal is to apply those methods to Laplace’s equation in the domains with extremely small holes, which are called actually punctured disks. By NFM, IFM, BIE, and CTM, numerical experiments are carried out, and comparisons are provided. This paper provides an in-depth overview of four methods, the error analysis of the NFM, and the intriguing computation, which are essential for the boundary methods.