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**Null field and interior field methods for Laplace’s equation in actually punctured disks.**
*(English)*
Zbl 1470.65203

Summary: For solving Laplace’s equation in circular domains with circular holes, the null field method (NFM) was developed by J.-T. Chen and W.-C. Shen [Numer. Methods Partial Differ. Equations 25, No. 1, 63–86 (2009; Zbl 1156.65097)]. In [the third author et al., Eng. Anal. Bound. Elem. 36, No. 3, 477–491 (2012; Zbl 1245.74098)], the explicit algebraic equations of the NFM were provided, where some stability analysis was made. For the NFM, the conservative schemes were proposed in [the second author et al., Eng. Anal. Bound. Elem. 37, No. 1, 95–106 (2013; Zbl 1351.74168)], and the algorithm singularity was fully investigated in [M.-G. Lee et al., “Algorithm singularity of the null-field method for Dirichlet problems of Laplace’s equation in annular and circular domains”, Eng. Anal. Bound. Elem. (submitted)]. 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.

### MSC:

65N80 | Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

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