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

Zi-Cai Li,Ming-Gong Lee,Hung-Tsai Huang,John Y. Chiang

doi:10.1155/2013/927873

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.

Highlights

For circular domains with circular holes, there exist a number of papers of boundary methods
In [17], we prove that when u ∈ H2(∂S) and u] ∈ H1(∂S), the null filed method (NFM) remains valid for the field nodes Q ∈ ∂S; that is, ρ = R on ∂SR and ρ = R1 on ∂SR1 and (23) and (24) hold
For the concentric circular domains, when ρ = R + ε ≠ 1, the leading coefficients are exact by the NFM, and the solution errors result only from the truncations of their Fourier expansions

Summary

Introduction

For circular domains with circular holes, there exist a number of papers of boundary methods. In [17], explicit algebraic equations of the NFM are derived, stability analysis is first made for the simple annular domain with concentric circular boundaries, and numerical experiments are performed to find the optimal field nodes. For the MFS, numerous computations are reviewed in Fairweather and Karageorghis [19] and Chen et al [20], but the error and stability analysis is developed by Li et al in [21, 22] Both the CTM and the MFS can be applied to arbitrary solution domains. The explicit discrete equations of NFM, IFM, CTM, and BIE are given, and their relations and overviews are explored.

The Null Field Method and Other Algorithms

Preliminary Analysis of the NFM

R1k bk

Numerical Experiments

Concluding Remarks