Dual null field method for Dirichlet problems of Laplace's equation in circular domains with circular holes

Dual techniques have been used in many engineering papers to deal with singularity and ill-conditioning of the boundary element method (BEM). Our efforts are paid to explore theoretical analysis, including error and stability analysis, to fill up the gap between theory and computation. Our group provides the analysis for Laplace’s equation in circular domains with circular holes and in this paper for elliptic domains with elliptic holes. The explicit algebraic equations of the first kind and second kinds of the null field method (NFM) and the interior field method (IFM) have been studied extensively. Traditionally, the first and the second kinds of the NFM are used for the Dirichlet and Neumann problems, respectively. To bypass the degenerate scales of Dirichlet problems, the second and the first kinds of the NFM are used for the exterior and the interior boundaries, simultaneously, called the dual null field method (DNFM) in this paper. Optimal convergence rates and good stability for the DNFM can be achieved from our analysis. This paper is the first part of the study and mostly concerns theoretical aspects; the second part is expected to be devoted to numerical experiments.

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

Abstract and Applied Analysis ◽

10.1155/2013/927873 ◽

2013 ◽

Vol 2013 ◽

pp. 1-15 ◽

Cited By ~ 1

Author(s):

Hung-Tsai Huang ◽

Ming-Gong Lee ◽

Zi-Cai Li ◽

John Y. Chiang

Keyword(s):

Error Analysis ◽

Numerical Solutions ◽

Field Method ◽

Boundary Integral ◽

Laplace’S Equation ◽

Algebraic Equations ◽

Field Methods ◽

Laplace's Equation ◽

Null Field ◽

Boundary Methods

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.

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Algorithm singularity of the null-field method for Dirichlet problems of Laplace׳s equation in annular and circular domains

Engineering Analysis with Boundary Elements ◽

10.1016/j.enganabound.2014.01.013 ◽

2014 ◽

Vol 41 ◽

pp. 160-172 ◽

Cited By ~ 5

Author(s):

Ming-Gong Lee ◽

Zi-Cai Li ◽

Liping Zhang ◽

Hung-Tsai Huang ◽

John Y. Chiang

Keyword(s):

Field Method ◽

Dirichlet Problems ◽

Null Field ◽

Circular Domains

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A Solution Procedure for Laplace’s Equation on Multiply Connected Circular Domains

Journal of Applied Mechanics ◽

10.1115/1.2899533 ◽

1992 ◽

Vol 59 (2) ◽

pp. 398-404 ◽

Cited By ~ 33

Author(s):

M. D. Bird ◽

C. R. Steele

Keyword(s):

Boundary Conditions ◽

Transformation Method ◽

Solution Procedure ◽

Laplace’S Equation ◽

Shape Functions ◽

Arbitrary Boundary ◽

Laplace's Equation ◽

Multiply Connected ◽

Circular Holes ◽

Circular Domains

A solution procedure is presented for the two-dimensional Laplace’s equation on circular domains with circular holes and arbitrary boundary conditions. The shape functions use the traditional trigonometric Fourier series on the boundaries with a power series decay into the domain thereby satisfying the governing equation exactly. The interaction of the boundaries is expressed simply and exactly resulting in quick processing time. The only simplification made is the use of a finite number of terms in the boundary conditions. The results are compared with a Green’s function method due to Naghdi (1991) and a Mo¨bius transformation method due to Honein et al. (1991).

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