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

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.

1. Introduction

For circular domains with circular holes, there exist a number of papers of boundary methods. In Barone and Caulk [1, 2] and Caulk [3], the Fourier functions are used for the circular holes for boundary integral equations. In Bird and Steele [4], the simple algorithms as the collocation Trefftz method (CTM) in [5, 6] are used. In Ang and Kang [7], complex boundary elements are studied. Recently, Chen and his research group have developed the null filed method (NFM), in which the field nodes 𝑄 are located outside of the solution domain 𝑆. The fundamental solutions (FS) can be expanded as the convergent series, and the Fourier functions are also used to approximate the Dirichlet and Neumann boundary conditions. Numerous papers have been published for different physical problems. Since error analysis and numerical experiments for four boundary methods are our main concern, we only cite [8–14]. More references of NFM are also given in [10–12, 14–17].

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. The field nodes can be located on the domain boundary: 𝑄∈𝜕𝑆, if the solutions are smooth enough to satisfy 𝑢∈𝐻²(𝜕𝑆) and 𝑢_𝜈∈𝐻¹(𝜕𝑆), where 𝑢_𝜈 is the normal derivative and 𝐻^𝑘(𝜕𝑆)(𝑘=1,2) are the Sobolev spaces; see the proof in [17]. It is discovered numerically that when the field nodes 𝑄∈𝜕𝑆, the NFM provides small errors and the smallest condition numbers, compared with all 𝑄∈𝑆^𝑐. Moreover for the NFM, the conservative schemes are proposed in [15], and the algorithm singularity is fully investigated in [16]. In fact, the explicit algebraic equations can also be derived from the Green representation formula with the field nodes inside the solution domain. This method is called the interior field method (IFM).

In addition to the NFM and IFM, the collocation Trefftz method (CTM) and the boundary integral equation method (BIE) are effective boundary methods too. Three goals are motivated in this paper. The first goal is to explore the intrinsic relations of NFM, IFM, CTM, and BIE with an in-depth overview. So far, there exists no error analysis for the NFM. The second goal is to derive error bounds of the numerical solutions by the NFM. The optimal convergence (or exponential) rates can be achieved. The third goal is to solve a challenging problem: Laplace's equation in the circular domains with extremely small holes, which are called the actually punctured disks in this paper. Four boundary methods, NFM, IFM, CTM, and BIE, are employed. Numerical experiments are carried out, and comparisons are provided. It is observed that the CTM is more advantageous in the applications than the others.

Besides, the method of fundamental solutions (MFS) is also popular in boundary methods, which originated from Kupradze and Aleksidze [18] in 1964. 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. However, the MFS incurs a severe numerical instability for very elongated domains [22]. Since the performance of the CTM is better than that of the MFS, reported elsewhere, we do not carry out the numerical computation of the MFS in this paper. Moreover, the null-field method with discrete source (NFM-DS) is effective and popular in light scattering (see Wriedt [23]), where the transition (𝑇) matrix is provided in Doicu and Wriedt [24]. In fact, the null field equation (NFE) of the Green representation formula in (9) can be employed on a source outside the solution domain 𝑆, without a need of the FS expansions, called the 𝑇 matrix method [24]. Hence, the 𝑇 matrix method is valid for arbitrary solution domains. There also occurs a severe numerical instability for very elongated holes (i.e., particles). To improve the stability for this case, different sources (i.e., discrete sources) may be utilized in the NFM-DS, by means of the idea of the MFS. The techniques for improving the stability by the NFM-DS are reported in many papers; we only cite [23, 25].

This paper is organized as follows. In the next section, the explicit discrete equations of NFM, IFM, CTM, and BIE are given, and their relations and overviews are explored. In Section 3, for the NFM some analysis is studied for circular domains with concentric circular boundaries. In Section 4, error bounds are provided without proof for the NFM with eccentric circular boundaries of simple annular domains. In Section 5, numerical experiments are carried out for Laplace's equation in the actually punctured disks. The results are reported with comparisons. In the last section, a few concluding remarks are addressed.

2. The Null Field Method and Other Algorithms

2.1. The Null Field Method

For simplicity in description of the NFM, we confine ourselves to Laplace's equation and choose the circular domain with one circular hole in this paper. Denote the disks 𝑆_𝑅 and 𝑆_𝑅1 with radii 𝑅 and 𝑅₁, respectively. Let 𝑆_𝑅1⊂𝑆_𝑅, and the eccentric circular domains 𝑆_𝑅 and 𝑆_𝑅1 may have different origins. Hence 2𝑅₁<𝑅. Choose the annular solution domain 𝑆=𝑆_𝑅⧵𝑆_𝑅1 with the exterior and the interior boundaries 𝜕𝑆_𝑅 and 𝜕𝑆_𝑅1, respectively. The following Dirichlet problems are discussed by Palaniappan [26]: 𝜕Δ𝑢=²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²=0in𝑆,𝑢=1on𝜕𝑆_𝑅,𝑢=0on𝜕𝑆_𝑅1. (1) In [11], 𝑅=2.5 and 𝑅₁=1 and the origins of 𝑆_𝑅 and 𝑆_𝑅1 are located at (0,0) and (−𝑅₁,0), respectively. In this paper, we fix 𝑅=2.5, while 𝑅₁ may be infinitesimal; that is, 𝑅₁≪1.

On the exterior boundary 𝜕𝑆_𝑅, there exist the approximations of Fourier expansions: 𝑢=𝑢₀∶=𝑎₀+^𝑀_𝑘=1𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃on𝜕𝑆_𝑅,𝜕𝑢𝜕𝜈=𝑞₀∶=𝑝₀+^𝑀_𝑘=1𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃on𝜕𝑆_𝑅, (2)(3) where 𝑎_𝑘,𝑏_𝑘,𝑝_𝑘, and 𝑞_𝑘 are coefficients. On the interior boundary 𝜕𝑆_𝑅1, we have similarly 𝑢=𝑢₀∶=𝑎₀+^𝑁_𝑘=1𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃on𝜕𝑆_𝑅1,𝜕𝑢𝜕𝜈𝜕=−𝑢𝜕𝑟=𝑞₀∶=𝑝₀+^𝑁_𝑘=1𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃on𝜕𝑆_𝑅1, (4)(5) where 𝑎_𝑘,𝑏_𝑘,𝑝_𝑘, and 𝑞_𝑘 are coefficients. In (2)–(5), 𝜃 and 𝜃 are the polar coordinates of 𝑆_𝑅 and 𝑆_𝑅1 with the origins (0,0) and (−𝑅₁,0), respectively, and 𝜈 and 𝜈 are the exterior normals of 𝜕𝑆_𝑅 and 𝜕𝑆_𝑅1, respectively. The Dirichlet, the Neumann conditions, and their mixed types on 𝜕𝑆_𝑅 may be given with known coefficients.

In 𝑆, denote two nodes 𝐱=𝑄=(𝑥,𝑦)=(𝜌,𝜃) and 𝐲=𝑃=(𝜉,𝜂)=(𝑟,𝜙), where 𝑥=𝜌cos𝜃,𝑦=𝜌sin𝜃,𝜉=𝑟cos𝜙, and 𝜂=𝑟sin𝜙. Then 𝜌=𝑥²+𝑦² and 𝑅=𝑟=𝜉²+𝜂². The FS of Laplace's equation is given by ln𝑃𝑄=ln{𝜌²−2𝜌𝑟cos(𝜃−𝜙)+𝑟²}. From the Green representation formula, we have different formulas for different locations of the field nodes 𝑄(𝐱): _𝜕𝑆||(𝐲)ln𝑃𝑄𝜕𝑢(𝜉)||𝜕𝜈−𝑢𝜕ln𝑃𝑄𝜕𝜈𝑑𝜎_𝜉=(𝑄)(𝑄)−2𝜋𝑢,𝑄∈𝑆,−𝜋𝑢,𝑄∈𝜕𝑆,0,otherwise, (6) where 𝑃(𝐲)∈(𝑆∪𝜕𝑆) and two kinds of series expansions of the FS ln|𝑃𝑄| are given by (see [27]) |||𝑃(𝐲)(𝐱𝑃=𝑈ln𝑃𝑄=ln−𝑄)|=ln𝑟,𝜙−𝑄𝜌,𝜃_𝑖(𝐱,𝐲)=ln𝑟−∑^∞_𝑛=11𝑛𝜌𝑟^𝑛𝑈cos𝑛𝜃−𝜙,𝜌<𝑟,_𝑒(𝐱,𝐲)=ln𝜌−∑^∞_𝑛=11𝑛𝑟𝜌^𝑛cos𝑛𝜃−𝜙,𝜌>𝑟, (7) where 𝐱=(𝜌,𝜃) and 𝐲=(𝑟,𝜙). Then we have two kinds of derivative expansions of FS 𝜕𝑈^𝑖(𝐱,𝐲)=1𝜕𝑟𝑟+^∞_𝑛=1𝜌^𝑛𝑟^𝑛+1cos𝑛𝜃−𝜙,𝜌<𝑟,𝜕𝑈^𝑒(𝐱,𝐲)𝜕𝑟=−^∞_𝑛=1𝑟^𝑛−1𝜌^𝑛cos𝑛𝜃−𝜙,𝜌>𝑟, (8) where the superscripts “𝑒” and “𝑖” designate the exterior and interior field nodes 𝐱, respectively. Note that the boundary element method (BEM) is based on the second equation of the Green formula (6), but the NFM is based on the third equation (i.e., the null field equation (NFE)) by using the FS expansions. We have _{𝜕𝑆𝑅∪𝜕𝑆𝑅1}𝑈()(𝐲)𝐱,𝐲𝜕𝑢𝜕𝜈𝑑𝜎_𝐲=_{𝜕𝑆𝑅∪𝜕𝑆𝑅1}𝑢(𝐲)𝜕𝑈(𝐱,𝐲)𝜕𝜈𝑑𝜎_𝐲,𝐱∈𝑆^𝑐, (9) where 𝑆^𝑐 is the complementary domain of 𝑆∪𝜕𝑆. Substituting the Fourier expansions (7)–(8) into (9) yields the basic algorithms of NFM, where the exterior normal of 𝜕𝑆_𝑅1 is given by 𝜕𝑈(𝐱,𝐲)/𝜕𝜈=−𝜕𝑈(𝐱,𝐲)/𝜕𝑟. In the Green formula (9), the field point 𝐱=(𝜌,𝜃) is supposed to locate outside of the solution domain 𝑆∪𝜕𝑆 only, so the algorithm of Chen is called the null field method (NFM) [8, 9, 11]. The field nodes can also be located on the domain boundary: 𝑄∈𝜕𝑆, if the solutions are smooth enough to satisfy 𝑢∈𝐻²(𝜕𝑆) and 𝑢_𝜈∈𝐻¹(𝜕𝑆), where 𝑢_𝜈 is the normal derivative and 𝐻^𝑘(𝜕𝑆)(𝑘=1,2) are the Sobolev spaces; see the rigorous proof in [17]. It is discovered numerically that when the field nodes 𝑄∈𝜕𝑆, the NFM provides small errors and condition numbers and has been widely implemented in many engineering problems.

Denote two systems of polar coordinates by (𝜌,𝜃) and (𝜌,𝜃) with the origins (0,0) and (𝑥₁,𝑦₁) for 𝑆_𝑅 and 𝑆_𝑅1, respectively. There exist the following conversion formulas:𝜌=𝜌cos𝜃+𝑥₁²+𝜌sin𝜃+𝑦₁²,tan𝜃=𝜌sin𝜃+𝑦₁𝜌cos𝜃+𝑥₁,𝜌=𝜌cos𝜃−𝑥₁²+𝜌sin𝜃−𝑦₁²,tan𝜃=𝜌sin𝜃−𝑦₁𝜌cos𝜃−𝑥₁. (10)(11)

First, consider the exterior field nodes 𝐱=(𝜌,𝜃) with 𝜌>𝑟=𝑅. The first explicit algebraic equations of the NFM are given for the exterior field nodes (see [17]) ℒ_ext𝜌,𝜃;𝜌,𝜃∶=−𝑅𝜋^𝑀_𝑘=1𝑅^𝑘−1𝜌^𝑘𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃+𝑅₁𝜋^𝑁_𝑘=1𝑅₁^𝑘−1𝜌^𝑘𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃−2𝜋𝑅ln𝜌𝑝₀−𝑅𝜋^𝑀_𝑘=11𝑘𝑅𝜌^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃+2𝜋𝑅₁ln𝜌𝑝₀−𝑅₁𝜋^𝑁_𝑘=11𝑘𝑅₁𝜌^𝑘𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃=0. (12) Next, consider the interior field nodes 𝐱=(𝜌,𝜃) with 𝜌<𝑟=𝑅₁. The second explicit algebraic equations of the NFM are given for the interior field nodes (see [17]) ℒ_int𝜌,𝜃;𝜌,𝜃∶=−2𝜋𝑎₀−𝑅₁𝜋^𝑁_𝑘=1𝜌^𝑘𝑅₁^𝑘+1𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃+2𝜋𝑎₀+𝑅𝜋^𝑀_𝑘=1𝜌^𝑘𝑅^𝑘+1𝑎_𝑘cos𝑘𝜃+𝑏_𝑘−sin𝑘𝜃2𝜋𝑅₁ln𝑅₁𝑝₀−𝑅₁𝜋^𝑁_𝑘=11𝑘𝜌𝑅₁^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃+2𝜋𝑅ln𝑅𝑝₀−𝑅𝜋^𝑀_𝑘=11𝑘𝜌𝑅^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃=0. (13)

Since one of Dirichlet or Neumann conditions is given on 𝜕𝑆_𝑅 and 𝜕𝑆_𝑅1, only 2(𝑀+𝑁)+2 coefficients in (2)–(5) are unknowns. We choose 2𝑀+1 and 2𝑁+1 field nodes located uniformly on the exterior and the interior circles, respectively, 𝜌,𝜃=(𝑅+𝜖,𝑖Δ𝜃),𝑖=0,1,…,2𝑀,𝜌,𝜃=𝑅₁−𝜖,𝑖Δ𝜃,𝑖=0,1,…,2𝑁, (14) where 𝜖≥0, 0≤𝜖<𝑅₁, Δ𝜃=2𝜋/(2𝑀+1), and Δ𝜃=2𝜋/(2𝑁+1). Denote the explicit equations (12) and (13) by ℒ_ext𝜌,𝜃;𝜌,𝜃=0,ℒ_int𝜌,𝜃;𝜌,𝜃=0. (15) We obtain 2(𝑀+𝑁)+2 discrete equations from (15) ℒ_ext𝑅+𝜖,𝑖Δ𝜃;𝜌_𝑖,𝜃_𝑖ℒ=0,𝑖=0,1,…,2𝑀,_int𝜌_𝑖,𝜃_𝑖;𝑅₁−𝜖,𝑖Δ𝜃=0,𝑖=0,1,…,2𝑁, (16) where the corresponding coordinates (𝜌_𝑖,𝜃_𝑖) and (𝜌_𝑖,𝜃_𝑖) are obtained from (10) and (11). Hence from (16), we obtain the following linear algebraic equations: 𝐅𝐱=𝐛, (17) where the matrices 𝐅(∈𝑅^𝑛×𝑛), the vector 𝐱(∈𝑅^𝑛), and 𝑛=2(𝑀+𝑁)+2. The unknown coefficients can be obtained from (17), if the matrix 𝐅 is nonsingular. In this paper, we confine the Dirichlet problems. The study of the Neumann problems will be reported in a subsequent paper.

Once all the coefficients are known, based on the first equation of the Green formula (6), the solution at the interior nodes: 𝐱=(𝜌,𝜃)∈𝑆 is expressed by 𝑢(𝐱)=−1=𝑢𝜌,𝜃2𝜋_{𝜕𝑆𝑅∪𝜕𝑆𝑅1}𝑈()(𝐲)𝐱,𝐲𝜕𝑢(𝜉)()𝜕𝜈−𝑢𝜕𝑈𝐱,𝐲𝜕𝑟𝑑𝜎_𝐲1=−2𝜋_𝜕𝑆𝑅𝑈^𝑖()(𝐲)𝐱,𝐲𝜕𝑢(𝐲)𝜕𝜈−𝑢𝜕𝑈^𝑖(𝐱,𝐲)𝜕𝑟𝑑𝜎_𝐲+_{𝜕𝑆𝑅1}𝑈^𝑒()(𝐲)𝐱,𝐲𝜕𝑢𝜕𝜈+𝑢(𝐲)𝜕𝑈^𝑒(𝐱,𝐲)𝜕𝑟𝑑𝜎_𝐲𝐱∈𝑆. (18) For (𝜌,𝜃)∈𝑆, from (2)–(5) and (7)-(8), (2.20) leads to (see [17]) 𝑢_𝑀−𝑁=𝑢_𝑀−𝑁𝜌,𝜃=𝑢_𝑀−𝑁𝜌,𝜃=𝑎₀−𝑅ln𝑅𝑝₀−𝑅₁ln𝜌𝑝₀+𝑅2^𝑀_𝑘=11𝑘𝜌𝑅^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘+𝑅sin𝑘𝜃2^𝑀_𝑘=1𝜌^𝑘𝑅^𝑘+1×𝑎_𝑘cos𝑘𝜃+𝑏_𝑘+𝑅sin𝑘𝜃¹2^𝑁_𝑘=11𝑘𝑅₁𝜌^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃+𝑅₁2^𝑁_𝑘=1𝑅₁^𝑘−1𝜌^𝑘×𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃,(𝑟,𝜃)∈𝑆, (19) where (𝜌,𝜃) are also given from (11).

2.2. Conservative Schemes

For some physical problems, the flux conservation is imperative and essential. The conservative schemes of NFM can be designed to satisfy exactly the flux conservation [15] _𝑆𝑅𝑢_𝑀𝜈+_𝑆𝑅1𝑢_𝑁𝜈=0. (20) Substituting (3) and (5) into (20) yields directly 𝑅𝑝₀+𝑅₁𝑝₀=0. (21) We may use (21) to remove one coefficient, say 𝑝₀, 𝑝₀𝑅=−𝑅₁𝑝₀. (22) By using (22), (12) and (13) lead to ℒ^𝐶_ext𝜌,𝜃;𝜌,𝜃∶=−𝑅𝜋^𝑀_𝑘=1𝑅^𝑘−1𝜌^𝑘𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃+𝑅₁𝜋^𝑁_𝑘=1𝑅₁^𝑘−1𝜌^𝑘𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃−𝜌2𝜋𝑅ln𝜌𝑝₀−𝑅𝜋^𝑀_𝑘=11𝑘𝑅𝜌^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃−𝑅₁𝜋^𝑁_𝑘=11𝑘𝑅₁𝜌^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃ℒ=0,_𝐶int𝜌,𝜃;𝜌,𝜃∶=−2𝜋𝑎₀−𝑅₁𝜋^𝑁_𝑘=1𝜌^𝑘𝑅₁^𝑘+1×𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃+2𝜋𝑎₀+𝑅𝜋^𝑀_𝑘=1𝜌^𝑘𝑅^𝑘+1𝑎_𝑘cos𝑘𝜃+𝑏_𝑘−sin𝑘𝜃−𝑅₁𝜋^𝑁_𝑘=11𝑘𝜌𝑅₁^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃𝑅+2𝜋𝑅ln𝑅₁𝑝₀−𝑅𝜋^𝑀_𝑘=11𝑘𝜌𝑅^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃=0. (23) Also the interior solution (19) leads to 𝑢^𝐶_𝑀−𝑁=𝑢^𝐶_𝑀−𝑁𝜌,𝜃=𝑢^𝐶_𝑀−𝑁𝜌,𝜃=𝑎₀𝑅−𝑅ln𝑅₁𝑝₀+𝑅2^𝑀_𝑘=11𝑘𝜌𝑅^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘+𝑅sin𝑘𝜃2^𝑀_𝑘=1𝜌^𝑘𝑅^𝑘+1×𝑎_𝑘cos𝑘𝜃+𝑏_𝑘+𝑅sin𝑘𝜃¹2^𝑁_𝑘=11𝑘𝑅₁𝜌^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃+𝑅₁2^𝑁_𝑘=1𝑅₁^𝑘−1𝜌^𝑘×𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃,(𝑟,𝜃)∈𝑆. (24) Hence, the total number of unknown coefficients is reduced to 2(𝑀+𝑁)+1. Based on the analysis in [15], to remain good stability, we still choose 2(𝑀+𝑁)+2 collocation nodes as in (16): 𝑤_𝑖ℒ^𝐶_ext𝑅+𝜀,𝑖Δ𝜃;𝜌_𝑖,𝜃_𝑖𝑤=0,𝑖=0,1,…,2𝑀,_𝑖ℒ^𝐶_int𝜌_𝑖,𝜃_𝑖;𝑅₁−𝜀,𝑖Δ𝜃=0,𝑖=1,2,…,2𝑁, (25) where the weights 𝑤₀=1, 𝑤_𝑖=√2 for 𝑖≥1, Δ𝜃=2𝜋/(2𝑀+1), and Δ𝜃=2𝜋/(2𝑁+1). Equation (25) form an overdetermined system, which can be solved by the QR method or the singular value decomposition.

2.3. The Interior Field Method

In [17], we prove that when 𝑢∈𝐻²(𝜕𝑆) and 𝑢_𝜈∈𝐻¹(𝜕𝑆), the NFM remains valid for the field nodes 𝑄∈𝜕𝑆; that is, 𝜌=𝑅 on 𝜕𝑆_𝑅 and 𝜌=𝑅₁ on 𝜕𝑆_𝑅1 and (23) and (24) hold. In fact, we may use (24) only, because (23) is obtained directly from the Dirichlet conditions on 𝜕𝑆_𝑅 and 𝜕𝑆_𝑅1, respectively. Interestingly, (24) is obtained from the interior (i.e., the first) Green formula in (6) only. For this reason, the interior field method (IFM) is named. Evidently, the IFM is equivalent to the special NFM. Based on this linkage, the new error analysis in Section 4 is explored.

2.4. The First Kind Boundary Integral Equations

We may also apply the series expansions of FS to the first kind boundary integral equations. Consider the Dirichlet problem Δ𝑢=0,inΩ=𝑅²𝑢(𝑥)|𝐱||𝐱|⧵Γ,𝑢=𝑓,onΓ,=𝑂log,as⟶∞, (26) where |𝐱| is the Euclidean distance. In (26), Γ(=∪^𝑑_𝑚=1Γ_𝑚) is an open arc, and each of its edges, Γ_𝑚(𝑚=1,…,𝑑), is assumed to be smooth. Let 𝐶_Γ be the logarithmic capacity of Γ. From the single layer potential theory [28–30], if 𝐶_Γ≠1, (26) can be converted to the first kind boundary integral equation (BIE), −12𝜋_Γ𝑣(𝐱)ln𝐱−𝑦𝑑𝑠_𝐱=𝑓(𝐲)(𝐲∈Γ), (27) where 𝑣(𝐱)(=(𝜕𝑢(𝐱)/𝜕𝑛⁻)−(𝜕𝑢(𝐱)/𝜕𝑛⁺)) is the unknown function and 𝜕𝑢/𝜕𝑛^± denote the normal derivatives along the positive and negative sides of Γ. If 𝐶_Γ≠1, there exists a unique solution of (27), see [28]. As soon as 𝑣(𝐱) is solved from (27), the solution 𝑢(𝐱) (𝐱∈Ω) of (26) can be evaluated by 𝑢(𝐱)1=−2𝜋_Γ𝑣(𝐱)ln|𝐱−𝐲|𝑑𝑠_𝐱(𝐲∈Ω). (28) For the smooth solution 𝑢, we have 𝑣(𝐱)=2(𝜕𝑢/𝜕𝜈), where 𝜈 is the normal of Γ. We may assume the Fourier expansions of 𝑣 on Γ𝑣(𝑠)=𝑣⁺(𝑠)=𝑞^⋆₀∶=𝑝^⋆₀+^𝑀_𝑘=1𝑝^⋆_𝑘cos𝑘𝜃+𝑞^⋆_𝑘sin𝑘𝜃on𝜕𝑆_𝑅,𝑣(𝑠)=𝑣⁻(𝑠)=𝑞^⋆₀∶=𝑝^⋆₀+^𝑁_𝑘=1𝑝^⋆_𝑘cos𝑘𝜃+𝑞^⋆_𝑘sin𝑘𝜃on𝜕𝑆_𝑅1, (29) where 𝑝^⋆_𝑘,𝑞^⋆_𝑘,𝑝^⋆_𝑘, and 𝑞^⋆_𝑘 are the coefficients. We have from [17] 𝑢(𝐱)1=𝑢𝜌,𝜃=−2𝜋_{𝜕𝑆𝑅∪𝜕𝑆𝑅1}𝑈(𝐱,𝐲)𝑣(𝐲)𝑑𝜎_𝐲1=−2𝜋_𝜕𝑆𝑅𝑈^𝑖(𝐱,𝐲)𝑣(𝐲)𝑑𝜎_𝐲+_{𝜕𝑆𝑅1}𝑈^𝑒(𝐱,𝐲)𝑣(𝐲)𝑑𝜎_𝐲,𝐱∈𝑆, (30) to give 𝑢_𝑀−𝑁𝜌,𝜃=−𝑅ln𝑅𝑝^⋆₀−𝑅₁ln𝜌𝑝^⋆₀+𝑅2^𝑀_𝑘=11𝑘𝜌𝑅^𝑘×𝑝^⋆_𝑘cos𝑘𝜃+𝑞^⋆_𝑘+𝑅sin𝑘𝜃¹2^𝑁_𝑘=11𝑘𝑅₁𝜌^𝑘×𝑝^⋆_𝑘cos𝑘𝜃+𝑞^⋆_𝑘sin𝑘𝜃,(𝑟,𝜃)∈𝑆. (31) Note that the derivation of (31) in the first kind BIE is simpler, because we do not need the series expansions of 𝜕𝑈^𝑖(𝐱,𝐲)/𝜕𝑟 and 𝜕𝑈^𝑒(𝐱,𝐲)/𝜕𝑟. This advantage is very important for elasticity problems, because the displacement conditions are much simpler than the traction ones.

2.5. The Collocation Trefftz Method

We also use the collocation Trefftz method (CTM). For (1), the particular solutions of CTM are given by (see [6]) 𝑢_𝑀−𝑁𝜌,𝜃;𝜌,𝜃=𝑎₀+^𝑀_𝑖=1𝜌𝑅^𝑖𝑎_𝑖cos𝑖𝜃+𝑏_𝑖+sin𝑖𝜃𝑎₀ln𝜌+^𝑁_𝑖=1𝑅₁𝜌^𝑖×𝑎_𝑖cos𝑖𝜃+𝑏_𝑖sin𝑖𝜃,𝜌≤𝑅,𝜌≥𝑅₁, (32) where 𝑎_𝑖,𝑏_𝑖,𝑎_𝑖, and 𝑏_𝑖 are the coefficients. Evidently, the admissible functions (19) of the IFM and (31) of the first kind BIE are the special cases of (32). Equation (31) may be written as (32) with the following relations of coefficients: 𝑎₀∶=−𝑅ln𝑅𝑝^⋆₀,𝑎₀∶=−𝑅₁𝑝^⋆₀,𝑎_𝑘𝑅∶=𝑝2𝑘^⋆𝑘,𝑏_𝑘𝑅∶=𝑞2𝑘^⋆𝑘,𝑎_𝑘𝑅∶=¹2𝑘𝑝^⋆_𝑘,𝑏_𝑘𝑅∶=¹2𝑘𝑞^⋆_𝑘. (33) Equation (19) can also be written as (32) with 𝑎₀∶=𝑎^IFM₀−𝑅ln𝑅𝑝^IFM₀,𝑎₀∶=−𝑅₁𝑝^IFM₀,𝑎_𝑘𝑅∶=𝑝2𝑘^IFM𝑘+12𝑎^IFM_𝑘,𝑏_𝑘𝑅∶=𝑞2𝑘^IFM𝑘+12𝑏^IFM_𝑘,𝑎_𝑘𝑅∶=¹2𝑘𝑝^IFM_𝑘+12𝑎^IFM_𝑘,𝑏_𝑘𝑅∶=¹2𝑘𝑞^IFM_𝑘+12𝑏^IFM_𝑘, (34) where 𝑝^IFM_𝑘,𝑞^IFM_𝑘,… are the coefficients in (19) of the IFM.

Therefore, we may classify the IFM and the first kind BIE into the TM family, and their analysis may follow the framework in [6]. However, the particular solutions (32) can be applied to arbitrary shaped domains, for example, simply or multiple-connected domains, but the functions (19) and (31) are confined themselves to the circular domains with circular holes only. The four boundary methods, NFM, IFM, BIE, and CTM, are described together, with their explicit algebraic equations. The relations of their expansion coefficients are discovered at the first time. Moreover, Figure 1 shows clear relations among NFM, IFM, BIE, and CTM. The intrinsic relations have been provided to fulfill the first goal of this paper.

Figure 1 

Relations among NFM, IFM, BIE, and CTM.

To close this section, we describe the CTM. Denote 𝑉_𝑀−𝑁 the set of (32), and define the energy 𝐼(𝑢)=_Γ𝑣−𝑓², (35) where Γ=𝜕𝑆 and 𝑓 is the known function of Dirichlet boundary conditions. Then the solution 𝑢_𝑀−𝑁 of the Trefftz methods (TM) can be obtained by 𝐼𝑢_𝑀−𝑁=min_{𝑣∈𝑉𝑀−𝑁}𝐼(𝑣). (36) The TM solution 𝑢_𝑀−𝑁 also satisfies 𝑢−𝑢_{𝑀−𝑁0,Γ}=min_{𝑣∈𝑉𝑀−𝑁}‖𝑢−𝑣‖_0,Γ. (37) When the integral in (35) involves numerical approximation, the modified energy is defined as 𝐼̂(𝑣)=∫_Γ𝑣−𝑓², (38) where ∫_Γ is the numerical approximations of ∫_Γ by some quadrature rules, such as the central or the Gaussian rule. Hence, the numerical solution 𝑢_𝑀−𝑁∈𝑉_𝑀−𝑁 is obtained by 𝐼̂𝑢_𝑀−𝑁=min_{𝑣∈𝑉𝑀−𝑁}𝐼̂(𝑣). (39) We may also establish the collocation equations directly from the Dirichlet condition to yield 𝑢_𝑀−𝑁𝑃_𝑗=𝑓_𝑀−𝑁𝑃_𝑗,𝑃_𝑗∈Γ. (40) Following [6], (40) is just equivalent to (38).

3. Preliminary Analysis of the NFM

In this section, a preliminary analysis of the NFM is made for concentric circular boundaries. In the next section, error analysis of the NFM with 𝜖=𝜖=0 is explored for eccentric circular boundaries. Consider the simple domains of 𝑆=𝑆_𝑅⧵𝑆_𝑅1, where 𝑆_𝑅 and 𝑆_𝑅1 have the same origin. For the same origin 𝑂 of 𝑆_𝑅 and 𝑆_𝑅1, the same polar coordinates (𝜌,𝜃) are used, and the general solutions in 𝑆_𝑅⧵𝑆_𝑅1 can be denoted by 𝑢𝜌,𝜃=𝑎^∗₀+^∞_𝑘=1𝜌^𝑘𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘sin𝑘𝜃+𝑎^∗₀+ln𝜌_∞𝑘=1𝜌^−𝑘𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘sin𝑘𝜃, (41) where 𝑎^∗_𝑖,𝑏^∗_𝑖,𝑎^∗_𝑖,𝑏^∗_𝑖 are true coefficients and 𝑅₁≤𝜌≤𝑅. Then their derivatives are given by 𝜕𝑢=𝜕𝜌𝜌,𝜃^∞_𝑘=1𝑘𝜌^𝑘−1𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘sin𝑘𝜃+𝑎^∗₀1𝜌−^∞_𝑘=1𝑘𝜌^−𝑘−1𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘sin𝑘𝜃. (42) When 𝜌=𝑅, from (41) and (42), we have 𝑢𝜌,𝜃_𝜌=𝑅=𝑎^∗₀+^∞_𝑘=1𝑅^𝑘𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘sin𝑘𝜃+𝑎^∗₀+ln𝑅_∞𝑘=1𝑅^−𝑘𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘,𝜕sin𝑘𝜃𝑢𝜕𝜌𝜌,𝜃_𝜌=𝑅=^∞_𝑘=1𝑘𝑅^𝑘−1𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘sin𝑘𝜃+𝑎^∗₀1𝑅−^∞_𝑘=1𝑘𝑅^−𝑘−1𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘sin𝑘𝜃. (43) Comparing (43) with (2) and (3), we have the following equalities of coefficients: 𝑎₀=𝑎^∗₀+𝑎^∗₀𝑎ln𝑅,_𝑘=𝑅^𝑘𝑎^∗_𝑘+𝑅^−𝑘𝑎^∗_𝑘,𝑏_𝑘=𝑅^𝑘𝑏^∗_𝑘+𝑅^−𝑘𝑏^∗_𝑘,𝑝₀=𝑎^∗₀1𝑅,𝑝_𝑘=𝑘𝑅^𝑘−1𝑎^∗_𝑘−𝑅^−𝑘−1𝑎^∗_𝑘,𝑞_𝑘=𝑘𝑅^𝑘−1𝑏^∗_𝑘−𝑅^−𝑘−1𝑏^∗_𝑘, (44)(45) where 𝑎_𝑘,𝑏_𝑘,𝑝_𝑘, and 𝑞_𝑘 are the coefficients of the NFM in Section 2.1.

Also, when 𝜌=𝑅₁, from (41) and (42), we have 𝑢𝜌,𝜃_𝜌=𝑅1=𝑎^∗₀+^∞_𝑘=1𝑅^𝑘₁𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘+sin𝑘𝜃𝑎^∗₀ln𝑅₁+^∞_𝑘=1𝑅₁^−𝑘𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘,𝜕sin𝑘𝜃𝑢𝜕𝜌𝜌,𝜃_𝜌=𝑅1=^∞_𝑘=1𝑘𝑅₁^𝑘−1𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘sin𝑘𝜃+𝑎^∗₀1𝑅₁−^∞_𝑘=1𝑘𝑅₁^−𝑘−1𝑎^∗_𝑘cos𝑘𝜃+𝑏^∗_𝑘sin𝑘𝜃. (46) Comparing (46) with (4) and (5), we have 𝑎₀=𝑎^∗₀+𝑎^∗₀ln𝑅₁,𝑎_𝑘=𝑅^𝑘₁𝑎^∗_𝑘+𝑅₁^−𝑘𝑎^∗_𝑘,𝑏_𝑘=𝑅^𝑘₁𝑏^∗_𝑘+𝑅₁^−𝑘𝑏^∗_𝑘,𝑝₀=−𝑎^∗₀1𝑅₁,𝑝_𝑘=−𝑘𝑅₁^𝑘−1𝑎^∗_𝑘−𝑅₁^−𝑘−1𝑎^∗_𝑘,𝑞_𝑘=−𝑘𝑅₁^𝑘−1𝑏^∗_𝑘−𝑅₁^−𝑘−1𝑏^∗_𝑘, (47)(48) where 𝑎_𝑘,𝑏_𝑘,𝑝_𝑘, and 𝑞_𝑘 are also the coefficients of the NFM in Section 2.1.

On the other hand, when (𝜌,𝜃)=(𝜌,𝜃), we have from the first original equation (12) −𝑅𝜋^𝑀_𝑘=1𝑅^𝑘−1𝜌^𝑘𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃+𝑅₁𝜋^𝑁_𝑘=1𝑅₁^𝑘−1𝜌^𝑘𝑎_𝑘cos𝑘𝜃+𝑏_𝑘𝑝sin𝑘𝜃=2𝜋𝑅ln𝜌₀−𝑅𝜋^𝑀_𝑘=11𝑘𝑅𝜌^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃+2𝜋𝑅₁ln𝜌𝑝₀−𝑅₁𝜋^𝑁_𝑘=11𝑘𝑅₁𝜌^𝑘𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃. (49) Then for 𝜌≥𝑅, we obtain the following equalities, based on the orthogonality of trigonometric functions: ln𝜌𝑅𝑝₀+𝑅₁𝑝₀𝑅=0,_𝑘𝑎_𝑘−𝑅^𝑘₁𝑎_𝑘=1𝑘𝑅^𝑘+1𝑝_𝑘+1𝑘𝑅₁^𝑘+1𝑝_𝑘,𝑅^𝑘𝑏_𝑘−𝑅^𝑘₁𝑏_𝑘=1𝑘𝑅^𝑘+1𝑞_𝑘+1𝑘𝑅₁^𝑘+1𝑞_𝑘. (50)(51)(52) Similarly, from the second equation (13), −2𝜋𝑎₀−𝑅₁𝜋^𝑁_𝑘=1𝜌^𝑘𝑅₁^𝑘+1𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃+2𝜋𝑎₀+𝑅𝜋^𝑀_𝑘=1𝜌^𝑘𝑅^𝑘+1𝑎_𝑘cos𝑘𝜃+𝑏_𝑘sin𝑘𝜃=2𝜋𝑅₁ln𝑅₁𝑝₀−𝑅₁𝜋^𝑁_𝑘=11𝑘𝜌𝑅₁^𝑘×𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃+2𝜋𝑅ln𝑅𝑝₀−𝑅𝜋^𝑀_𝑘=11𝑘𝜌𝑅^𝑘𝑝_𝑘cos𝑘𝜃+𝑞_𝑘sin𝑘𝜃. (53) Then for 𝜌≤𝑅₁, we obtain 𝑎₀−𝑎₀=𝑅ln𝑅𝑝₀+𝑅₁ln𝑅₁𝑝₀,1𝑅^𝑘𝑎_𝑘−1𝑅^𝑘₁𝑎_𝑘1=−𝑘1𝑅^𝑘−1𝑝_𝑘−1𝑘1𝑅₁^𝑘−1𝑝_𝑘,1𝑅^𝑘𝑏_𝑘−1𝑅^𝑘₁𝑏_𝑘1=−𝑘1𝑅^𝑘−1𝑞_𝑘−1𝑘1𝑅₁^𝑘−1𝑞_𝑘. (54)(55)(56) Below, we prove that the true coefficients can be obtained directly from the NFM based on (50)–(52) for 𝜌≥𝑅 and on (54)–(56) for 𝜌≤𝑅₁. Outline of the proof is as follows. We will prove that the true solutions satisfy (50)–(52) and (54)–(56) of the NFM. Based on the analysis in [16], when 𝑅≠1, there exists a unique solution of the special NFM with 𝜖=𝜖=0. Therefore, the true coefficients can be determined by the IFM uniquely.

First to show (50). The consistent condition is given by _𝜕𝑆𝑅𝜕𝑢+𝜕𝜈_{𝜕𝑆𝑅1}𝜕𝑢𝜕𝜈=2𝜋𝑅𝑝₀+2𝜋𝑅₁𝑝₀=0. (57) Equation (57) can also be obtained from (45) and (48). Equations (57) and (50) are equivalent if ln𝜌≠0 (i.e., 𝜌≠1), which is also the necessary condition of nonsingularity of matrix 𝐅 in (17) [16]. Based on (57), the conservative schemes are proposed in [15]. Equation (54) is shown next. We have from (44) and (47) 𝑎₀−𝑎₀=𝑎^∗₀ln𝑅−𝑎^∗₀ln𝑅₁=𝑅ln𝑅𝑎^∗₀𝑅+𝑅₁ln𝑅₁−𝑎^∗₀𝑅₁=𝑅ln𝑅𝑝₀+𝑅₁ln𝑅₁𝑝₀, (58) where we have used (45) and (48).

Equations (51) and (55) are shown below. Denote them in matrix form 𝑅^𝑘−𝑅^𝑘₁1𝑅^𝑘−1𝑅^𝑘₁𝑎_𝑘𝑎_𝑘=1𝑘𝑅^𝑘+1𝑅₁^𝑘+1−1𝑅^𝑘−1−1𝑅₁^𝑘−1𝑝_𝑘𝑝_𝑘, (59) and denote from (44) and (47) 𝑎_𝑘𝑎_𝑘=𝑅^𝑘𝑅^−𝑘𝑅^𝑘₁𝑅₁^−𝑘𝑎^∗_𝑘𝑎^∗_𝑘, (60) where 𝑎^∗_𝑘 and 𝑎^∗_𝑘 are true expansion coefficients. Also denote from (45) and (48) 𝑝_𝑘𝑝_𝑘𝑅=𝑘^𝑘−1−𝑅^−𝑘−1−𝑅₁^𝑘−1𝑅₁^−𝑘−1𝑎^∗_𝑘𝑎^∗_𝑘. (61) By substituting (60) and (61) into (59), its left-hand side leads to 𝑅^𝑘−𝑅^𝑘₁1𝑅^𝑘−1𝑅^𝑘₁𝑎_𝑘𝑎_𝑘=𝑅^𝑘−𝑅^𝑘₁1𝑅^𝑘−1𝑅^𝑘₁𝑅^𝑘𝑅^−𝑘𝑅^𝑘₁𝑅₁^−𝑘𝑎^∗_𝑘𝑎^∗_𝑘=𝑅^2𝑘−𝑅₁^2𝑘00𝑅^−2𝑘−𝑅₁^−2𝑘𝑎^∗_𝑘𝑎^∗_𝑘. (62) The right-hand side of (59) leads to 1𝑘𝑅^𝑘+1𝑅₁^𝑘+1−1𝑅^𝑘−1−1𝑅₁^𝑘−1𝑝_𝑘𝑝_𝑘=1𝑘𝑅^𝑘+1𝑅₁^𝑘+1−1𝑅^𝑘−1−1𝑅₁^𝑘−1𝑘𝑅^𝑘−1−𝑅^−𝑘−1−𝑅₁^𝑘−1𝑅₁^−𝑘−1𝑎^∗_𝑘𝑎^∗_𝑘=𝑅^2𝑘−𝑅₁^2𝑘00𝑅^−2𝑘−𝑅₁^−2𝑘𝑎^∗_𝑘𝑎^∗_𝑘. (63) The second equality of the right-hand sides of (62) and (63) yield (59). The proof for the validity of (52) and (56) is similar. We write these important results as a proposition.

Proposition 1. 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.

4. Error Bounds of the NFM with 𝜖=𝜖=0

The NFM with the field nodes 𝑄∈𝜕𝑆 (i.e., 𝜖=𝜖=0) located on the domain boundary is the most important application for Chen's publications (see [8–14]). We will provide the errors bounds under the Sobolev norms of this special NFM for circular domains with eccentric circular boundaries without proof. Based on the equivalence of the special NFM and the CTM, we may follow the framework of analysis of Treffez method in [6]. The Sobolev norms for Fourier functions are provided in Kreiss and Oliger [31], Pasciak [32], and Canuto and Quarteroni [33].

Let the domain 𝑆 be divided into two subdomains 𝑆^ext and 𝑆^int with an interface boundary Γ₀∈𝑆. We have 𝑆=𝑆^ext∪𝑆^int∪Γ₀ and 𝑆^ext∩𝑆^int=∅, where 𝜕𝑆^ext=𝜕𝑆_𝑅∪Γ₀ and 𝜕𝑆^int=𝜕𝑆_𝑅1∪Γ₀. We assume that the true solutions have different regularities 𝑢∈𝐻^𝑝+(1/2)𝑆^ext,𝑢∈𝐻^𝜎+(1/2)𝑆^int, (64) where 𝑝≥2 and 𝜎≥2. Then there are different regularities on the boundary 𝑢∈𝐻^𝑝𝜕𝑆^ext,𝑢_𝜈∈𝐻^𝑝−1𝜕𝑆^ext,𝑢∈𝐻^𝜎𝜕𝑆^int,𝑢_𝜈∈𝐻^𝜎−1𝜕𝑆^int, (65) where 𝜈 and 𝜈 are the exterior normal to 𝜕𝑆^ext and 𝜕𝑆^int, respectively. Therefore, the true solutions can be expressed by the Fourier expansions on 𝜕𝑆_𝑅𝑢𝜌,𝜃_𝜕𝑆𝑅=𝑎^∘₀+^∞_𝑘=1𝑎^∘_𝑘cos𝑘𝜃+𝑏^∘_𝑘,𝜕sin𝑘𝜃𝑢𝜕𝜌𝜌,𝜃_𝜕𝑆𝑅=𝑝^∘₀+^∞_𝑘=1𝑝^∘_𝑘cos𝑘𝜃+𝑞^∘_𝑘sin𝑘𝜃, (66)(67) where 𝑎^∘_𝑘,𝑏^∘_𝑘,𝑝^∘_𝑘,𝑞^∘_𝑘, are the true boundary coefficients. Similarly, we have 𝑢𝜌,𝜃_{𝜕𝑆𝑅1}=𝑎^∘₀+^∞_𝑘=1𝑎^∘_𝑘cos𝑘𝜃+𝑏^∘_𝑘sin𝑘𝜃,𝜕𝜕𝜈𝑢𝜌,𝜃_{𝜕𝑆𝑅1}=𝑝^∘₀+^∞_𝑘=1𝑝^∘_𝑘cos𝑘𝜃+𝑞^∘_𝑘sin𝑘𝜃, (68) where 𝑎^∘_𝑘,𝑏^∘_𝑘,𝑝^∘_𝑘,𝑞^∘_𝑘, are the true boundary coefficients.

Denote finite terms of the Fourier expansions on 𝜕𝑆_𝑅 in (66) and (67) by 𝑢^𝑀=𝑢^𝑀𝜌,𝜃_𝜕𝑆𝑅=𝑎^∘₀+^𝑀_𝑘=1𝑎^∘_𝑘cos𝑘𝜃+𝑏^∘_𝑘sin𝑘𝜃,𝑢^𝑀_𝜌=𝜕𝜕𝜌𝑢^𝑀𝜌,𝜃_𝜕𝑆𝑅=𝑝^∘₀+^𝑀_𝑘=1𝑝^∘_𝑘cos𝑘𝜃+𝑞^∘_𝑘sin𝑘𝜃; (69) also denote the circle ℓ_𝑟={(𝜌,𝜃)∣𝜌=𝑟,0≤𝜃≤2𝜋}. For 𝜕𝑆_𝑅=ℓ_𝑅, for the solution (66), the Sobolev norms are defined as |𝑢|_0,ℓ𝑅=𝜋𝑅𝑎^∘₀²+^∞_𝑘=1𝑎^∘_𝑘²+𝑏^∘_𝑘^21/2,|𝑢|_𝑝,ℓ𝑅=𝜋𝑅^∞_𝑘=1𝑘^2𝑝𝑎^∘_𝑘²+𝑏^∘_𝑘^21/2‖𝑢‖,𝑝≥1,_𝑝,ℓ𝑅=^𝑝_𝑘=0|𝑢|²_𝑘,ℓ𝑅^1/2. (70) We have the following lemma, whose proof can be found in Canuto et al. [33, 34].

Lemma 2. Let (64) be given, for 𝜕𝑆_𝑅=ℓ_𝑅; there exist the bounds of the remainders of (69) 𝑢−𝑢^𝑀_{𝑞,𝜕𝑆𝑅}1≤𝐶𝑀^𝑝−𝑞|𝑢|_{𝑝,𝜕𝑆𝑅}𝑢,0≤𝑞≤𝑝,_𝜌−𝑢^𝑀_{𝜌𝑞,𝜕𝑆𝑅}1≤𝐶𝑀^{𝑝−𝑞−1}𝑢_{𝜌𝑝−1,𝜕𝑆𝑅},0≤𝑞≤𝑝−1, (71) where 𝐶 is a constant independent of 𝑀.

Also denote the finite terms of the Fourier expansions on 𝜕𝑆_𝑅1 in (68) by 𝑢^𝑁=𝑢^𝑁𝜌,𝜃_{𝜕𝑆𝑅1}=𝑎^∘₀+^𝑁_𝑘=1𝑎^∘_𝑘cos𝑘𝜃+𝑏^∘_𝑘sin𝑘𝜃,𝑢^𝑁_𝜈=𝜕𝜕𝜈𝑢𝜌,𝜃_{𝜕𝑆𝑅1}=𝑝^∘₀+^𝑁_𝑘=1𝑝^∘_𝑘cos𝑘𝜃+𝑞^∘_𝑘sin𝑘𝜃. (72)

We can prove the following lemma similarly.

Lemma 3. Let (64) be given, for 𝜕𝑆_𝑅1=ℓ_𝑅1; there exist the bounds of the remainders of (72) 𝑢−𝑢^𝑁_{𝑞,𝜕𝑆𝑅1}1≤𝐶𝑁^𝑝−𝑞|𝑢|_{𝑝,𝜕𝑆𝑅1}𝑢,0≤𝑞≤𝑝,_𝜈−𝑢^𝑁_{𝜈𝑞,𝜕𝑆𝑅1}1≤𝐶𝑁^{𝑝−𝑞−1}𝑢_{𝜈𝑝−1,𝜕𝑆𝑅1},0≤𝑞≤𝑝−1, (73) where 𝐶 is a constant independent of 𝑁.

We have the following theorem.

Theorem 4. Let (64) and 𝑅≠1 hold. For the solution 𝑢_𝑁,𝑀 from the TM in (36), there exists the error bound 𝑢−𝑢_{𝑁,𝑀0,𝜕𝑆}1≤𝐶𝑀^𝑝|𝑢|_{𝑝,𝜕𝑆𝑅}+1𝑁^𝜎|𝑢|_{𝜎,𝜕𝑆𝑅1}, (74) where 𝐶 is a constant independent of 𝑁 and 𝑀.

Next, we study the errors of the interpolant solutions from (16) of the NFM with 𝜖=𝜖=0, ℒ_ext𝑅,𝑖Δ𝜃;𝜌_𝑖,𝜃_𝑖ℒ=0,𝑖=0,1,…,2𝑀,_int𝜌_𝑖,𝜃_𝑖;𝑅₁,𝑖Δ𝜃=0,𝑖=0,1,…,2𝑁, (75) where the uniform nodes Δ𝜃=2𝜋/(2𝑀+1) and Δ𝜃=2𝜋/(2𝑁+1). Equation (75) is equivalent to 𝑢_𝑀−𝑁𝑅,𝑖Δ𝜃;𝜌_𝑖,𝜃_𝑖=𝑢₀𝜃_𝑖𝑢,𝑖=0,1,…,2𝑀,_𝑀−𝑁𝜌_𝑖,𝜃_𝑖;𝑅₁,𝑖Δ𝜃=𝑢₀𝜃_𝑖,𝑖=0,1,…,2𝑁, (76) where 𝑢₀ and 𝑢₀ are given in (2) and (4). We have the following theorem.

Theorem 5. Let (64) and 𝑅≠1 hold. For the NFM with 𝜖=𝜖=0 and the uniform nodes, the interpolant solutions 𝑢_𝑀−𝑁 from (76) have the same error bound of (74) 𝑢−𝑢_{𝑁,𝑀0,𝜕𝑆}1≤𝐶𝑀^𝑝|𝑢|_{𝑝,𝜕𝑆𝑅}+1𝑁^𝜎|𝑢|_{𝜎,𝜕𝑆𝑅1}, (77) where 𝐶 is a constant independent of 𝑁 and 𝑀.

5. Numerical Experiments

5.1. IFM and Its Conservative Schemes

In this paper, we choose the NFM with 𝜖=𝜖=0, which is equivalent to the IFM, and its conservative schemes of [15]. For (1) with symmetry, the explicit interior solution (24) is simplified as 𝑢^𝐶_𝑀−𝑁𝜌,𝜃=𝑎₀𝑅−𝑅ln𝑅₁𝑝₀+𝑅2^𝑀_𝑘=11𝑘𝜌𝑅^𝑘𝑝_𝑘+𝑅cos𝑘𝜃₁2^𝑁_𝑘=11𝑘𝑅₁𝜌^𝑘𝑝_𝑘cos𝑘𝜃,𝜌,𝜃∈𝑆. (78) In [17], when 𝑢∈𝐻²(𝑆), we may choose the field nodes to be located on the solution boundary for (78): (𝜌,𝜃)∈𝜕𝑆_𝑅 and (𝜌,𝜃)∈𝜕𝑆_𝑅1. Then we obtain two boundary equations of the conservative schemes of the IFM from (78), (2), and (4) ℒ^𝐶_ext𝑅,𝜃;𝜌,𝜃𝑅=−𝑅ln𝑅₁𝑝₀+𝑅2^M_𝑘=11𝑘𝑝_𝑘+𝑅cos𝑘𝜃₁2^𝑁_𝑘=11𝑘𝑅₁𝜌^𝑘𝑝_𝑘cos𝑘𝜃=0,(𝑟,𝜃)∈𝜕𝑆_𝑅,ℒ^𝐶_int𝜌,𝜃;𝑅₁,𝜃∶=𝑎₀−𝑎₀𝑅−𝑅ln𝑅₁𝑝₀+𝑅2^𝑀_𝑘=11𝑘𝜌𝑅^𝑘𝑝_𝑘+𝑅cos𝑘𝜃₁2^𝑁_𝑘=11𝑘𝑝_𝑘cos𝑘𝜃=0,𝑟,𝜃∈𝜕𝑆_𝑅1. (79) The coefficients 𝑝₀,𝑝_𝑘,𝑝_𝑘 are unknowns, and the total number of unknowns is 𝑀+𝑁+1. Based on [15], to bypass the pseudosingularity, we still choose 𝑀+𝑁+2 equations from (79) 𝑤_𝑖ℒ^𝐶_ext𝑅,𝑖Δ𝜃;𝜌_𝑖,𝜃_𝑖𝑤=0,𝑖=0,1,…,𝑀,_𝑖ℒ^𝐶_int𝜌_𝑖,𝜃_𝑖;𝑅₁,𝑖Δ𝜃=0,𝑖=0,1,…,𝑁, (80) where 𝜖≥0, 0≤𝜖<𝑅₁, Δ𝜃=2𝜋/(2𝑀+1) and Δ𝜃=2𝜋/(2𝑁+1). The weights 𝑤₀=1 and 𝑤_𝑖=√2 are defined for 𝑖≥1, based on the stability analysis in [17]. The overdetermined system of (80) is denoted by the linear algebraic equations 𝐅𝐱=𝐛, (81) where 𝐅∈𝑅^𝑚×𝑛 with 𝑛=𝑀+𝑁+1 and 𝑚=𝑀+𝑁+2. The traditional condition number and the effective condition number in [35] are defined by 𝜎Cond=^max𝜎_min‖𝐛‖,Cond_eff=𝜎_min‖𝐱‖, (82) where 𝜎_max and 𝜎_min are the maximal and the minimal singular values of the matrix 𝐅 in (81), respectively.

Next, we use the original IFM (i.e., the original NFM with 𝜖=𝜖=0). The particular solutions (78) are replaced by 𝑢_𝑀−𝑁𝜌,𝜃=𝑎₀−𝑅ln𝑅𝑝₀−𝑅₁ln𝜌𝑝₀+𝑅2^𝑀_𝑘=11𝑘𝜌𝑅^𝑘𝑝_𝑘+𝑅cos𝑘𝜃₁2^𝑁_𝑘=11𝑘𝑅₁𝜌^𝑘𝑝_𝑘cos𝑘𝜃,(𝑟,𝜃)∈𝑆. (83) In (83), both 𝑝₀, 𝑝₀ are also unknown variables, and the total number of unknowns is now 𝑀+𝑁+2. Then 𝑚=𝑛=𝑀+𝑁+2 in (81).

Consider the model problem with 𝑅=2.5 and 𝑅₁=1 and then shrink the interior hole 𝑆_𝑅1 by decreasing radius 𝑅₁ from 1 down to 10⁻⁴. This reflects that Laplace's equation may occur in an actually punctured disk, where there may be a very small hole but not as a solitary point. For the conservative schemes of the IFM, the errors, condition numbers, and the leading coefficients are listed in Tables 1 and 2, where 𝛿=𝑢−𝑢_𝑀−𝑁. For 𝑅₁=0.1,0.01,0.001,0.0001, the optimal results are marked in bold. We also note that when 𝑅₁ decreases, the errors decrease and the condition numbers increase. Table 2 lists the leading coefficients, 𝑝₀,𝑝₁, and 𝑝₁. All tables are computed by MATLAB with double precision.

Table 1 

The errors and condition numbers by the conservative schemes of the IFM, where 𝑅=2.5 and 𝛿=𝑢−𝑢_𝑀−𝑁.

Table 2 

The leading coefficients by the conservative schemes of the IFM, where 𝑅=2.5.

As for the computations by the original IFM, the errors, condition numbers, and the leading coefficients are listed in Tables 3 and 4, where only the optimal results are listed. Comparing Table 3 with Table 1, the differences in terms of errors and condition number are insignificant, but the effective condition numbers are much smaller by the original IFM. Strictly speaking, the conservative schemes satisfy the flux conservative law exactly, but the original IFM does not.

Table 3 

The errors and condition numbers by the original IFM, where 𝑅=2.5 and 𝛿=𝑢−𝑢_𝑀−𝑁.

Table 4 

The leading coefficients by the original IFM, where 𝑅=2.5.

5.2. The CTM and the BIE

By means of symmetry, we choose the simple particular solutions in the CTM 𝑢_𝑀−𝑁𝜌,𝜃;𝜌,𝜃=𝑎₀+^𝑀_𝑖=1𝜌𝑅^𝑖𝑎_𝑖cos𝑖𝜃+𝑎₀ln𝜌+^𝑁_𝑖=1𝑅₁𝜌^𝑖𝑎_𝑖cos𝑖𝜃,𝜌≤𝑅,𝜌≥𝑅₁, (84) where 𝑎_𝑖 and 𝑎_𝑖 are the true coefficients and (𝜌,𝜃) and (𝜌,𝜃) are the polar coordinates with the origins (0,0) and (−1,0), respectively. We have also carried out the computation by CTM and BIE and have given their results in Tables 5, 6, 7, and 8. Comparing Table 7 of the BIE with Table 3 of the original IFM, the errors and the condition numbers are the same, but the effective condition numbers are slightly different. Then we conclude that the performance of the original IFM and BIE is the same. For comparisons of different methods, we draw their curves of errors and condition numbers in Figures 2 and 3, and it is clear that CTM is the best.

Table 5 

The errors and condition numbers by the simple particular solutions of the CTM, where 𝑅=2.5 and 𝛿=𝑢−𝑢_𝑀−𝑁.

Table 6 

The leading coefficients by the CTM, where 𝑅=2.5.

Table 7 

The errors and condition numbers by the BIE, where 𝑅=2.5 and 𝛿=𝑢−𝑢_𝑀−𝑁.

Table 8 

The leading coefficients by the BIE, where 𝑅=2.5.

Figure 2 

The curves of ‖𝛿‖∞,𝜕𝑆 via 𝑅1 by the conservative schemes, the original IFM, and the CTM.

Figure 3 

The curves of Cond via 𝑅1 by the conservative schemes, the original IFM, and the CTM.

6. Concluding Remarks

To close this paper, let us make a few concluding remarks.

(1) By following [17] for the NFM, we propose the interior field method (IFM). Since all boundary methods can be applied to any annular domains, they may be used for circular domains with circular holes; in this paper, we employ the first kind boundary integral equation (BIE) in [30] and the collocation Trefftz method (CTM) in [6]. The relations of expansion coefficients among NFM, IFM, BIE, and CTM are found. The intrinsic relations among them are discovered, to show that the IFM and the BIE are special cases of CTM. Section 2 yields an in-depth overview of four methods for circular domains with circular holes.

(2) For the NFM, some stability analysis in [17] was made for concentric circular boundaries. The error analysis of the NFM is challenging. Sections 3 and 4 are devoted to the error analysis of the NFM. In Section 3, a preliminary analysis is provided. In Section 4, for the special NFM with 𝜖=𝜖=0, the error bounds are provided without proof. The optimal convergence rates can be achieved. The error analysis is important and valid in wide applications, because the special NFM offers the best numerical performance in convergence and stability; see [17].

(3) Numerical experiments are carried out for a challenging problem of the actually punctured disks. We choose NFM, IFM, CTM, and BIE and their conservative schemes. Numerical results are reported from 𝑅₁=1 down to 𝑅₁=10⁻⁴. Note that the popular methods, such as the finite element method (FEM), the finite difference method (FDM), and the boundary element method (BEM), may fail to handle this problem. The actually punctured disks may be regarded as a kind of singularity problems, and the local mesh refinements and other innovations of FEM, FDM, and BEM are indispensable. However, their algorithms are complicated and troublesome; see [5]. Consequently, the computation of this paper enriches the boundary methods [6].

(4) Numerical comparisons of different methods are imperative in real application. Though their numerical performances are basically the same, the CTM is best in accuracy, stability, and simplicity of algorithms. Moreover, the CTM can always circumvent the degenerate scale problems encountered in NFM, IFM, and BIE. More importantly, the CTM can be applied to any shape domains and singularity problems (see [5, 6]). In summary, three goals motivated have been fulfilled.

Acknowledgments

The authors are grateful to T. Wriedt and J. T. Chen for their valuable comments and suggestions and indebted to I. S. Lin for the computation in this paper.