#### 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 are located outside of the solution domain . 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: , if the solutions are smooth enough to satisfy and , where is the normal derivative and are the Sobolev spaces; see the proof in [17]. It is discovered numerically that when the field nodes , the NFM provides small errors and the smallest condition numbers, compared with all . 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 () 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 , without a need of the FS expansions, called the matrix method [24]. Hence, the 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 and with radii and , respectively. Let , and the eccentric circular domains and may have different origins. Hence . Choose the annular solution domain with the exterior and the interior boundaries and , respectively. The following Dirichlet problems are discussed by Palaniappan [26]: In [11], and and the origins of and are located at and , respectively. In this paper, we fix , while may be infinitesimal; that is, .

On the exterior boundary , there exist the approximations of Fourier expansions: where , and are coefficients. On the interior boundary , we have similarly where , and are coefficients. In (2)–(5), and are the polar coordinates of and with the origins and , respectively, and and are the exterior normals of and , respectively. The Dirichlet, the Neumann conditions, and their mixed types on may be given with known coefficients.

In , denote two nodes and , where , and . Then and . The FS of Laplace's equation is given by . From the Green representation formula, we have different formulas for different locations of the field nodes : where and two kinds of series expansions of the FS are given by (see [27]) where and . Then we have two kinds of derivative expansions of FS where the superscripts “” and “” designate the exterior and interior field nodes , 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 where is the complementary domain of . Substituting the Fourier expansions (7)–(8) into (9) yields the basic algorithms of NFM, where the exterior normal of is given by . In the Green formula (9), the field point is supposed to locate outside of the solution domain 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: , if the solutions are smooth enough to satisfy and , where is the normal derivative and are the Sobolev spaces; see the rigorous proof in [17]. It is discovered numerically that when the field nodes , the NFM provides small errors and condition numbers and has been widely implemented in many engineering problems.

Denote two systems of polar coordinates by and with the origins and for and , respectively. There exist the following conversion formulas:

First, consider the exterior field nodes with . The first explicit algebraic equations of the NFM are given for the exterior field nodes (see [17]) Next, consider the interior field nodes with . The second explicit algebraic equations of the NFM are given for the interior field nodes (see [17])

Since one of Dirichlet or Neumann conditions is given on and , only coefficients in (2)–(5) are unknowns. We choose and field nodes located uniformly on the exterior and the interior circles, respectively, where , , , and . Denote the explicit equations (12) and (13) by We obtain discrete equations from (15) where the corresponding coordinates and are obtained from (10) and (11). Hence from (16), we obtain the following linear algebraic equations: where the matrices , the vector , and . The unknown coefficients can be obtained from (17), if the matrix 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: is expressed by For , from (2)–(5) and (7)-(8), (2.20) leads to (see [17]) where 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] Substituting (3) and (5) into (20) yields directly We may use (21) to remove one coefficient, say , By using (22), (12) and (13) lead to Also the interior solution (19) leads to Hence, the total number of unknown coefficients is reduced to . Based on the analysis in [15], to remain good stability, we still choose collocation nodes as in (16): where the weights , for , and . 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 and , the NFM remains valid for the field nodes ; that is, on and on and (23) and (24) hold. In fact, we may use (24) only, because (23) is obtained directly from the Dirichlet conditions on and , 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 where is the Euclidean distance. In (26), is an open arc, and each of its edges, , is assumed to be smooth. Let be the logarithmic capacity of . From the single layer potential theory [28–30], if , (26) can be converted to the first kind boundary integral equation (BIE), where is the unknown function and denote the normal derivatives along the positive and negative sides of . If , there exists a unique solution of (27), see [28]. As soon as is solved from (27), the solution ( of (26) can be evaluated by For the smooth solution , we have , where is the normal of . We may assume the Fourier expansions of on where , and are the coefficients. We have from [17] to give Note that the derivation of (31) in the first kind BIE is simpler, because we do not need the series expansions of and . 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]) where , and 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: Equation (19) can also be written as (32) with where 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.

To close this section, we describe the CTM. Denote the set of (32), and define the energy where and is the known function of Dirichlet boundary conditions. Then the solution of the Trefftz methods (TM) can be obtained by The TM solution also satisfies When the integral in (35) involves numerical approximation, the modified energy is defined as where is the numerical approximations of by some quadrature rules, such as the central or the Gaussian rule. Hence, the numerical solution is obtained by We may also establish the collocation equations directly from the Dirichlet condition to yield 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 is explored for eccentric circular boundaries. Consider the simple domains of , where and have the same origin. For the same origin of and , the same polar coordinates are used, and the general solutions in can be denoted by where are true coefficients and . Then their derivatives are given by When , from (41) and (42), we have Comparing (43) with (2) and (3), we have the following equalities of coefficients: where , and are the coefficients of the NFM in Section 2.1.

Also, when , from (41) and (42), we have Comparing (46) with (4) and (5), we have where , and are also the coefficients of the NFM in Section 2.1.

On the other hand, when , we have from the first original equation (12) Then for , we obtain the following equalities, based on the orthogonality of trigonometric functions: Similarly, from the second equation (13), Then for , we obtain Below, we prove that the true coefficients can be obtained directly from the NFM based on (50)–(52) for and on (54)–(56) for . 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 , there exists a unique solution of the special NFM with . Therefore, the true coefficients can be determined by the IFM uniquely.

First to show (50). The consistent condition is given by Equation (57) can also be obtained from (45) and (48). Equations (57) and (50) are equivalent if (i.e., ), which is also the necessary condition of nonsingularity of matrix in (17) [16]. Based on (57), the conservative schemes are proposed in [15]. Equation (54) is shown next. We have from (44) and (47) where we have used (45) and (48).

Equations (51) and (55) are shown below. Denote them in matrix form and denote from (44) and (47) where and are true expansion coefficients. Also denote from (45) and (48) By substituting (60) and (61) into (59), its left-hand side leads to The right-hand side of (59) leads to 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 , 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

The NFM with the field nodes (i.e., ) 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 be divided into two subdomains and with an interface boundary . We have and , where and . We assume that the true solutions have different regularities where and . Then there are different regularities on the boundary where and are the exterior normal to and , respectively. Therefore, the true solutions can be expressed by the Fourier expansions on where , are the true boundary coefficients. Similarly, we have where , are the true boundary coefficients.

Denote finite terms of the Fourier expansions on in (66) and (67) by also denote the circle . For , for the solution (66), the Sobolev norms are defined as We have the following lemma, whose proof can be found in Canuto et al. [33, 34].

Lemma 2. *Let (64) be given, for ; there exist the bounds of the remainders of (69)
**
where is a constant independent of . *

Also denote the finite terms of the Fourier expansions on in (68) by

We can prove the following lemma similarly.

Lemma 3. *Let (64) be given, for ; there exist the bounds of the remainders of (72)
**
where is a constant independent of . *

We have the following theorem.

Theorem 4. *Let (64) and hold. For the solution from the TM in (36), there exists the error bound
**
where is a constant independent of and .*

Next, we study the errors of the interpolant solutions from (16) of the NFM with , where the uniform nodes and . Equation (75) is equivalent to where and are given in (2) and (4). We have the following theorem.

Theorem 5. *Let (64) and hold. For the NFM with and the uniform nodes, the interpolant solutions from (76) have the same error bound of (74)
**
where is a constant independent of and . *

#### 5. Numerical Experiments

##### 5.1. IFM and Its Conservative Schemes

In this paper, we choose the NFM with , which is equivalent to the IFM, and its conservative schemes of [15]. For (1) with symmetry, the explicit interior solution (24) is simplified as In [17], when , we may choose the field nodes to be located on the solution boundary for (78): and . Then we obtain two boundary equations of the conservative schemes of the IFM from (78), (2), and (4) The coefficients are unknowns, and the total number of unknowns is . Based on [15], to bypass the pseudosingularity, we still choose equations from (79) where , , and . The weights and are defined for , based on the stability analysis in [17]. The overdetermined system of (80) is denoted by the linear algebraic equations where with and . The traditional condition number and the effective condition number in [35] are defined by where and are the maximal and the minimal singular values of the matrix in (81), respectively.

Next, we use the original IFM (i.e., the original NFM with ). The particular solutions (78) are replaced by In (83), both , are also unknown variables, and the total number of unknowns is now