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

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.


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][9][10][11][12][13][14].More references of NFM are also given in [10][11][12][14][15][16][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  ∈  2 () and  ] ∈  1 (), where  ] is the normal derivative and   () ( = 1, 2) 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],

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   1 with radii  and  1 , respectively.Let   1 ⊂   , and the eccentric circular domains   and   1 may have different origins.Hence 2 1 < .Choose the annular solution domain  =   \   1 with the exterior and the interior boundaries   and   1 , respectively.The following Dirichlet problems are discussed by Palaniappan [26]: 2   2 = 0 in ,  = 1 on   ,  = 0 on   1 . (1) In [11],  = 2.5 and  1 = 1 and the origins of   and   1 are located at (0, 0) and (− 1 , 0), respectively.In this paper, we fix  = 2.5, while  1 may be infinitesimal; that is,  1 ≪ 1.
On the exterior boundary   , there exist the approximations of Fourier expansions: where   ,   ,   , and   are coefficients.In (2)- (5),  and  are the polar coordinates of   and   1 with the origins (0, 0) and (− 1 , 0), respectively, and ] and ] are the exterior normals of   and   1 , respectively.The Dirichlet, the Neumann conditions, and their mixed types on   may be given with known coefficients.

The Interior Field Method.
In [17], we prove that when  ∈  2 () and  ] ∈  1 (), the NFM remains valid for the field nodes  ∈ ; that is,  =  on   and  =  1 on   1 and ( 23) and ( 24) hold.In fact, we may use ( 24) only, because ( 23) is obtained directly from the Dirichlet conditions on   and   1 , 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.

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 |x| is the Euclidean distance.In (26), ) is an open arc, and each of its edges, Γ  ( = 1, . . ., ), is assumed to be smooth.Let  Γ be the logarithmic capacity of Γ. From the single layer potential theory [28][29][30], if  Γ ̸ = 1, (26) can be converted to the first kind boundary integral equation (BIE), where V(x)(= ((x)/ − ) − ((x)/ + )) is the unknown function and / ± denote the normal derivatives along the positive and negative sides of Γ.If  Γ ̸ = 1, there exists a unique solution of (27), see [28].As soon as V(x) is solved from (27), the solution (x) (x ∈ Ω) of ( 26) can be evaluated by For the smooth solution , we have V(x) = 2(/]), where ] is the normal of Γ.We may assume the Fourier expansions of 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   (x, y)/ and   (x, y)/.This advantage is very important for elasticity problems, because the displacement conditions are much simpler than the traction ones.

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  IFM  ,  IFM  , . . .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 Γ =  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      −  −    0,Γ = min When the integral in (35) involves numerical approximation, the modified energy is defined as where ∫Γ is the numerical approximations of ∫ Γ by some quadrature rules, such as the central or the Gaussian rule.Hence, the numerical solution û− ∈  − 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).

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  =  = 0 is explored for eccentric circular boundaries.Consider the simple domains of  =   \   1 , where   and   1 have the same origin.For the same origin  of   and   1 , the same polar coordinates (, ) are used, and the general solutions in   \   1 can be denoted by 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  =  1 , 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  ≤  1 , 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  ≤  1 .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  ̸ = 1, there exists a unique solution of the special NFM with  =  = 0. Therefore, the true coefficients can be determined by the IFM uniquely.
We write these important results as a proposition.
Proposition 1.For the concentric circular domains, when  = + ̸ = 1, the leading coefficients are exact by the NFM, and the solution errors result only from the truncations of their Fourier expansions.

Error Bounds of the NFM with 𝜖 = 𝜖 = 0
The NFM with the field nodes  ∈  (i.e.,  =  = 0) located on the domain boundary is the most important application for Chen's publications (see [8][9][10][11][12][13][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  ext and  int with an interface boundary Γ 0 ∈ .We have  =  ext ∪  int ∪ Γ 0 and  ext ∩  int = 0, where  ext =   ∪ Γ 0 and  int =   1 ∪ Γ 0 .We assume that the true solutions have different regularities where  ≥ 2 and  ≥ 2. Then there are different regularities on the boundary where ] and ] are the exterior normal to  ext and  int , respectively.Therefore, the true solutions can be expressed by the Fourier expansions on
Lemma 2. Let (64) be given, for   = ℓ  ; 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   1 in (68) by We can prove the following lemma similarly.
Lemma 3. Let (64) be given, for   1 = ℓ  1 ; there exist the bounds of the remainders of (72) where  is a constant independent of .
We have the following theorem.
where  is a constant independent of  and .
where  is a constant independent of  and .
The traditional condition number and the effective condition number in [35] are defined by where  max and  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  =  = 0).The particular solutions (78) are replaced by In (83), both  0 ,  0 are also unknown variables, and the total number of unknowns is now  +  + 2. Then  =  =  +  + 2 in (81).
Consider the model problem with  = 2.5 and  1 = 1 and then shrink the interior hole   1 by decreasing radius  1 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  = − − .For  1 = 0.1, 0.01, 0.001, 0.0001, the optimal results are marked in bold.We also note that when  1 decreases, the errors decrease and the condition numbers increase.Table 2 lists the leading coefficients,  0 ,  1 , and  1 .All tables are computed by MATLAB with double precision.
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.where   and   are the true coefficients and (, ) and (, ) 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.

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  =  = 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  1 = 1 down to  1 = 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.

Figure 2 :
Figure 2: The curves of ‖‖ ∞, via  1 by the conservative schemes, the original IFM, and the CTM.

5. 2 .𝑎 1 Figure 3 :
Figure 3: The curves of Cond via  1 by the conservative schemes, the original IFM, and the CTM.

Table 1 :
The errors and condition numbers by the conservative schemes of the IFM, where  = 2.5 and  =  −  − .

Table 3 :
The errors and condition numbers by the original IFM, where  = 2.5 and  =  −  − .

Theorem 4 .
Let (64)and  ̸ = 1 hold.For the solution  , from the TM in (36), there exists the error bound

Table 4 :
The leading coefficients by the original IFM, where  = 2.5.

Table 5 :
The errors and condition numbers by the simple particular solutions of the CTM, where  = 2.5 and  =  −  − .

Table 6 :
The leading coefficients by the CTM, where  = 2.5.