#### Abstract

The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.

#### 1. Introduction

The meshless method is an important numerical method after the finite element method, which has been widely used in scientific engineering calculation [1–5]. Based on different methods of constructing the shape function and problem discretization, scholars have proposed a series of meshless methods, such as the smoothed particle hydrodynamics (SPH) method [6, 7], element-free Galerkin (EFG) method [8], the singular boundary method (SBM) [9], radial basis functions [10, 11], boundary knot method (BKM) [12], meshless local Petrov–Galerkin (MLPG) method [13], complex variable element-free Galerkin (ICVFG) method [14, 15], point interpolation method (PIM) [16], method of fundamental solutions [17], and dimension splitting meshless method [18].

The traditional least-squares method (LSM) is a commonly used method in numerical computation, which obtains the optimal solution by seeking the minimum deviation of all data [19, 20]. When studying surface fitting, based on the LSM, Lancaster and Salkauskas proposed the moving least-squares (MLS) method [21], which is one of the most important methods to construct the trial function in the meshless method. The meshless method based on the MLS method has high numerical accuracy [22, 23]. However, the shape function of the MLS method does not satisfy the property of the Kronecker *δ* function [24], which increases the difficulty of dealing with the boundary conditions, and also raises a lot of extra computational burden. The meshless method of the MLS method class cannot apply essential boundary conditions directly and simply as the finite element method [25–27]. To solve this problem, by using the singular weight function, Lancaster and Salkauskas proposed the interpolation moving least square (IMLS) method [21], in which the shape function meets the interpolation characteristics at the nodes. The meshless method based on the IMLS method can directly and simply impose the essential boundary conditions [28], which improve calculation accuracy and efficiency. Based on the IMLS method, different meshless methods are proposed to solve various problems [29–33].

Based on the MLS method and boundary integral equation method, the boundary element-free method (BEFM) was developed by Cheng and Peng [34]. The BEFM is a meshless boundary integral equation method. Compared with the boundary point method, the boundary integral equation adopted in BEFM is a common and more regular form, and there is need neither to set Gauss points in the neighbourhood of the source point nor to choose the so-called “evaluation point,” so the solution process of the BEFM is more straightforward. The BEFM uses the real solutions of the node variables as the unknown quantities, while the boundary point method and the local boundary integral equation method both use the approximate solutions of the node variables as the unknown, so the BEFM is a direct numerical mesh-free method of the boundary integral equation and has greater computational precision [35–38].

In the BEFM, the essential boundary conditions are directly imposed. However, because the shape function of the MLS method does not satisfy the interpolation characteristics at the nodes, the essential boundary conditions are only approximately satisfied, which may lead to reduction in calculation accuracy. In order to improve this deficiency, by taking into account the interpolation characteristics of the IMLS method, Ren et al. also proposed the interpolating boundary element-free method (IBEFM) [39], which has higher calculation accuracy than the BEFM [40, 41]. By using the improved IMLS method with a nonsingular weight function, Wang et al. also developed an enhanced IBEFM for the potential problems [42]. As a direct numerical mesh-free method, the IBEFM has both the advantages of high accuracy and the ability to directly and accurately apply boundary conditions.

The BEFM has been widely used in various fields [37, 39, 43]. However, there is still little research on the error estimation of the BEFM. Based on the error estimation of the IMLS method and related matrix theory, this paper will study the error estimation of the IBEFM for the two-dimensional potential problems. The relationship between the errors and the influence radius and the condition number of the coefficient matrix is studied, and some numerical examples are presented to verify the correctness of the theoretical results.

#### 2. The IBEFM for the Potential Problems

Consider the following two-dimensional Poisson equation,with the Dirichlet boundary condition,and the Neumann boundary condition,where is the problem field, represents the potential of the field function, is the given source function, is the boundary of satisfying , denotes the known potential function on the Dirichlet boundary , is the known potential gradient on the Neumann boundary , and is the cosine of the outer normal direction on the boundary .

The integral equation can be obtained using the weighted residual method:where and represent, respectively, the field point and source point and is the basic solution of the Poisson equation (1) satisfyingwhere is the Dirac function. Then, it follows that

Performing the partial integration on (4) and using the properties of the basic solution of the Poisson equation, when the source point is within , the following boundary integral equation can be obtained:when the source point is on the boundary , and the boundary integral equation iswhere is a free term determined by the geometry of the point , andwhere and according to the boundary integration direction, and are the angles between the boundary tangent and the coordinate axis away from the corner point and near the corner point, respectively.

For the two-dimensional potential problem, the points on the boundary can be seen as points in one-dimensional space. can be expressed by curvilinear coordinates , that is . In this article, the curvilinear coordinates are taken as the arc length of the curve. Let denote the coordinate set of all nodes on the boundary, where represents the total number of nodes. Suppose the arc length coordinate of to be , and this node is also called a node . The union of the influence domains of all nodes must cover the entire boundary, and for , it is assumed that the influence domain iswhere is the influence radius of node .

Suppose are the nodes with the influence domain covering the . Then, from the IMLS method, the approximation functions can be obtained as follows:where and .

Substituting (13) and (14) into (10), it follows that

To calculate the definite integral, the boundary is divided into some integral subdomains , . is the total number of boundary integral subdomains, and only one point of any adjacent subdomains is connected. These integral subdomains are independent of the nodes distribution on the boundary, and they are only used to calculate the numerical integral.

To calculate , the field is divided into integral subfields. is the area of the subfield , and is the number of Gauss integral points in the integral subfield . is the integral weight coefficient corresponding to the -th Gauss integral point, and is the value of . Therefore, equation (15) can be written as

By substituting each node on the boundary into (16) and combining the equations, the following equation can be obtained:where , , , and

In the calculation of the boundary integral equation (16), in most cases, the boundary source point is not on the integral subdomain , and the integration has no singularity. The Gauss integration method can be used directly. If the source point coincides with the integration point, the basic solution will lead to the integration singularity. At this time, the Gauss integration point can be rearranged to eliminate the singularity.

Because the shape function of the IMLS method satisfies the property of the Kronecker *δ* function, the boundary condition can be easily and directly applied. Substituting the potential and potential gradient at the nodes on the boundary into (17), the potential and potential gradient at all the nodes on the boundary can be solved. Then, from (9), the potential at can be obtained as

Calculating differential for (19), the potential gradient of the interior point can be obtained:

The abovementioned equation is the IBEFM for two-dimensional potential problems.

#### 3. Error Estimations of the IBEFM for Two-Dimensional Potential Problems

For simplicity, let be the set of all nodes on the Dirichlet boundary, and be the set of all nodes on the Neumann boundary. For , assume that there is a constant such that , and the basis functions used in the IMLS method are complete polynomials of order .

Since there is no intersection between and , then in (17), when the -th element in is unknown, the -th element in must be known; otherwise, when the -th element in is known, the -th element in must be unknown. By shifting the term, all the unknown quantities are moved to the left end, and the known quantities are moved to the right end. Suppose the unknown quantities to be solved at the right end be , and its coefficient matrix be , and at the right end, the known column vector is denoted by after the combining operation. Then, (17) can be transformed into the following linear equation:where is an matrix and and are column vectors.

Suppose and are the numerical solutions of the potential and potential gradient obtained by the IBEFM. Since the IMLS method satisfies interpolation characteristics, it follows on the Dirichlet and Neumann boundaries that

Let be the solution of the linear system (21), that is,

Let and be the exact solutions to the potential and potential gradient, respectively. Also, let

Therefore, at the boundary node, the following theorem exists between the exact solution and the numerical solution of the IBEFM.

Theorem 1. *When the boundary is sufficiently smooth and the radius of the influence domain of the boundary node is small enough, there exists a constant independent with such thatwhere represents the maximum norm of a vector (matrix) and*

*Proof. *Letwhere and are the approximation functions obtained by the IMLS method.

Then, from the error estimates of the IMLS method [28, 44, 45], there existIt follows from (10) thatwhere is the source point on the boundary .

Then,where is a boundary node.

The same numerical integration is used for (16) and (31). Also, using the interpolation properties of the IMLS method, we havewhereand is an column vector with the element given byBy adopting the same shifting arrangement of (21) for (32), there existswhereAccording to matrix theory, it follows from (23) and (35) thatwhere represents the maximum norm of the vector, e.g., .

From (28), (29), and (34), we haveThen,whereWhen the Gaussian point and the source point do not coincide, it can be seen from equations (6) and (7) that the basic solutions are bounded. Also, from the errors of the IMLS method [28, 45], there exists constant independent of such thatThen, on the boundary, when or is given, the expression or must be bounded. Thus, it can be seen from (34)–(36) that the element of the vector must be a bounded value that is not a constant equal to 0. From (38), when the radius of the influence domain is small enough, the element value of is much smaller than that of , which is almost negligible. Then, the theorem can be directly obtained from (39).

When the error caused by numerical integration is ignored, the following error estimate can be obtained.

Theorem 2. *Suppose is the solution of (1) determined by (9) and is the numerical solution of the IBEFM solved from (19). When the boundary is sufficiently smooth and the radius is small enough, there is a constant independent of such that*

*Proof. *From (19) and (9), we haveThen,From the errors of the IMLS method [28, 45], there exists constant independent of such thatFrom (25), it follows thatThen, using trigonometric inequality and Hölder’s inequality and substituting (28), (29), (46), and (47) into (44), the theorem holds.

The value of contained in Theorem 2 is determined by the value of the potential function and its potential gradient on the boundary. Therefore, when the solution of the problem (1) is sufficiently smooth, there are three main aspects that affect the errors of the numerical solution: the size of the influence domain of nodes, that is, the density of nodes; the order of the complete polynomials basis function of the IMLS method; and the condition number of the coefficient matrix .

#### 4. Numerical Examples

Two numerical examples will be given to verify the error estimation of the interpolated boundary element-free method in this section. The linear basis function is used in all calculations. The radius of influence domain is , where is the arc length between two adjacent nodes.

*Example 1. *Consider the following Laplace equation:where is a circular domain with boundary : . The essential boundary condition isand the derivative boundary condition is are the known functions determined by the following exact solution:When 30 regular nodes are used (as shown in Figure 1), in Figure 2, the exact and numerical results on the inner circle with radii of 0.5, 0.7, and 0.9 are given. From the figure, it can be seen that the numerical solution obtained by the IBEFM has high accuracy. Figure 3 shows the absolute errors of the solutions obtained by the BEFM and IBEFM on the half circle with a radius 0.5, respectively, in which the BEFM and IBEFM adopt the same boundary integral grid. It can be seen that the solutions of the IBEFM have better accuracy, which also shows that the IBEFM has better advantages than the BEFM. The MATLAB programs of the BEFM and IBEFM run 100 times, respectively, and their average CPU time is 2.2403 s and 1.9519 s. This shows that the IBEFM has high computational efficiency.

To study the convergence of the IFEFM with respect to the radius , when 10, 20, 30, 40, and 50 regular nodes are used, the corresponding absolute errors between the analytical and numerical solutions at points and are as shown in Figures 4 and 5, respectively. As can be seen from these figures, the solution of the IBEFM has a good convergence order. It has a better convergence order than the error estimate given by this paper, which also shows the correctness of the theorem.

To study the influence of the condition number of the coefficient matrix on the error, we apply the random function “” of MATLAB software to randomly generate 50 kinds of irregular node distribution with 30 nodes. Then, under each irregular node distribution, the IBEFM is used to solve the numerical solutions at internal points and .

Figures 6 and 7 show, respectively, the changes of the absolute errors at and for the different matrix condition numbers. As can be seen from the figure, although the error is not strictly worse as the condition number becomes more massive, on the whole, there is a great correlation between the condition number and the error. When the value of the condition number becomes significantly larger, the error will deteriorate. This means that when the condition number of the coefficient matrix becomes considerably larger and the error will become larger, which is consistent with the theoretical results in the paper.

*Example 2. *Consider the Poisson equation with a square temperature field:The Dirichlet and Neumann boundary conditions are, respectively,where are known functions obtained from the exact solution:When 40 regular nodes are used (as shown in Figure 8), Figure 9 shows the exact and numerical solutions on the lines and . From this figure, it can be seen that the numerical solutions are very consistent with the analytical results, which also shows that the solution of the IBEFM has good accuracy.

In Figure 10, the absolute errors of the numerical solutions obtained by BEFM and IBEFM on the line are given. It can be seen that the solution of the IBEFM still has good accuracy, which also shows the superiority of the IBEFM. The MATLAB programs of the BEFM and IBEFM are repeatedly run 100 times, and their corresponding average CPU time is 3.0335 s and 3.0170 s, which also shows that the IBEFM has high computational efficiency.

In order to study the convergence of the IFEFM with respect to the radius , when the boundary nodes distributions of 16, 24, 40, and 60 nodes are, respectively, used, the corresponding absolute errors for different influence domain radii at the interior point and are given in Figures 11 and 12, respectively. It can be seen from these figures that the solution of the IBEFM has a good convergence order, which is consistent with the theoretical results.

#### 5. Conclusions

The IBEFM has both the advantages of the high accuracy of the boundary element method and meshless method, and it can also directly apply the boundary conditions. Based on the error estimation of the IMLS method and related matrix theory, this paper studies the error estimation of the IBEFM for two-dimensional potential problems. The relationship between the error and the influence radius and the condition number of the coefficient matrix is studied. The theoretical results show that the upper error limit of the IBEFM solution will decrease with the decrease in the influence domain radius of the boundary nodes and rise with the increase in the condition number of the coefficient matrix. Two numerical examples are provided. The numerical results are in good agreement with the theoretical results in this paper, which verifies the correctness of the theoretical results in the article.

#### Data Availability

The data used to support the findings of this study are available within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11702153), the Natural Science Foundation of Zhejiang Province, China (Grant nos. LY20A010021 and LY20G030025), the Natural Science Foundation of Ningbo City, China (Grant nos. 2018A610196, 2019A610095, and 2019A610048), and National Social Science Foundation of China (18BGL184).