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The interpolating moving least-squares (IMLS) method is discussed in detail, and a simpler formula of the shape function of the IMLS method is obtained. Then, based on the IMLS method and the Galerkin weak form, an interpolating element-free Galerkin (IEFG) method for two-point boundary value problems is presented. The IEFG method has high computing speed and precision. Then error analysis of the IEFG method for two-point boundary value problems is presented. The convergence rates of the numerical solution and its derivatives of the IEFG method are presented. The theories show that, if the original solution is sufficiently smooth and the order of the basis functions is big enough, the solution of the IEFG method and its derivatives are convergent to the exact solutions in terms of the maximum radius of the domains of influence of nodes. For the purpose of demonstration, two selected numerical examples are given to confirm the theories.

Conventional computational methods, such as the finite element method (FEM) and the boundary element method (BEM), cannot be applied well to some engineering problems. For the extremely large deformation and crack growth problems, the remeshing technique must be used. Meshless methods have been developed in recent years. The most important common feature of meshless methods is that the trial function is constructed from a set of nodes with no meshing at all. Then some complex problems, such as the large deformation and crack growth problems, can be simulated with the method without the remeshing techniques [

Many kinds of meshless methods have been proposed, such as element-free Galerkin (EFG) method [

The element-free Galerkin (EFG) method is one of the most powerful meshless methods [

The EFG method is constructed based on the moving least-squares (MLS) approximation. The shape function that is formed with MLS approximation can obtain a solution with high precision. However, a disadvantage of the MLS approximation is that its shape function does not satisfy the property of Kronecker

To overcome this problem, Most and Bucher designed a regularized weight function with a regularization parameter

Error estimation for meshless method is certainly important to increase the reliability and reduce the cost of numerical computations in many engineering problems. Some error analyses have been done for the MLS approximation and the meshless method based on it [

Two-point boundary value problems occur in applied mathematics, theoretical physics, engineering, and optimization theory. Since it is usually impossible to obtain analytical solutions to two-point boundary value problems met in practice, these problems must be attacked by numerical methods. Many numerical methods have been proposed for the solutions of these problems, such as the Galerkin and collocation methods, boundary value method, variational iteration method, and meshless method based radial basis functions.

In this paper, the IMLS method is discussed in detail. The computation of the shape function of this paper is simpler than the corresponding expression presented by Lancaster and Salkauskas. Then based on the IMLS method of this paper and the Galerkin weak form, an IEFG method for two-point boundary value problems is presented. Since the shape function of the IMLS method satisfies the property of Kronecker

Then the error analysis of the IEFG method for two-point boundary value problems is presented. The convergence rates of the numerical solution and its derivatives of the IEFG method are presented. The theoretical results show that if the exact solution is smooth enough and the order of the polynomial basis functions is big enough, then the solution of the IEFG method and its derivatives are convergent to the exact solutions in terms of the maximum radius of the domains of influence of nodes. For the purpose of demonstration, some selected numerical examples are given to confirm the theory.

Let

Let

For a given point

Let

Define a local approximation as

Then define a functional as

By minimizing the functional

From (

In [

If the weight function of (

Then (

Equation (

From (

Then the local approximation function is obtained as

Thus the global interpolating approximation function of

Equation (

Equations (

Consider the following two-point boundary value problem:

The Galerkin weak form of (

From the IMLS method, the unknown solution

Substituting (

Since the shape function of the IMLS method satisfies the property of Kronecker

To evaluate the integrals in (

The numerical procedure of IEFG method for two-point boundary value problems is listed as follows.

Looping over background cells to determine all Gauss points to find out its location and weight.

Looping over Gauss points for integration of (

determine the support domain for specified Gauss point and select neighboring nodes based on a defined criterion;

compute shape function and its derivatives for each Gauss point;

assemble the contribution of each Gauss point to form system equation.

Enforcing essential (displacement) boundary conditions.

Solving the system equation to obtain nodal displacements.

Thus the IEFG method is presented for two-point boundary value problems.

In this section, the error analysis of the IEFG method for two-point boundary value problems is presented. The convergence rates of the numerical solution and its derivatives of the IEFG method are presented.

Let

Define the

The Sobolev space

Define

Then the variational problem in accordance with (

Suppose that the bilinear form

Since the shape function of the IMLS method satisfies the property of Kronecker

Then the IEFG method according to (

Obviously, there exists

Suppose that

In fact, the approximation function of the IMLS method provides a linear operator

If

Let

Suppose that

From Theorem

By using the Aubin-Nitsche method, the following error estimates in the

Suppose that

For

Let

For arbitrary

It follows from (

In (

From (

To study the error estimates of the high derivatives of the numerical solution of the IEFG method, we need to firstly prove the following inverse estimates of the function in the shape function space.

Suppose that

From (

It is obvious that there exist bounded functions

Then we have

Then it follows from (

From (

From (

Then from (

By using the inverse estimates, the following error estimates of the high derivatives of the numerical solution can be obtained.

Suppose that

From Theorem

In this section, two numerical examples are presented to show the applicability and the theoretical error estimates of the IEFG method of this paper. In our numerical computation, the nodes are arranged regularly, and the radius of the domain of influence of node

The first example considered is a linear elastostatics problem. A one-dimensional bar of unit length is subjected to a body force of magnitude

The analytical solution to the above problem is

Let

The analytical and numerical displacement.

The analytical and numerical strain.

The absolute errors of the displacements obtained by the IEFG and EFG methods are shown in Figure

The absolute error of the EFG and IEFG methods.

The error norms of

Error norms of

Error norms of

The second example considers the following equilibrium equation:

When the quadratic basis functions and 21 regular distributed nodes are used, the analytical and numerical solutions of

Analytical and numerical solutions of

Analytical and numerical solutions of

The absolute errors of

The absolute error of the EFG and IEFG methods.

The convergence rates of

Error norms of

Error norms of

In this paper, the IMLS method is discussed in detail. The computation of the shape function of this paper is simpler than the corresponding expression presented by Lancaster and Salkauskas. Then, based on the IMLS method of this paper and the Galerkin weak form, an IEFG method for two-point boundary value problems is presented. Since the shape function of the IMLS method satisfies the property of Kronecker

The error analysis of the IEFG method for two-point boundary value problems is presented. The convergence rates of the numerical solution and its derivatives of the IEFG method are presented. The theories of this paper show that if the analytical solution is sufficiently smooth and the order of the polynomial basis functions is big enough, then the solution of the IEFG method and its derivatives are convergent to the analytical solutions in terms of the maximum radius of the domains of influence of nodes. For the purpose of demonstration, some selected numerical examples are given to confirm the theories.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (no. 11171208), Shanghai Leading Academic Discipline Project (no. S30106), and the Natural Science Foundation of Ningbo (no. 2013A610103).