MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 641592 10.1155/2014/641592 641592 Research Article The Error Estimates of the Interpolating Element-Free Galerkin Method for Two-Point Boundary Value Problems Wang J. F. 1, 2 Hao S. Y. 3 Cheng Y. M. 2 Peng Miaojuan 1 Ningbo Institute of Technology Zhejiang University Ningbo 315100 China zju.edu.cn 2 Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai 200072 China shu.edu.cn 3 School of Traffic and Transportation Lanzhou Jiaotong University Lanzhou 730070 China lzjtu.edu.cn 2014 942014 2014 23 12 2013 19 02 2014 9 4 2014 2014 Copyright © 2014 J. F. Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

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 [1, 2].

Many kinds of meshless methods have been proposed, such as element-free Galerkin (EFG) method , complex variable meshless method , mesh-free reproducing kernel particle Ritz method , radial basis function (RBF) method , finite point method (FPM) , meshless local Petrov-Galerkin (MLPG) method , reproducing kernel particle method (RKPM) , meshless manifold method , boundary element-free method (BEFM) , and local boundary integral equation (LBIE) method [41, 42].

The element-free Galerkin (EFG) method is one of the most powerful meshless methods . The EFG method can obtain a solution with high precision. Various problems have been successfully analyzed by the EFG method. By using the orthogonal function system with a weight function as the basis function, Zhang et al. presented the improved element-free Galerkin method , which has high computational efficiency. By combining the complex variable moving least-squares (CVMLS) approximation and the EFG method, Peng et al. presented the complex variable element-free Galerkin (CVEFG) method . Compared with the conventional EFG methods, the CVEFG method has greater computational precision and efficiency.

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 δ function. Then the EFG method based on the MLS approximation cannot apply the essential boundary conditions directly and easily. The essential boundary conditions need to be introduced by additional approaches, such as Lagrange multipliers  and penalty methods . However, for Lagrange multipliers, the corresponding discrete system will introduce additional unknowns which are not directly associated with the solution themselves. Furthermore, the banded structure of the matrix equation system is seriously worsened, as well as the conditioning properties. And, for penalty methods, the optimal value of penalty factor is hard to be set, which always affects the accuracy of the final solution.

To overcome this problem, Most and Bucher designed a regularized weight function with a regularization parameter ε , by which the MLS approximation can almost fulfill the interpolation with high accuracy . Most and Bucher enhanced the regularized weighting function to obtain a true interpolation MLS approximation . Another possible solution for this problem is the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas . The IMLS method is established based on the MLS approximation by using singular weight functions. The shape function of the IMLS method satisfies the property of Kronecker δ function. Thus the meshless methods based on the IMLS method can apply the essential boundary condition directly without any additional numerical effort. Based on the IMLS method, Kaljević and Saigal  presented an improved EFG method, in which the boundary condition is applied directly. Ren improved the expression of the shape function of the IMLS method and then presented the interpolating element-free Galerkin (IEFG) method and the interpolating boundary element-free (IBEFG) method for two-dimensional potential and elasticity problems . To overcome the singularity of the weight function in the IMLS method, Wang et al. presented the improved interpolating moving least-squares (IIMLS) method, in which nonsingular weight function is used . In the IEFG method, the essential boundary conditions are applied directly and easily, and the number of unknown coefficients in the trial function of the IMLS method is less than that in the trial function of the MLS approximation. Therefore, the IEFG method based on the IMLS method has high computational efficiency and precision.

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 . Krysl and Belytschko studied the convergence of the continuous and discontinuous shape functions of the second-order elliptic partial differential equations . Chung and Belytschko proposed the local and global error estimates using the difference between the values of the projected stress and these given directly by the EFG solution . Dolbow and Belytschko studied the integration error . Gavete et al. presented a procedure to estimate the error in elliptic equations and then proposed a posteriori error approximation [78, 79]. R. J. Cheng and Y. M. Cheng studied the error estimate of the finite point method based on the MLS approximation  and the error estimates of element-free Galerkin method for potential and elasticity problems [81, 82]. For the IEFG method, since the essential boundary condition is applied directly, then the error estimate of the IEFG method is no doubt different from that of the EFG method. However, until now the error analysis of the IEFG method has not been seen in the recent literature.

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 δ function, then the IEFG method can apply the essential boundary condition directly. And as the number of the coefficients in the trial function of the IMLS method is less than that in the MLS approximation, then fewer nodes are selected in the entire domain in the IEFG method than in the conventional EFG method. Hence, the IEFG method has high computing speed and precision.

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.

2. Interpolating Moving Least-Squares Method

Let X = { x 1 , x 2 , , x M } be a set of all nodes in the bounded domain Ω R n , where M is the number of nodes. The parameter ρ I denotes the radius of the domain of influence of node x I , and · denotes the Euclidean norm. The domain of influence of x I is defined by (1) Ω I = { x x - x I ρ I , x Ω } .

Let u ( x ) be the function of the field variable defined in Ω . The approximation function of u ( x ) is denoted by u h ( x ) . In this paper, the following weight function is used: (2) w ( x , x I ) = m I ( x ) x - x I ρ I - α , where m I ( x ) = m ( x - x I ) C l ( Ω ) satisfying m I ( x ) > 0 for x - x I < ρ I and m I ( x ) = 0 for x - x I ρ I . In general, m I ( x ) can be chosen to be any weight function of the MLS approximation.

For a given point x , the inner product is defined as (3) ( f , g ) x = I = 1 n w ( x , x I ) f ( x I ) g ( x I ) , f , g C 0 ( Ω ) , where n is the number of nodes whose compact support domains cover x .

Let p 0 ( x ) 1 and let p 1 ( x ) , , p m ( x ) be given basis functions. Then a new set of basis functions will be generated from these given basis. Let (4) p ~ 0 ( x ; x - ) = 1 [ I = 1 n w ( x , x I ) ] 1 / 2 , (5) p ~ i ( x ; x - ) = p i ( x - ) - S p i ( x ) , i = 1,2 , , m , where S is a linear operator defined as (6) S p i ( x ) = I = 1 n v ( x , x I ) p i ( x I ) , (7) v ( x , x I ) = w ( x , x I ) J = 1 n w ( x , x J ) .

Define a local approximation as (8) u h ( x , x - ) = p ~ 0 ( x ; x - ) a 0 ( x ) + i = 1 m p ~ i ( x ; x - ) a i ( x ) , where x - is the point in the domain of influence of x    , and a i ( x ) ( i = 0,1 , , m ) are the unknown coefficients of basis functions.

Then define a functional as (9) κ = I = 1 n w ( x , x I ) [ u h ( x , x I ) - u I ] 2 , where w ( x , x I ) shown in (2) is a weight function with compact support, x I are the nodes with domains of influence that cover the point x , and u I = u ( x I ) .

By minimizing the functional κ , we have (10) ( u ( · ) - u h ( x , · ) , p ~ 0 ) x = 0 , (11) ( u ( · ) - u h ( x , · ) , p ~ i ) x = 0 , i = 1,2 , , m .

From (4) and (10), we have (12) p ~ 0 ( x ; x - ) a 0 ( x ) = I = 1 n v ( x , x I ) u = S u . Then (11) reduces to (13) i = 1 m a i ( x ) ( p ~ i , p ~ j ) x = ( u - S u , p ~ j ) x , j = 1,2 , , m .

In , the unknown parameters a i ( x ) ( i = 1,2 , , m ) are solved from (13). In fact, (13) can be simplified.

If the weight function of (2) is used, x Ω , it can be proved that there exists (14) ( S u , p ~ i ) x = 0 , i = 1,2 , , m .

Then (13) can be simplified as (15) i = 1 m a i ( x ) ( p ~ i , p ~ j ) x = ( u , p ~ j ) x , j = 1,2 , , m .

Equation (15) is simpler than the expression (13) presented in  and can be rewritten as (16) A ( x ) a ( x ) = F W ( x ) u , where (17) a T ( x ) = ( a 1 ( x ) , a 2 ( x ) , , a m ( x ) ) , (18) u T = ( u 1 , u 2 , , u n ) , (19) A ( x ) = F W ( x ) F T ( x ) , (20) F ( x ) = [ p ~ 1 ( x ; x 1 ) p ~ 1 ( x ; x 2 ) p ~ 1 ( x ; x n ) p ~ 2 ( x ; x 1 ) p ~ 2 ( x ; x 2 ) p ~ 2 ( x ; x n ) p ~ m ( x ; x 1 ) p ~ m ( x ; x 2 ) p ~ m ( x ; x n ) ] , and F W ( x ) = ( ϖ k J ( x ) ) m × n is a m × n matrix, and (21) ϖ k J ( x ) = { w ( x , x J ) p ~ k ( x ; x J ) , x x J ; I = 1 , I J n w ( x J , x I ) [ p k ( x J ) - p k ( x I ) ] , x = x J .

From (16), we have (22) a ( x ) = A - 1 ( x ) F W ( x ) u .

Then the local approximation function is obtained as (23) u h ( x , x - ) = S u + i = 1 m a i ( x ) p ~ i ( x ; x - ) .

Thus the global interpolating approximation function of u ( x ) can be obtained as (24) u h ( x ) = S u + i = 1 m a i ( x ) g i ( x ) 𝚽 T ( x ) u , where 𝚽 T ( x ) is the shape function vector as (25) 𝚽 T ( x ) = v T + p T ( x ) A - 1 ( x ) F W ( x ) , (26) v T = ( v ( x , x 1 ) , v ( x , x 2 ) , , v ( x , x n ) ) , (27) p T ( x ) = ( g 1 ( x ) , g 2 ( x ) , , g m ( x ) ) , (28) g i ( x ) = p i ( x ) - S p i ( x ) .

Equation (25) is the shape function of the IMLS method, and then the IMLS method is presented.

Equations (13) and (15) show that the computation of the shape function of this paper is simpler than the corresponding expression in . The reduction of computational amount is related to the order of the operations. And, in any case, the calculation amount of the shape functions of this paper is at least n 2 multiplication operations less than that of the IMLS method in .

3. Interpolating Element-Free Galerkin Method for Two-Point Boundary Value Problems

Consider the following two-point boundary value problem: (29) - d d x ( p d u d x ) + q d u d x + g u = f , x Γ = ( a , b ) , u ( a ) = u ( b ) = 0 , where p , q , g , and f are known sufficiently smooth functions and p ( x ) p min > 0 . f C ( Γ ) , suppose that problem (29) has a unique solution.

The Galerkin weak form of (29) is (30) Γ δ u , x p u , x d x + Γ δ u q u , x d x + Γ δ u g u d x = Γ δ u f d x , where u , x = d u / d x .

From the IMLS method, the unknown solution u ( x ) at arbitrary field point x in the interval Γ can be expressed as (31) u ( x ) 𝚽 T ( x ) u = I = 1 n Φ I ( x ) u I , where n is the number of nodes whose compact support domains cover the point x .

Substituting (31) into (30) yields (32) Γ δ u T 𝚽 , x 𝚽 , x T u p d x + Γ δ u T 𝚽 𝚽 , x T u q d x + Γ δ u T 𝚽 𝚽 T u g d x = Γ δ u T 𝚽 f d x . Because the nodal test function δ u is arbitrary, the final discretized equation of (29) is obtained as (33) K u = F , where (34) K = Γ p 𝚽 , x 𝚽 , x T d x + Γ q 𝚽 𝚽 , x T d x + Γ g 𝚽 𝚽 T d x , F = Γ 𝚽 f d x .

Since the shape function of the IMLS method satisfies the property of Kronecker δ function, then the essential boundary conditions can be applied directly. Substituting the boundary conditions into (33) directly, we can obtain the unknowns at nodes by solving the linear equations (33).

To evaluate the integrals in (34), it is necessary to generate integration cells over the whole domain of the problem. These cells can be defined arbitrarily, but a sufficient number of quadrature points must be used to obtain a well-conditioned and nonsingular system of (33). In one dimension, one example of a cell structure is to set the quadrature cells equal to the intervals between the nodes. Once the cells and corresponding quadrature points are defined, the discrete equations are assembled by looping over each quadrature point.

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 (34) to

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.

4. Error Estimates

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 z = { x 1 , x 2 , , x M } be a set of all nodes in the interval [ a , b ] , where M is the number of nodes. Let ε = min x I , x J z , I J { x I - x J } and ρ = max x I z { ρ I } , where ρ I is the radius of the domain of influence of node x I . For a given x , ρ x denotes the maximum radius of the influence domains of nodes whose compact support domains cover x . And suppose that there exist constants c ε and c I such that ρ c ε ε and ρ c I ρ I , respectively.

Define the L p Lebesgue space as (35) L p ( Γ ) : = { f ( x ) : f L p ( Γ ) < } , 1 p < , where (36) f L p ( Γ ) : = ( Γ f p ( x ) d x ) 1 / p .

The Sobolev space H k ( Γ ) is defined as (37) H k ( Γ ) : = { f L loc 1 ( Γ ) : f H k ( Γ ) < } , where (38) f H k ( Γ ) : = ( s k d s f ( x ) d x s L 2 ( Γ ) ) 1 / 2 .

Define (39) H 0 1 ( Γ ) = { v v H 1 ( Γ ) , v ( a ) = v ( b ) = 0 } , hhhhhhhhhhhhhhhhhhh Γ = [ a , b ] .

Then the variational problem in accordance with (29) is to find u H 0 1 ( Γ ) such that (40) a ( u , v ) = ( f , v ) , v H 0 1 ( Γ ) , where (41) a ( u , v ) = Γ p u , x v , x d x + Γ q u , x v d x + Γ g u v d x , ( f , v ) = Γ f v d x .

Suppose that the bilinear form a ( · , · ) on the Sobolev space H 0 1 ( Γ ) is bounded and coercive; that is, there exist constants α - > 0 and M - < such that (42) | a ( u , v ) | M - u H 1 ( Γ ) v H 1 ( Γ ) , u , v H 0 1 ( Γ ) , a ( v , v ) α - v H 1 ( Γ ) 2 , v H 0 1 ( Γ ) .

Since the shape function of the IMLS method satisfies the property of Kronecker δ function, then the finite-dimensional solution space of the IEFG method for two-point boundary value problems is defined as (43) V ρ ( Γ ) = { v v span { Φ I ( x ) , 1 I M } , v ( a ) = v ( b ) = 0 } .

Then the IEFG method according to (29) is to find u h V ρ ( Γ ) such that (44) a ( u h , v ) = ( f , v ) , v V ρ ( Γ ) .

Obviously, there exists V ρ ( Γ ) H 0 1 ( Γ ) . Hence, the IEFG method for two-point boundary value problems has a unique solution. And the following theorem can be obtained.

Theorem 1.

Suppose that u is the solution of the variational problem (40) and u h is the solution of the IEFG method (44). Then there exist

a ( u - u h , v ) = 0 , v V ρ ( Γ ) ;

a ( u - u h , u - u h ) = inf v V ρ ( Γ ) a ( u - v , u - v ) ;

u - u h H 1 ( Γ ) C inf v V ρ ( Γ ) u - v H 1 ( Γ ) .

In fact, the approximation function of the IMLS method provides a linear operator 𝒜 defined as (45) 𝒜 u S u + i = 1 m a i ( x ) g i ( x ) = 𝚽 T ( x ) u .

If u H m + 1 ( Γ ) , then we have proved that there exist bounded function C k ( x ) and constant C k such that (46) d k d x k Φ I ( x ) = C k ( x ) ρ x - k , (47) 𝒜 u - u H k ( Γ ) C k ρ m + 1 - k u H m + 1 ( Γ ) , hhhhhhhhh 0 k m .

Let u - u h L 2 = a ( u - u h , u - u h ) . Then the following error estimates of the energy norm and the H 1 norm can be obtained.

Theorem 2.

Suppose that u H m + 1 ( Γ ) . Let u and u h be, respectively, the solutions of the problems (40) and (44). Then there exist C 1 and C 2 , which are independent with the parameter ρ , such that (48) u - u h L C 1 ρ m u H m + 1 ( Γ ) , u - u h H 1 ( Γ ) C 2 ρ m u H m + 1 ( Γ ) .

Proof.

From Theorem 1 and (47) we have (49) u - u h L 2 = a ( u - u h , u - u h ) = inf v V ρ ( Γ ) a ( u - v , u - v ) a ( u - 𝒜 u , u - 𝒜 u ) M u - 𝒜 u H 1 ( Γ ) 2 C 1 ρ 2 m u H m + 1 ( Γ ) 2 , u - u h H 1 ( Γ ) C inf v V ρ ( Γ ) u - v H 1 ( Γ ) C 𝒜 u - u H 1 ( Γ ) C 2 ρ m u H m + 1 ( Γ ) . Then this theorem holds.

By using the Aubin-Nitsche method, the following error estimates in the L 2 norm can be obtained.

Theorem 3.

Suppose that u H m + 1 ( Γ ) . Let u and u h be, respectively, the solutions of the problems (40) and (44). Then there exists a constant C , which is independent with the parameter ρ , such that (50) u - u h L 2 ( Γ ) C ρ m + 1 u H m + 1 ( Γ ) .

Proof.

For g L 2 ( Γ ) , let φ H 0 1 H 2 be the solution to the adjoint problems (51) a ( φ , v ) = ( g , v ) , v H 0 1 ( Γ ) . If the coefficients of the bilinear form a ( · , · ) are sufficiently smooth, then there exists the following estimate: (52) φ H 2 ( Γ ) g L 2 ( Γ ) .

Let v = u - u h . Then we have (53) a ( φ , u - u h ) = ( g , u - u h ) .

For arbitrary v h V ρ , from Theorem 1, we have (54) a ( v h , u - u h ) = 0 .

It follows from (53) and (54) that (55) a ( φ - v h , u - u h ) = ( g , u - u h ) .

In (55), if we let 𝒱 𝓀 = 𝒜 φ   and   g = u - u h , then there exists (56) u - u h L 2 ( Γ ) 2 = ( u - u h , u - u h ) = a ( φ - 𝒜 φ , u - u h ) M φ - 𝒜 φ H 1 ( Γ ) u - u h H 1 ( Γ ) M C 1 ρ φ H 2 ( Γ ) C 2 ρ m u H m + 1 ( Γ ) .

From (52) and (56), we have (57) u - u h L 2 ( Γ ) C ρ m + 1 u H m + 1 ( Γ ) . Then this theorem is proved.

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.

Theorem 4.

Suppose that Φ I ( x ) is defined by (25). Then v h ( x ) span { Φ I ( x ) , 1 I M } ; there exists a constant C , which is independent with the parameter ρ , such that (58) v h ( x ) H k C ρ n - k v h ( x ) H n , 0 k m , - m n m .

Proof.

From (46), there exists a bounded function C k ( x ) independent with ρ such that (59) k v h ( x ) = C k ( x ) ρ x - k , where k v h ( x ) = ( d k / d x k ) v h ( x ) .

It is obvious that there exist bounded functions C 1 ( x ) and C 2 ( x ) independent with ρ such that (60) C 1 ( x ) ρ ρ x C 2 ( x ) ρ .

Then we have (61) | v h ( x ) | H k 2 = Γ [ k v h ( x ) ] 2 d x Γ [ C k ( x ) C 2 ( x ) ρ - k ] 2 d x , hhhhhhhhhhh h 0 k m , | v h ( x ) | H n 2 = Γ [ n v h ( x ) ] 2 d x Γ [ C k ( x ) C 2 ( x ) ρ - n ] 2 d x . hhhhhhhhhhh h 0 n m .

Then it follows from (61) that (62) v h ( x ) H k C ρ n - k v h ( x ) H n , 0 k , n m , where C is independent with ρ .

From (62), we have (63) v h ( x ) H k C ρ - k v h ( x ) H 0 , 0 k m . And there exists (64) v h ( x ) H 0 2 v h ( x ) H n v h ( x ) H - n v h ( x ) H n · C ρ n v h ( x ) H 0 , - m n 0 .

From (63) and (64), we have (65) v h ( x ) H k C ρ n - k v h ( x ) H n , 0 k m , - m n 0 .

Then from (62) and (65), this theorem holds.

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

Theorem 5.

Suppose that u H m + 1 ( Γ ) . Let u and u h be, respectively, the solutions of the problems (40) and (44). Then there exists a constant C independent with ρ , such that (66) u - u h H s ( Γ ) C ρ m + 1 - s u H m + 1 ( Γ ) , 0 s m .

Proof.

From Theorem 4 we have (67) u h - 𝒜 u H s ( Γ ) C h 1 - s u h - 𝒜 u H 1 ( Γ ) , 0 s m . Then it follows from Theorem 2 and (47) that (68) u h - 𝒜 u H s ( Γ ) C h 1 - s [ u h - u H 1 ( Γ ) + u - 𝒜 u H 1 ( Γ ) ] C h m + 1 - s u H m + 1 ( Γ ) . There certainly exits (69) u - u h H s ( Γ ) u - 𝒜 u H s ( Γ ) + 𝒜 u - u h H s ( Γ ) . From (68), (69), and (47), we have (70) u - u h H s ( Γ ) C ρ m + 1 - s u H m + 1 ( Γ ) . Then this theorem holds.

5. Numerical Examples

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 x I is determined by ρ I = d max · | x I - x I - 1 | , where d max is a positive scalar. The value of d max must be chosen so that the solution of (22) exists. In our following examples, d max = 2.5 and α = 4 . Define the error norms as (71) | e k | d k d x k u h ( x ) - d k d x k u ( x ) L 2 ( Γ ) = [ Γ ( d k d x k u h - d k d x k u ) 2 d Γ ] 1 / 2 , where u and u h are, respectively, the analytical and numerical solutions. The integration in (71) is obtained numerically by fourth-order Gaussian quadrature. The m I ( x ) in (2) is chosen to be the cubic spline weight function.

The first example considered is a linear elastostatics problem. A one-dimensional bar of unit length is subjected to a body force of magnitude x . The displacement of the bar is fixed at the left end, and the right end is traction free. The bar has a constant cross sectional area of the unit value, and the elastic modulus is E . Then the equilibrium equation and boundary conditions of this problem are (72) E u , x x + x = 0 , 0 < x < 1 , u ( 0 ) = 0 , u , x ( 1 ) = 0 .

The analytical solution to the above problem is (73) u ( x ) = 1 E ( x 2 - x 3 6 ) .

Let E = 1 . Under the quadratic basis functions and 21 regular distributed nodes, the analytical and numerical solutions of the displacement and strains along the bar are shown, respectively, in Figures 1 and 2, where the numerical values of the IEFG method are in good agreement with the exact ones.

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 3. Here, the essential boundary conditions are enforced by the penalty method in the EFG method, and the penalty factor is chosen to be 10 8 . The CPU times to obtain these results by using the IEFG and EFG methods are, respectively, 0.0267 s and 0.0253 s. It can be seen that the IEFG method in this paper has higher precision than the EFG method under the similar CPU time.

The absolute error of the EFG and IEFG methods.

The error norms of | e 0 | , | e 1 | , and | e 2 | under the quadratic and linear basis functions are, respectively, shown in Figures 4 and 5. The convergence rates of | e 0 | , | e 1 | , and | e 2 | with quadratic basis are, respectively, about 3, 2, and 1, and the convergence rates with linear basis are 2, 1, and - 0.1 . It is also shown that the second derivatives of the numerical solution of the IEFG method are not convergent to the exact values on the radius ρ when the linear basis is used. It can be seen that these numerical results are in excellent agreement with the ones of the theories of the paper.

Error norms of | e 0 | , | e 1 | , and | e 2 | with quadratic basis functions.

Error norms of | e 0 | , | e 1 | , and | e 2 | with linear basis functions.

The second example considers the following equilibrium equation: (74) u , x x + π 2 u = 2 π 2 sin ( π x ) , 0 < x < 1 , with the boundary conditions (75) u ( 0 ) = u ( 1 ) = 0 . The analytical solution of this example is (76) u ( x ) = sin ( π x ) .

When the quadratic basis functions and 21 regular distributed nodes are used, the analytical and numerical solutions of u and u , x are shown, respectively, in Figures 6 and 7, where the numerical values of the IEFG method are also in accordance well with the exact ones.

Analytical and numerical solutions of u .

Analytical and numerical solutions of u , x .

The absolute errors of u obtained by the IEFG and EFG methods are shown in Figure 8. In the EFG method, the cubic spline weight function is used, and the essential boundary conditions are enforced by the penalty method. The penalty factor is chosen to be 10 10 . The CPU times to obtain these results by the IEFG and EFG methods are, respectively, 0.0361 s and 0.0325 s. Again, the IEFG method has higher precision than the EFG method under the similar CPU time.

The absolute error of the EFG and IEFG methods.

The convergence rates of | e 0 | , | e 1 | , and | e 2 | are shown in Figures 9 and 10, respectively, with the quadratic and linear basis functions. The convergence rates of | e 0 | , | e 1 | , and | e 2 | with the quadratic basis are, respectively, 3, 2, and 1. And the corresponding rates with linear basis are, respectively, 2, 1, and - 0.004 . Figure 10 also shows that the second derivatives of the numerical solution of the IEFG method are not convergent to the exact values on the radius ρ under the linear basis. It is also evident that these numerical results agree well with the ones of the theories of the paper.

Error norms of | e 0 | , | e 1 | , and | e 2 | with quadratic basis functions.

Error norms of | e 0 | , | e 1 | , and | e 2 | with linear basis functions.

6. Conclusions

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 δ function, then the IEFG method can apply the essential boundary condition directly. And as the number of the coefficients in the trial function of the IMLS method is less than that in the MLS approximation, then fewer nodes are selected in the entire domain in the IEFG method than in the conventional EFG method. Hence, the IEFG method has high computing speed and precision.

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.

Conflict of Interests

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

Acknowledgments

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).

Liew K. M. Ren J. Reddy J. N. Numerical simulation of thermomechanical behaviours of shape memory alloys via a non-linear mesh-free Galerkin formulation International Journal for Numerical Methods in Engineering 2005 63 7 1014 1040 2-s2.0-20844434156 10.1002/nme.1320 Liu G. R. Gu Y. T. A point interpolation method for two-dimensional solids International Journal for Numerical Methods in Engineering 2001 50 4 937 951 ZBL1050.74057 Belytschko T. Lu Y. Y. Gu L. Element-free Galerkin methods International Journal for Numerical Methods in Engineering 1994 37 2 229 256 10.1002/nme.1620370205 MR1256818 ZBL0796.73077 ZBL0796.73077 Ju-Feng W. Feng-Xin S. Rong-Jun C. Element-free Galerkin method for a kind of KdV equation Chinese Physics B 2010 19 6 2-s2.0-77953593286 10.1088/1674-1056/19/6/060201 060201 Cheng R. J. Cheng Y. M. A meshless method for the compound KdV-Burgers equation Chinese Physics B 2011 20 7 2-s2.0-79960791825 10.1088/1674-1056/20/7/070206 070206 Cheng Y. M. Peng M. J. Li J. H. The complex variable moving least-square approximation and its application Chinese Journal of Theoretical and Applied Mechanics 2005 37 6 719 723 MR2223274 Cheng Y. M. Li J. H. A meshless method with complex variables for elasticity Acta Physica Sinica 2005 54 10 4463 4471 MR2201966 ZBL1202.74163 ZBL1202.74163 Cheng Y. M. Li J. A complex variable meshless method for fracture problems Science in China G: Physics Astronomy 2006 49 1 46 59 2-s2.0-33644653009 10.1007/s11433-004-0027-y ZBL1147.74410 Liew K. M. Feng C. Cheng Y. M. Kitipornchai S. Complex variable moving least-squares method: a meshless approximation technique International Journal for Numerical Methods in Engineering 2007 70 1 46 70 10.1002/nme.1870 MR2307709 ZBL1194.74554 ZBL1194.74554 Wang J. F. Cheng Y. M. A new complex variable meshless method for transient heat conduction problems Chinese Physics B 2012 21 12 120206 Wang J. F. Cheng Y. M. New complex variable meshless method for advection-diffusion problems Chinese Physics B 2013 22 3 030208 Cheng R. J. Liew K. M. Analyzing two-dimensional sine-Gordon equation with the mesh-free reproducing kernel particle Ritz method Computer Methods in Applied Mechanics and Engineering 2012 245-246 132 143 10.1016/j.cma.2012.07.010 MR2969191 Dai B. D. Cheng Y. M. Local boundary integral equation method based on radial basis functions for potential problems Acta Physica Sinica 2007 56 2 597 603 MR2309608 ZBL1150.31300 ZBL1150.31300 Cheng R. J. Cheng Y. M. A meshless method for solving the inverse heat conduction problem with a source parameter Acta Physica Sinica 2007 56 10 5569 5574 MR2493027 ZBL1150.80304 ZBL1150.80304 Cheng R. J. Cheng Y. M. The meshless method for a two-dimensional inverse heat conduction problem with a source parameter Chinese Journal of Theoretical and Applied Mechanics 2007 39 6 843 847 2-s2.0-37349102322 Wang J. F. Bai F.-N. Cheng Y. M. A meshless method for the nonlinear generalized regularized long wave equation Chinese Physics B 2011 20 3 2-s2.0-79955666054 10.1088/1674-1056/20/3/030206 030206 Atluri S. N. Zhu T. A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics Computational Mechanics 1998 22 2 117 127 10.1007/s004660050346 MR1649420 ZBL0932.76067 ZBL0932.76067 Chen L. Cheng Y. M. Reproducing kernel particle method with complex variables for elasticity Acta Physica Sinica 2008 57 1 1 10 MR2415775 Chen L. Cheng Y. M. Complex variable reproducing kernel particle method for transient heat conduction problems Acta Physica Sinica 2008 57 10 6047 6055 MR2492810 ZBL1187.80046 ZBL1187.80046 Chen L. Cheng Y. M. The complex variable reproducing kernel particle method for elasto-plasticity problems Science China: Physics, Mechanics & Astronomy 2010 53 5 954 965 2-s2.0-77953149569 10.1007/s11433-010-0186-y Chen L. Cheng Y. M. The complex variable reproducing kernel particle method for two-dimensional elastodynamics Chinese Physics B 2010 19 9 2-s2.0-78649338771 10.1088/1674-1056/19/9/090204 090204 Weng Y. J. Cheng Y. M. Analyzing variable coefficient advection-diffusion problems via complex variable reproducing kernel particle method Chinese Physics B 2013 22 9 090204 Li S. C. Cheng Y. M. Meshless numerical manifold method based on unit partition Acta Mechanica Sinica 2004 36 4 496 500 Li S. C. Cheng Y. M. Numerical manifold method and its applications in rock mechanics Advances in Mechanics 2004 34 4 446 454 Li S. Cheng Y. M. Wu Y.-F. Numerical manifold method based on the method of weighted residuals Computational Mechanics 2005 35 6 470 480 2-s2.0-17444369765 10.1007/s00466-004-0636-3 ZBL1109.74373 Li S. C. Li S. C. Cheng Y. M. Enriched meshless manifold method for two-dimensional crack modeling Theoretical and Applied Fracture Mechanics 2005 44 3 234 248 2-s2.0-29544440199 10.1016/j.tafmec.2005.09.002 Li S.-C. Cheng Y. M. Li S.-C. Meshless manifold method for dynamic fracture mechanics Acta Physica Sinica 2006 55 9 4760 4766 2-s2.0-33750033641 Gao H. F. Cheng Y. M. Complex variable numerical manifold method for elasticity Chinese Journal of Theoretical and Applied Mechanics 2009 41 4 480 488 MR2589292 Gao H. Cheng Y. M. A complex variable meshless manifold method for fracture problems International Journal of Computational Methods 2010 7 1 55 81 10.1142/S0219876210002064 MR2646928 ZBL1267.74124 ZBL1267.74124 Cheng Y. M. Chen M. J. A boundary element-free method for linear elasticity Acta Mechanica Sinica 2003 35 2 181 186 Cheng Y. M. Peng M. Boundary element-free method for elastodynamics Science in China G: Physics Astronomy 2005 48 6 641 657 2-s2.0-33644660251 10.1360/142004-25 Liew K. M. Cheng Y. M. Kitipornchai S. Boundary Element-Free Method (BEFM) for two-dimensional elastodynamic analysis using Laplace transform International Journal for Numerical Methods in Engineering 2005 64 12 1610 1627 2-s2.0-28344442464 10.1002/nme.1417 ZBL1122.74533 Kitipornchai S. Liew K. M. Cheng Y. M. A Boundary Element-Free Method (BEFM) for three-dimensional elasticity problems Computational Mechanics 2005 36 1 13 20 2-s2.0-17744365462 10.1007/s00466-004-0638-1 ZBL1109.74372 Liew K. M. Cheng Y. M. Kitipornchai S. Boundary Element-Free Method (BEFM) and its application to two-dimensional elasticity problems International Journal for Numerical Methods in Engineering 2006 65 8 1310 1332 2-s2.0-32444439975 10.1002/nme.1489 ZBL1147.74047 Qin Y.-X. Cheng Y. M. Reproducing kernel particle boundary element-free method Acta Physica Sinica 2006 55 7 3215 3222 2-s2.0-33747469895 Liew K. M. Cheng Y. M. Kitipornchai S. Analyzing the 2D fracture problems via the enriched boundary element-free method International Journal of Solids and Structures 2007 44 11-12 4220 4233 2-s2.0-33947577913 10.1016/j.ijsolstr.2006.11.018 ZBL05198864 Qin Y. X. Cheng Y. M. Reproducing kernel particle boundary element-free method for potential problems Chinese Journal of Theoretical and Applied Mechanics 2009 41 6 898 905 MR2667866 Liew K. M. Cheng Y. M. Complex variable boundary element-free method for two-dimensional elastodynamic problems Computer Methods in Applied Mechanics and Engineering 2009 198 49–52 3925 3933 10.1016/j.cma.2009.08.020 MR2557480 ZBL1231.74502 ZBL1231.74502 Peng M. Cheng Y. M. A Boundary Element-Free Method (BEFM) for two-dimensional potential problems Engineering Analysis with Boundary Elements 2009 33 1 77 82 10.1016/j.enganabound.2008.03.005 MR2466280 ZBL1160.65348 ZBL1160.65348 Cheng Y. M. Liew K. M. Kitipornchair S. Reply to “Comments on Boundary Element-Free Method (BEFM) and its application to two-dimensional elasticity problems’ International Journal for Numerical Methods in Engineering 2009 78 10 1258 1260 2-s2.0-65449132743 10.1002/nme.2544 Atluri S. N. Sladek J. Sladek V. Zhu T. The Local Boundary Integral Equation (LBIE) and it's meshless implementation for linear elasticity Computational Mechanics 2000 25 2-3 180 198 2-s2.0-0033885238 10.1007/s004660050468 Dai B. Cheng Y. M. An improved local boundary integral equation method for two-dimensional potential problems International Journal of Applied Mechanics 2010 2 2 421 436 2-s2.0-83455207970 10.1142/S1758825110000561 Zhang Z. Liew K. M. Cheng Y. M. Lee Y. Y. Analyzing 2D fracture problems with the improved element-free Galerkin method Engineering Analysis with Boundary Elements 2008 32 3 241 250 2-s2.0-38349104986 10.1016/j.enganabound.2007.08.012 ZBL1244.74240 Zhang Z. Liew K. M. Cheng Y. M. Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems Engineering Analysis with Boundary Elements 2008 32 2 100 107 2-s2.0-36649017509 10.1016/j.enganabound.2007.06.006 ZBL1244.74204 Zhang Z. Li D.-M. Cheng Y. M. Liew K. M. The improved element-free Galerkin method for three-dimensional wave equation Acta Mechanica Sinica 2012 28 3 808 818 10.1007/s10409-012-0083-x MR2950238 Zhang Z. Wang J. F. Cheng Y. M. Liew K. M. The improved element-free Galerkin method for three-dimensional transient heat conduction problems Science China: Physics, Mechanics & Astronomy 2013 56 8 1568 1580 10.1007/s11433-013-5135-0 Zhang Z. Hao S. Y. Liew K. M. Cheng Y. M. The improved element-free Galerkin method for two-dimensional elastodynamics problems Engineering Analysis with Boundary Elements 2013 37 12 1576 1584 10.1016/j.enganabound.2013.08.017 MR3124864 Peng M. J. Li R. X. Cheng Y. M. Analyzing three-dimensional viscoelasticity problems via the Improved Element-Free Galerkin (IEFG) method Engineering Analysis with Boundary Elements 2014 40 104 113 10.1016/j.enganabound.2013.11.018 MR3161271 Peng M. Liu P. Cheng Y. M. The Complex Variable Element-Free Galerkin (CVEFG) method for two-dimensional elasticity problems International Journal of Applied Mechanics 2009 1 2 367 385 2-s2.0-77953162678 10.1142/S1758825109000162 Peng M. Li D. Cheng Y. M. The Complex Variable Element-Free Galerkin (CVEFG) method for elasto-plasticity problems Engineering Structures 2011 33 1 127 135 2-s2.0-78649445574 10.1016/j.engstruct.2010.09.025 Cheng Y. M. Wang J. F. Bai F. N. A new complex variable element-free Galerkin method for two-dimensional potential problems Chinese Physics B 2012 21 9 090203 10.1088/1674-1056/21/9/090203 Cheng Y. M. Li R. X. Peng M. J. Complex Variable Element-Free Galerkin (CVEFG) method for viscoelasticity problems Chinese Physics B 2012 21 9 090205 10.1088/1674-1056/21/9/090205 Cheng Y. M. Wang J. F. Li R. X. The Complex Variable Element-Free Galerkin (CVEFG) method for two-dimensional elastodynamics problems International Journal of Applied Mechanics 2012 4 4 23 125004 10.1142/S1758825112500421 Li D. Bai F. Cheng Y. M. Liew K. M. A novel complex variable element-free Galerkin method for two-dimensional large deformation problems Computer Methods in Applied Mechanics and Engineering 2012 233–236 1 10 10.1016/j.cma.2012.03.015 MR2924016 ZBL1253.74106 ZBL1253.74106 Bai F.-N. Li D.-M. Wang J. F. Cheng Y. M. An improved complex variable element-free Galerkin method for two-dimensional elasticity problems Chinese Physics B 2012 21 2 2-s2.0-84857344990 10.1088/1674-1056/21/2/020204 020204 Li D. M. Liew K. M. Cheng Y. M. An improved complex variable element-free Galerkin method for two-dimensional large deformation elastoplasticity problems Computer Methods in Applied Mechanics and Engineering 2014 269 72 86 10.1016/j.cma.2013.10.018 MR3144633 Zhu T. Atluri S. N. A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method Computational Mechanics 1998 21 3 211 222 10.1007/s004660050296 MR1633725 ZBL0947.74080 ZBL0947.74080 Most T. Bucher C. A moving least squares weighting function for the element-free Galerkin method which almost fulfills essential boundary conditions Structural Engineering and Mechanics 2005 21 3 315 332 2-s2.0-28044450607 10.12989/sem.2005.21.3.315 Most T. Bucher C. New concepts for moving least squares: an interpolating non-singular weighting function and weighted nodal least squares Engineering Analysis with Boundary Elements 2008 32 6 461 470 2-s2.0-43649104938 10.1016/j.enganabound.2007.10.013 ZBL1244.74228 Lancaster P. Salkauskas K. Surfaces generated by moving least squares methods Mathematics of Computation 1981 37 155 141 158 10.2307/2007507 MR616367 ZBL0469.41005 ZBL0469.41005 Kaljević I. Saigal S. An improved element free Galerkin formulation International Journal for Numerical Methods in Engineering 1997 40 16 2953 2974 MR1461997 ZBL0895.73079 ZBL0895.73079 Ren H.-P. Cheng Y. M. Zhang W. An Improved Boundary Element-Free Method (IBEFM) for two-dimensional potential problems Chinese Physics B 2009 18 10 4065 4073 2-s2.0-70350510162 10.1088/1674-1056/18/10/002 Hongping R. Yumin C. Wu Z. An Interpolating Boundary Element-Free Method (IBEFM) for elasticity problems Science China: Physics, Mechanics & Astronomy 2010 53 4 758 766 2-s2.0-77953167582 10.1007/s11433-010-0159-1 Ren H. P. Cheng Y. M. The Interpolating Element-Free Galerkin (IEFG) method for two-dimensional elasticity problems International Journal of Applied Mechanics 2011 3 4 735 758 10.1142/S1758825111001214 Ren H. Cheng Y. M. The Interpolating Element-Free Galerkin (IEFG) method for two-dimensional potential problems Engineering Analysis with Boundary Elements 2012 36 5 873 880 10.1016/j.enganabound.2011.09.014 MR2880734 Wang J. F. Sun F. X. Cheng Y. M. An improved interpolating element-free Galerkin method with nonsingular weight function for two-dimensional potential problems Chinese Physics B 2012 21 9 090204 Wang J. Wang J. Sun F. Cheng Y. M. An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems International Journal of Computational Methods 2013 10 6 23 1350043 10.1142/S0219876213500436 MR3056497 Sun F. X. Wang J. F. Cheng Y. M. An improved interpolating element-free Galerkin method for elasticity Chinese Physics B 2013 22 12 120203 Levin D. The approximation power of moving least-squares Mathematics of Computation 1998 67 224 1517 1531 10.1090/S0025-5718-98-00974-0 MR1474653 ZBL0911.41016 ZBL0911.41016 Armentano M. G. Durán R. G. Error estimates for moving least square approximations Applied Numerical Mathematics 2001 37 3 397 416 10.1016/S0168-9274(00)00054-4 MR1822748 ZBL0984.65096 ZBL0984.65096 Zuppa C. Error estimates for moving least square approximations Bulletin of the Brazilian Mathematical Society 2003 34 2 231 249 10.1007/s00574-003-0010-7 MR1992639 ZBL1040.65034 ZBL1040.65034 Li X. Zhu J. A Galerkin boundary node method and its convergence analysis Journal of Computational and Applied Mathematics 2009 230 1 314 328 10.1016/j.cam.2008.12.003 MR2532313 ZBL1189.65291 ZBL1189.65291 Li X. Meshless Galerkin algorithms for boundary integral equations with moving least square approximations Applied Numerical Mathematics 2011 61 12 1237 1256 10.1016/j.apnum.2011.08.003 MR2851120 ZBL1232.65160 ZBL1232.65160 Li X. The meshless Galerkin boundary node method for Stokes problems in three dimensions International Journal for Numerical Methods in Engineering 2011 88 5 442 472 10.1002/nme.3181 MR2842410 ZBL1242.76244 ZBL1242.76244 Krysl P. Belytschko T. Element-free Galerkin method: convergence of the continuous and discontinuous shape functions Computer Methods in Applied Mechanics and Engineering 1997 148 3-4 257 277 10.1016/S0045-7825(96)00007-2 MR1465821 ZBL0918.73125 ZBL0918.73125 Chung H.-J. Belytschko T. An error estimate in the EFG method Computational Mechanics 1998 21 2 91 100 10.1007/s004660050286 MR1616071 ZBL0910.73060 ZBL0910.73060 Dolbow J. Belytschko T. Numerical integration of the Galerkin weak form in meshfree methods Computational Mechanics 1999 23 3 219 230 10.1007/s004660050403 MR1691704 ZBL0963.74076 ZBL0963.74076 Gavete L. Cuesta J. L. Ruiz A. A procedure for approximation of the error in the EFG method International Journal for Numerical Methods in Engineering 2002 53 3 677 690 10.1002/nme.307 MR1876499 ZBL1112.74564 ZBL1112.74564 Gavete L. Gavete M. L. Alonso B. Martín A. J. A posteriori error approximation in EFG method International Journal for Numerical Methods in Engineering 2003 58 15 2239 2263 10.1002/nme.850 MR2020718 ZBL1047.74081 ZBL1047.74081 Cheng R. J. Cheng Y. M. Error estimates for the finite point method Applied Numerical Mathematics 2008 58 6 884 898 10.1016/j.apnum.2007.04.003 MR2420624 ZBL1145.65086 ZBL1145.65086 Cheng R. J. Cheng Y. M. Error estimates of the element-free Galerkin method for potential problems Acta Physica Sinica 2008 57 10 6037 6046 MR2492809 Cheng R. J. Cheng Y. M. Error estimate of element-free Galerkin method for elasticity Acta Physica Sinica 2011 60 7 2-s2.0-79961107471 070206