Mechanical Quadrature Methods and Extrapolation for Solving

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Apr 24, 2018 - This paper will study the high accuracy numerical solutions for elastic equations with nonlinear boundary value conditions. The equations willΒ ...
Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 6932164, 8 pages https://doi.org/10.1155/2018/6932164

Research Article Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity Pan Cheng

and Ling Zhang

School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China Correspondence should be addressed to Pan Cheng; cheng [email protected] Received 25 December 2017; Accepted 24 April 2018; Published 31 May 2018 Academic Editor: Nunzio Salerno Copyright Β© 2018 Pan Cheng and Ling Zhang. 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. This paper will study the high accuracy numerical solutions for elastic equations with nonlinear boundary value conditions. The equations will be converted into nonlinear boundary integral equations by the potential theory, in which logarithmic singularity and Cauchy singularity are calculated simultaneously. Mechanical quadrature methods (MQMs) are presented to solve the nonlinear equations where the accuracy of the solutions is of three orders. According to the asymptotical compact convergence theory, the errors with odd powers asymptotic expansion are obtained. Following the asymptotic expansion, the accuracy of the solutions can be improved to five orders with the Richardson extrapolation. Some results are shown regarding these approximations for problems by the numerical example.

1. Introduction This paper will describe the isolated elastic equations on a bounded planar region Ξ© in the plane with nonlinear boundary value conditions: πœŽπ‘–π‘—,𝑗 = 0,

in Ξ©,

𝑝 = βˆ’π‘” (π‘₯, 𝑒) + 𝑓 (π‘₯) ,

on Ξ“, 𝑖, 𝑗 = 1, 2,

(1)

where Ξ© βŠ‚ 𝑅2 is a connected domain with a smooth closed curve Ξ“, the stress tensors are πœŽπ‘–π‘— , 𝑛 = (𝑛1 , 𝑛2 ) is the unit outward normal vector on Ξ“, the tractor vector is assumed given 𝑝 = (𝑝1 , 𝑝2 )𝑇 with 𝑝𝑖 = πœŽπ‘–1 𝑛1 + πœŽπ‘–2 𝑛2 , 𝑓(π‘₯) = (𝑓1 (π‘₯), 𝑓2 (π‘₯))𝑇 and continuous on Ξ“, and 𝑔(π‘₯, 𝑒) = (𝑔1 (π‘₯, 𝑒), 𝑔2 (π‘₯, 𝑒))𝑇 , where 𝑔𝑖 (π‘₯, 𝑒) is a nonlinear function corresponding to the displaced 𝑒. Following vector computational rules, the repeated subscripts imply the summation from 1 to 2. The problems are studied in many applications, e.g., the circular ring problems and the instance problems [1, 2], and the bending of prismatic bars [3]. Some methods have been proposed for solving the elasticity. The nonconforming mixed finite element methods are established by Hu and Shi [4]

to solve the elasticity with a linear boundary. Talbot and Crampton [5] use a pseudo-spectral method to approach matrix eigenvalue problems which is transformed from the governing partial differential equations. The least-square methods are introduced by Cai and Starke [6] for obtaining the solution of the elastic problems. Chen and Hong [7] solved the hyper singular integral equations applying the dual boundary element methods in elasticity. Kuo et al. [8] solve true and spurious eigensolutions of the circular cavity problems that used dual methods. Li and Nie [9] researched the stressed axis-symmetric rods problems by a high-order integration factor method. And the mechanical quadrature methods (MQMs) are adopted by Cheng et al. [10] to solve Steklov eigensolutions in elasticity to obtain high accuracy solutions. Equations (1) are converted into the boundary integral equations [10–12] (BIEs) with the Cauchy and logarithmic singularities by the variational method. 𝛿𝑖𝑗 2

𝑒𝑗 (𝑦) + ∫ π‘˜π‘–π‘—βˆ— (𝑦, π‘₯) 𝑒𝑗 (π‘₯) 𝑑𝑠π‘₯ Ξ“

=∫

Ξ“

β„Žπ‘–π‘—βˆ—

(𝑦, π‘₯) 𝑝𝑗 (π‘₯) 𝑑𝑠π‘₯ ,

(2) 𝑦 = (𝑦1 , 𝑦2 ) ∈ Ξ“,

2

Mathematical Problems in Engineering

where 𝛿𝑖𝑗 is the Kronecker delta for 𝑖, 𝑗 = 1, 2, and the kernels: β„Žπ‘–π‘—βˆ— =

1 [βˆ’ (3 βˆ’ 4]) 𝛿𝑖𝑗 ln π‘Ÿ + π‘Ÿβ‹…π‘– π‘Ÿβ‹…π‘— ] , 8πœ‹πœ‡ (1 βˆ’ ])

π‘˜π‘–π‘—βˆ— =

1 πœ•π‘Ÿ [ ((1 βˆ’ 2]) 𝛿𝑖𝑗 + 2π‘Ÿβ‹…π‘– π‘Ÿβ‹…π‘— ) 4πœ‹ (1 βˆ’ ]) π‘Ÿ πœ•π‘›

where βˆ— = π‘˜π‘–π‘—π‘™

(3)

βˆ— = β„Žπ‘–π‘—π‘™

[(1 βˆ’ 2]) (π‘Ÿ.𝑗 𝛿𝑙𝑖 + π‘Ÿ.𝑖 𝛿𝑙𝑗 βˆ’ π‘Ÿ.𝑙 𝛿𝑖𝑗 ) + 2π‘Ÿ.𝑖 π‘Ÿ.𝑗 π‘Ÿ.𝑙 ] [4πœ‹ (1 βˆ’ ]) π‘Ÿ]

,

πœ‡ πœ•π‘Ÿ (2 [(1 βˆ’ 2]) π‘Ÿ.𝑙 𝛿𝑖𝑗 2 2πœ‹ (1 βˆ’ ]) π‘Ÿ πœ•π‘›

+ ] (π‘Ÿ.𝑗 𝛿𝑖𝑙 + π‘Ÿ.𝑖 𝛿𝑗𝑙 ) βˆ’ 4π‘Ÿ.𝑖 π‘Ÿ.𝑗 π‘Ÿ.𝑙 ] + 2] (𝑛𝑖 π‘Ÿ.𝑗 π‘Ÿ.𝑙

+ (1 βˆ’ 2]) (𝑛𝑖 π‘Ÿβ‹…π‘— βˆ’ 𝑛𝑗 π‘Ÿβ‹…π‘– )] .

(6)

+ 𝑛𝑗 π‘Ÿ.𝑖 π‘Ÿ.𝑙 ) + (1 βˆ’ 2]) (2𝑛𝑙 π‘Ÿ.𝑗 π‘Ÿ.𝑖 + 𝑛𝑗 𝛿𝑖𝑙 + 𝑛𝑖 𝛿𝑗𝑙 ) βˆ’ (1 The kernels are Kelvin’s fundamental solutions [11]. The Poisson ratio is ] with ] = πœ†/[2(πœ† + πœ‡)], π‘Ÿβ‹…π‘– means the derivative to π‘₯𝑖 , and π‘Ÿ = √(𝑦1 βˆ’ π‘₯1

)2

)2

+ (𝑦2 βˆ’ π‘₯2 is βˆ— distance of π‘₯ and 𝑦. The parts βˆ«Ξ“ π‘˜π‘–π‘— (𝑦, π‘₯)𝑒𝑗 (π‘₯)𝑑𝑠π‘₯ of (2) the Cauchy singularity and the parts βˆ«Ξ“ β„Žπ‘–π‘—βˆ— (𝑦, π‘₯)𝑒𝑗 (π‘₯)𝑑𝑠π‘₯

the are are

the logarithmic singularity. The nonlinear integral equations are obtained after the boundary conditions are substituted into (2): 𝛿𝑖𝑗 2

𝑒𝑗 (𝑦) + ∫ π‘˜π‘–π‘—βˆ— (𝑦, π‘₯) 𝑒𝑗 (π‘₯) 𝑑𝑠π‘₯ Ξ“

= ∫ β„Žπ‘–π‘—βˆ— (𝑦, π‘₯) 𝑔𝑗 (π‘₯, 𝑒) 𝑑𝑠π‘₯

(4)

Ξ“

+ ∫ β„Žπ‘–π‘—βˆ— (𝑦, π‘₯) 𝑓𝑗 (π‘₯) 𝑑𝑠π‘₯ . Ξ“

This kind of nonlinear integral equation has been discussed in many papers. Rodriguez [12] established sufficient conditions for the existence of solutions to nonlinear, discrete boundary value problems. Wavelet-Galerkin’s methods constructed from Legendre wavelet functions are used by Maleknejad and Mesgarani [13] to approximate the solution. Atkinson and Chandler [14] presented product Simpson’s rule methods and two-grid methods to solve the nonlinear equations. Pao [15] gave a systematic treatment of a class of nonlinear elliptic differential equations and their applications of these problems. Abels et al. [16] discussed the convergence in the systems of nonlinear boundary conditions with HΒ¨older space. Since this is a nonlinear system including the logarithmic singularity and the Cauchy singularity, the difficulty is to obtain the discrete equations appropriately. The displacement vector and stress tensor in Ξ© can be calculated [17, 18] as follows after the discrete nonlinear equations are solved: 𝑒𝑖 (𝑦) = ∫ β„Žπ‘–π‘—βˆ— (𝑦, π‘₯) 𝑝𝑗 (π‘₯) 𝑑𝑠π‘₯ Ξ“

βˆ’ ∫ π‘˜π‘–π‘—βˆ— (𝑦, π‘₯) 𝑒𝑗 (π‘₯) 𝑑𝑠π‘₯ , Ξ“

πœŽπ‘–π‘— (𝑦) = ∫

Ξ“

βˆ’βˆ«

Ξ“

(𝑦, π‘₯) 𝑝𝑙 (π‘₯) 𝑑𝑠π‘₯

βˆ— π‘˜π‘–π‘—π‘™

Richardson extrapolation algorithms (EAs) are pretty effective parallel algorithms which are based on asymptotic expansion about errors. The solutions, which are solved on coarse grid and fine grid, are used to construct high accuracy solutions. The method is known to be of good stability and optimal computational complexity. Cheng et al. [19] harnessed extrapolation algorithms to obtain high accuracy order for Steklov eigenvalue of Laplace equations. Huang and LΒ¨u established extrapolation algorithms to obtain high accuracy solutions for solving Laplace equations on arcs [20] and plane problem in elasticity [21]. The reminder of this paper is stated as follows: In Section 2 the MQMs are constructed for the nonlinear boundary integral equations and obtain a discrete nonlinear system. In Section 3 the asymptotically compact convergence is proved. In Section 4 an asymptotic expansion of the error about π‘’β„Ž is analyzed and the Richardson extrapolation is applied to achieve the accuracy orders 𝑂(β„Ž5 ). In Section 5 some results are shown regarding approximations for problems by the numerical example.

2. Mechanical Quadrature Methods We firstly define the notation of the integral operators on boundary Ξ“ as follows to simplify the equations: (𝐾𝑖𝑗 𝑒) (𝑦) = ∫ π‘˜π‘–π‘—βˆ— (𝑦, π‘₯) 𝑒 (π‘₯) 𝑑𝑠π‘₯ Ξ“

(𝐻𝑖𝑗 𝑒) (𝑦) = ∫

Ξ“

(𝑦, π‘₯) 𝑒𝑙 (π‘₯) 𝑑𝑠π‘₯ , βˆ€π‘¦ ∈ Ξ©,

𝑦 ∈ Ξ“, 𝑖, 𝑗 = 1, 2, (7)

β„Žπ‘–π‘—βˆ—

(𝑦, π‘₯) 𝑒 (π‘₯) 𝑑𝑠π‘₯

𝑦 ∈ Ξ“, 𝑖, 𝑗 = 1, 2.

So (2) can be simplified as the following operator equations: 1 𝐾12 𝐼0 + 𝐾11 𝑒1 (2 )( ) 1 𝑒2 𝐾21 𝐼 + 𝐾22 2 0

βˆ€π‘¦ ∈ Ξ©, (5)

βˆ— β„Žπ‘–π‘—π‘™

βˆ’ 4]) 𝑛𝑙 𝛿𝑖𝑗 ) .

=(

𝐻11 𝐻12 𝐻21 𝐻22

)(

𝑔1 + 𝑓1 𝑔2 + 𝑓2

(8)

),

where 𝐼0 is an identity operator. Suppose 𝐢2π‘š [0, 2πœ‹] be the set of 2π‘š times differentiable periodic functions in which the periods of all functions are

Mathematical Problems in Engineering

3

2πœ‹. A regular parameter mapping [0, 2πœ‹] β†’ Ξ“ is introduced, so the smooth closed curve π‘₯(𝑠) = (π‘₯1 (𝑠), π‘₯2 (𝑠)) satisfying |π‘₯σΈ€  (𝑠)|2 = |π‘₯1σΈ€  (𝑠)|2 + |π‘₯2σΈ€  (𝑠)|2 > 0 with π‘₯𝑖 (𝑠) ∈ 𝐢2π‘š [0, 2πœ‹], 𝑖 = 1, 2. The singularity of the kernels β„Žπ‘–π‘—βˆ— , π‘˜π‘–π‘—βˆ— will be analyzed as 𝑑 β†’ 𝑠 before the application of discrete methods. Since it is stated in the page 497 of the paper [21] that 𝑛𝑖 π‘Ÿβ‹…π‘— βˆ’ 𝑛𝑗 π‘Ÿβ‹…π‘– π‘Ÿ

= (βˆ’1)𝑖

1 + 𝑂 (𝑑 βˆ’ 𝑠) , (𝑑 βˆ’ 𝑠) + 𝑂 (𝑑 βˆ’ 𝑠)

Then the equivalent equation of (8) will be shown as 1 ( 𝐼 + 𝐢 + 𝑀) 𝑒 = (𝐴 + 𝐡) 𝑔 (𝑒) + 𝑓, 2

where 𝑒(𝑑) = (𝑒1 (π‘₯(𝑑)), 𝑒2 (π‘₯(𝑑))), 𝑓(𝑑) = (𝑓1 , 𝑓2 )𝑇 with 𝑓𝑖 = 𝐻𝑖,𝑗 𝑓𝑗 (π‘₯(𝑑)), 𝑔(𝑒(𝑑)) = 𝑔(π‘₯(𝑑), 𝑒(𝑑)), and 𝐼=(

𝑖 =ΜΈ 𝑗,

(9)

so the Cauchy singularity of π‘˜π‘–π‘—βˆ— comes from the part (𝑛𝑖 π‘Ÿβ‹…π‘— βˆ’ 𝑛𝑗 π‘Ÿβ‹…π‘– )/π‘Ÿ. Moreover, the logarithmic singularity of β„Žπ‘–π‘—βˆ— comes from the component log |π‘₯ βˆ’ 𝑦| and the other parts of the operators are smooth. So the operators 𝐻𝑖,𝑗 and 𝐾𝑖,𝑗 will be divided into several parts. The logarithmic singular operator 𝐻𝑖𝑗 will be divided into three parts on 𝐢2π‘š [0, 2πœ‹] as follows: 2πœ‹ 󡄨 󡄨 (𝐴 0 𝑒) (𝑑) = ∫ π‘Ž0 (𝑑, 𝜏) 𝑒 (𝜏) 󡄨󡄨󡄨󡄨π‘₯σΈ€  (𝜏)󡄨󡄨󡄨󡄨 π‘‘πœ, 0

(10)

with π‘Ž0 (𝑑, 𝜏) = 𝑐0 ln |2π‘’βˆ’1/2 sin((𝑑 βˆ’ 𝜏)/2)|, 𝑐0 = βˆ’(3 βˆ’ 4])/[8πœ‹πœ‡(1 βˆ’ ])], and 2πœ‹ 󡄨 󡄨 (𝐡0 𝑒) (𝑑) = ∫ 𝑏0 (𝑑, 𝜏) 𝑒 (𝜏) 󡄨󡄨󡄨󡄨π‘₯σΈ€  (𝜏)󡄨󡄨󡄨󡄨 π‘‘πœ, 0

(11)

with 𝑏0 (𝑑, 𝜏) = 𝑐0 (ln |π‘₯(𝑑) βˆ’ π‘₯(𝜏)| βˆ’ ln |2π‘’βˆ’1/2 sin((𝑑 βˆ’ 𝜏)/2)|), and 2πœ‹

󡄨 󡄨 (𝐡𝑖𝑗 𝑒) (𝑑) = ∫ 𝑏𝑖𝑗 (𝑑, 𝜏) 𝑒 (𝜏) 󡄨󡄨󡄨󡄨π‘₯σΈ€  (𝜏)󡄨󡄨󡄨󡄨 π‘‘πœ, 0

0

0

(13)

(14)

with π‘šπ‘–π‘– (𝑑, 𝜏) = 𝑐3 (πœ•π‘Ÿ/πœ•π‘›)[(1βˆ’2V)+2π‘Ÿβ‹…π‘– π‘Ÿβ‹…π‘– ]/π‘Ÿ, 𝑐3 = βˆ’1/[4πœ‹(1βˆ’ ])], and

0

𝐡=( 𝑀=(

0

), 𝐢0

βˆ’πΆ0 0

𝐴0 0 0 𝐴0

),

),

(17)

𝐡0 + 𝐡11

𝐡12

𝐡21

𝐡0 + 𝐡22

𝑀11 𝑀12 𝑀21 𝑀22

),

).

Suppose the boundary Ξ“ will be divided into 2𝑛, (𝑛 ∈ 𝑁) equal parts, so the mesh width will be β„Ž = πœ‹/𝑛 and the nodes will be 𝑑𝑗 = πœπ‘— = π‘—β„Ž, (𝑗 = 0, 1, . . . , 2𝑛 βˆ’ 1). The smooth integral operators with the period 2πœ‹ can easily obtain high accuracy NystrΒ¨om’s approximations [22]. For example, the approximation operator 𝐡0β„Ž of 𝐡0 can be approximated as follows: 2π‘›βˆ’1

(𝐡0β„Ž 𝑒) (𝑑) = β„Ž βˆ‘ 𝑏0 (𝑑, πœπ‘— ) 𝑒 (πœπ‘— ) ,

(18)

𝑗=0

(𝐡0 𝑒) (𝑑) βˆ’ (𝐡0β„Ž 𝑒) (𝑑) = 𝑂 (β„Ž2π‘š ) .

(19)

NystrΒ¨om’s approximation π΅π‘–π‘—β„Ž of 𝐡𝑖𝑗 and π‘€π‘–π‘—β„Ž of 𝑀𝑖𝑗 can be approximated similarly. In order to approximate the logarithmic singular operator 𝐴 0 , the continuous approximation kernel π‘Žπ‘ (𝑑, 𝜏) can be defined as follows: for |𝑑 βˆ’ 𝜏| β‰₯ β„Ž, π‘Ž (𝑑, 𝜏) , { { 0 󡄨 󡄨󡄨 (20) π‘Žπ‘ (𝑑, 𝜏) = { {𝑐 β„Ž ln σ΅„¨σ΅„¨σ΅„¨π‘’βˆ’1/2 β„Ž 󡄨󡄨󡄨󡄨 , for |𝑑 βˆ’ 𝜏| < β„Ž. 0 󡄨 󡄨 󡄨󡄨 2πœ‹ 󡄨󡄨 { By Sidi’s quadrature rules [3], the approximation operator can be constructed as follows: 2π‘›βˆ’1 󡄨 󡄨 (π΄β„Ž0 𝑒) (𝑑) = β„Ž βˆ‘ π‘Žπ‘ (𝑑, πœπ‘— ) 𝑒 (πœπ‘— ) 󡄨󡄨󡄨󡄨π‘₯σΈ€  (πœπ‘— )󡄨󡄨󡄨󡄨 ,

(21)

𝑗=0

2πœ‹

󡄨 󡄨 (𝑀𝑖𝑗 𝑒) (𝑑) = ∫ π‘šπ‘–π‘— (𝑑, 𝜏) 𝑒 (𝜏) 󡄨󡄨󡄨󡄨π‘₯σΈ€  (𝜏)󡄨󡄨󡄨󡄨 π‘‘πœ,

𝐴=(

0 𝐼0

and the error will be

with 𝑐0 (𝑑, 𝜏) = 𝑐2 (𝑛1 π‘Ÿβ‹…2 βˆ’ 𝑛2 π‘Ÿβ‹…1 )/π‘Ÿ, 𝑐2 = βˆ’(1 βˆ’ 2])/[4πœ‹(1 βˆ’ ])], and 2πœ‹ 󡄨 󡄨 (𝑀𝑖𝑖 𝑒) (𝑑) = ∫ π‘šπ‘–π‘– (𝑑, 𝜏) 𝑒 (𝜏) 󡄨󡄨󡄨󡄨π‘₯σΈ€  (𝜏)󡄨󡄨󡄨󡄨 π‘‘πœ, 𝑖 = 1, 2,

𝐢=(

𝐼0 0

(12)

with 𝑏𝑖𝑗 (𝑑, 𝜏) = 𝑐1 π‘Ÿβ‹…π‘– π‘Ÿβ‹…π‘— , 𝑐1 = 1/[8πœ‹πœ‡(1 βˆ’ ])]. The Cauchy singular operator 𝐾𝑖𝑗 will also be divided into three parts on 𝐢2π‘š [0, 2πœ‹] as follows: 2πœ‹ 󡄨 󡄨 (𝐢0 𝑒) (𝑑) = ∫ 𝑐0 (𝑑, 𝜏) 𝑒 (𝜏) 󡄨󡄨󡄨󡄨π‘₯σΈ€  (𝜏)󡄨󡄨󡄨󡄨 π‘‘πœ,

(16)

(15)

𝑖, 𝑗 = 1, 2, 𝑖 =ΜΈ 𝑗, with π‘šπ‘–π‘— (𝑑, 𝜏) = 𝑐3 (πœ•π‘Ÿ/πœ•π‘›)(2π‘Ÿβ‹…π‘– π‘Ÿβ‹…π‘— )/π‘Ÿ. We can find that 𝐡0 , 𝐡𝑖,𝑗 , 𝑀𝑖,𝑗 are smooth operators, 𝐴 0 is a logarithmic singular operator, and 𝐢0 is a Cauchy singular operator.

by which the error estimate will be (𝐴 0 𝑒) (𝑑) βˆ’ (π΄β„Ž0 𝑒) (𝑑) π‘šβˆ’1 σΈ€ 

𝜍 (βˆ’2πœ‡) (2πœ‡) 𝑒 (𝑑) β„Ž2πœ‡+1 + 𝑂 (β„Ž2π‘š ) , (2πœ‡)! πœ‡=1

= 2βˆ‘

where πœσΈ€  (𝑑) is the derivative of Riemann zeta function.

(22)

4

Mathematical Problems in Engineering

For the Cauchy singular operator 𝐢0 , LΒ¨u and Huang [21] and Sidi [22] have proposed the operator 𝐢0β„Ž to approximate it as shown in Lemma 1.

Theorem 3. The approximate operator sequences {πΏβ„Ž1 } and {πΏβ„Ž2 } are asymptotically compact sequences and convergent to 𝐿 1 and 𝐿 2 in 𝑉(0) , respectively; i.e.,

Lemma 1. The NystrΒ¨om approximate operator 𝐢0β„Ž can be defined as

πΏβ„Ž1 󳨀→ 𝐿 1 ,

2π‘›βˆ’1

(𝐢0β„Ž 𝑒) (𝑑𝑖 ) = 2𝑐2 π‘Ž1 (𝑑𝑖 , 𝑑𝑗 ) β„Ž βˆ‘ cot (

(𝑑𝑗 βˆ’ 𝑑𝑖 )

𝑗=0

2

a.c πΏβ„Ž2 󳨀→

) 𝑒 (𝑑𝑗 )

(23)

󡄨 󡄨 β‹… 󡄨󡄨󡄨󡄨π‘₯σΈ€  (𝑑𝑗 )󡄨󡄨󡄨󡄨 πœ€π‘–π‘— , where π‘Ž1 (𝑑, 𝑠) = 2[(π‘‘βˆ’π‘ )+𝑂(π‘‘βˆ’π‘ )]βˆ’1 tan((π‘‘βˆ’π‘ )/2) when 𝑠 β†’ 𝑑, and 󡄨 󡄨 {1, if 󡄨󡄨󡄨𝑖 βˆ’ 𝑗󡄨󡄨󡄨 is odd number, πœ€π‘–π‘— = { 󡄨 󡄨 0, if 󡄨󡄨󡄨𝑖 βˆ’ 𝑗󡄨󡄨󡄨 is even number, {

(𝐢0 𝑒) (𝑑𝑖 ) βˆ’

(29)

𝐿 2,

a.c

where 󳨀󳨀→ means the asymptotically compact convergence. Proof. An asymptotically compact sequence will be proved for operator {πΏβ„Ž2 : 𝑉(0) β†’ 𝑉(0) }, firstly. We can find that the kernels of 𝐡0 and 𝐡𝑖𝑗 (𝑖, 𝑗 = 1, 2, ) are continuous functions, so the collectively compact convergence [10, 20] is true:

(24) c.c

𝐡0β„Ž 󳨀→ 𝐡0 ,

so the error estimate will be (𝐢0β„Ž 𝑒) (𝑑𝑖 )

a.c

c.c

π΅π‘–π‘—β„Ž 󳨀→ 𝐡𝑖𝑗

2π‘š

= 𝑂 (β„Ž ) .

(25)

(30) in 𝐢 [0, 2πœ‹] , as 𝑛 󳨀→ ∞.

Thus (16) can be rewritten as follows: 1 ( 𝐼 + πΆβ„Ž + π‘€β„Ž ) π‘’β„Ž βˆ’ (π΄β„Ž + π΅β„Ž ) 𝑔 (π‘’β„Ž ) = π‘“β„Ž , 2

(26)

where π΄β„Ž , π΅β„Ž , πΆβ„Ž , and π‘€β„Ž are the approximate matrixes corresponding to the operators 𝐴, 𝐡, 𝐢, and 𝑀, respectively.

3. Asymptotically Compact Convergence The Cauchy approximate operator combined with the identity operator will be studied for the reversibility. A property about the operator (1/2)𝐼 + πΆβ„Ž will be presented [10]. βˆ’1

Lemma 2. (1/2)𝐼 + πΆβ„Ž is invertible and ((1/2)𝐼 + πΆβ„Ž ) uniformly bounded.

is

From Lemma 2, we can find that the eigenvalues of both 𝐢 and πΆβ„Ž do not include βˆ’1/2. So (16) and (26) can be rewritten as follows: find 𝑒 ∈ 𝑉(0) satisfying Μ‚ (𝐼 + 𝐿 1 ) 𝑒 + 𝐿 2 𝑔 (𝑒) = 𝑓,

(27)

where 𝐿 1 = (𝐼/2 + 𝐢)βˆ’1 𝑀, 𝐿 2 = (𝐼/2 + 𝐢)βˆ’1 (𝐴 + 𝐡), 𝑓̂ = (𝐼/2 + 𝐢)βˆ’1 𝑓, and the space 𝑉(π‘š) = 𝐢(π‘š) [0, 2πœ‹] Γ— 𝐢(π‘š) [0, 2πœ‹], π‘š = 0, 1, 2, . . .. Equation (26) can be rewritten as follows: (𝐼 + πΏβ„Ž1 ) π‘’β„Ž + πΏβ„Ž2 𝑔 (π‘’β„Ž ) = π‘“Μ‚β„Ž , βˆ’1

βˆ’1

(28)

with πΏβ„Ž1 = (𝐼/2 + πΆβ„Ž ) π‘€β„Ž , πΏβ„Ž2 = (𝐼/2 + πΆβ„Ž ) (π΄β„Ž + π΅β„Ž ), and βˆ’1 𝑓̂ = (𝐼/2 + πΆβ„Ž ) 𝑓 . β„Ž

β„Ž

Since we have constructed the continuous approximate function of π‘Ž(𝑑, 𝜏) of π‘Žπ‘ (𝑑, 𝜏), the approximate operator {π΄β„Ž0 } is the a.c

asymptotically compact convergent to 𝐴 0 ; i.e., π΄β„Ž0 󳨀󳨀→ 𝐴 0 in 𝐢[0, 2πœ‹], as 𝑛 β†’ ∞. a.c a.c Then we have π΄β„Ž 󳨀󳨀→ 𝐴 and π΅β„Ž 󳨀󳨀→ 𝐡 in 𝑉(0) . It can be concluded that for any bounded sequence {π‘¦π‘š ∈ 𝑉(0) } a convergent subsequence must be found in {(π΄β„Ž + π΅β„Ž )π‘¦π‘š }. Without loss of generality, we assume (π΄β„Ž + π΅β„Ž )π‘¦π‘š β†’ 𝑧, as π‘š β†’ ∞. According to the asymptotically compact convergence and the errors quadrature formulae [17, 21], we obtain σ΅„©σ΅„© βˆ’1 σ΅„© σ΅„©σ΅„© 1 σ΅„©σ΅„© β„Ž 󡄩󡄩𝐿 2 π‘¦π‘š βˆ’ ( 𝐼 + 𝐢) 𝑧󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„©σ΅„© 2 σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© βˆ’1 σ΅„© σ΅„© 1 σ΅„© σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©σ΅„©( 𝐼 + πΆβ„Ž ) σ΅„©σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©σ΅„©(π΄β„Ž + π΅β„Ž ) π‘¦π‘š βˆ’ 𝑧󡄩󡄩󡄩󡄩 σ΅„©σ΅„© 2 σ΅„©σ΅„© σ΅„©σ΅„© βˆ’1 βˆ’1 σ΅„© σ΅„©σ΅„© 1 σ΅„© 1 + σ΅„©σ΅„©σ΅„©σ΅„©( 𝐼 + πΆβ„Ž ) (𝐢 βˆ’ πΆβ„Ž ) ( 𝐼 + 𝐢) 𝑧󡄩󡄩󡄩󡄩 2 σ΅„©σ΅„© 2 σ΅„©σ΅„© 󳨀→ 0,

(31)

as π‘š 󳨀→ ∞, β„Ž 󳨀→ 0,

where β€– β‹… β€– is the norm of m(𝑉(0) , 𝑉(0) ). We proved that {πΏβ„Ž2 : 𝑉(0) β†’ 𝑉(0) } is an asymptotically compact sequence. Furthermore, we will show that {πΏβ„Ž2 } will be pointwise a.c convergent to 𝐿 2 , as 𝑛 β†’ ∞. We can see that π΄β„Ž + π΅β„Ž 󳨀󳨀→ 𝐴 + π΅βˆ€π‘¦ ∈ 𝑉(0) , so we obtain σ΅„©σ΅„© β„Ž σ΅„© σ΅„©σ΅„©(𝐴 + π΅β„Ž ) 𝑦 βˆ’ (𝐴 + 𝐡) 𝑦󡄩󡄩󡄩 󳨀→ 0, σ΅„© σ΅„©

as β„Ž 󳨀→ 0.

(32)

Mathematical Problems in Engineering

5

βˆ’1

From Lemma 2, ((1/2)𝐼 + πΆβ„Ž ) is uniformly bounded, and the errors of the approximate operators of π΄β„Ž , π΅β„Ž , πΆβ„Ž will be substituted into the following equations, then σ΅„© βˆ’1 σ΅„© σ΅„©σ΅„© σ΅„© σ΅„©σ΅„© 1 σ΅„©σ΅„© β„Ž 󡄩󡄩𝐿 2 𝑦 βˆ’ 𝐿 2 𝑦󡄩󡄩󡄩 ≀ σ΅„©σ΅„©σ΅„©( 𝐼 + πΆβ„Ž ) σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©σ΅„© 2 σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© β‹… σ΅„©σ΅„©σ΅„©σ΅„©(π΄β„Ž + π΅β„Ž ) 𝑦 βˆ’ (𝐴 + 𝐡) 𝑦󡄩󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„©σ΅„© βˆ’1 βˆ’1 1 σ΅„© σ΅„© 1 + σ΅„©σ΅„©σ΅„©σ΅„©( 𝐼 + πΆβ„Ž ) (πΆβ„Ž βˆ’ 𝐢) ( 𝐼 + 𝐢) (𝐴 + 𝐡) 𝑦󡄩󡄩󡄩󡄩 2 σ΅„©σ΅„© σ΅„©σ΅„© 2

Proof. Let (16) subtract (26) at 𝑑 = 𝑑𝑖 , then 𝐼 ( + 𝐢 + 𝑀) 𝑒 βˆ’ (𝐴 + 𝐡) 𝑔 (𝑒) 2 𝐼 βˆ’ ( + πΆβ„Ž + π‘€β„Ž ) π‘’β„Ž + (π΄β„Ž + π΅β„Ž ) 𝑔 (π‘’β„Ž ) 2

(36)

= 𝑓 βˆ’ π‘“β„Ž ; (33)

that is, (𝑒 βˆ’ π‘’β„Ž ) + (𝐢𝑒 βˆ’ πΆβ„Ž π‘’β„Ž ) + (𝑀𝑒 βˆ’ π‘€β„Ž π‘’β„Ž ) 2

󳨀→ 0, as β„Ž 󳨀→ 0.

(37)

βˆ’ ((𝐴 + 𝐡) 𝑔 (𝑒) βˆ’ (π΄β„Ž + π΅β„Ž ) 𝑔 (π‘’β„Ž )) = 𝑓 βˆ’ π‘“β„Ž .

So we proved that {πΏβ„Ž2 } is pointwise convergent to 𝐿 2 , as 𝑛 β†’ ∞. Since the series {πΏβ„Ž2 } are asymptotically compact sequences and pointwise convergent to 𝐿 2 , it has been a.c proved that πΏβ„Ž2 󳨀󳨀→ 𝐿 2 . Similar proof can be done for {πΏβ„Ž1 }, so the proof of Theorem 3 is completed. In order to obtain the asymptotically compact convergence of the nonlinear equations, some assumptions should be given [14, 23] firstly.

Consider the components 𝐢𝑒 βˆ’ πΆβ„Ž π‘’β„Ž in the equations 𝐢𝑒 βˆ’ πΆβ„Ž π‘’β„Ž = (𝐢𝑒 βˆ’ πΆβ„Ž 𝑒) + (πΆβ„Ž 𝑒 βˆ’ πΆβ„Ž π‘’β„Ž ) = (𝐢 βˆ’ πΆβ„Ž ) 𝑒 + πΆβ„Ž (𝑒 βˆ’ π‘’β„Ž ) ,

and similar results can be obtained as 𝑀𝑒 βˆ’ π‘€β„Ž π‘’β„Ž = (𝑀 βˆ’ π‘€β„Ž )𝑒 + π‘€β„Ž (𝑒 βˆ’ π‘’β„Ž ). Moreover, using the mean value theorem of differentials, we obtain ((𝐴 + 𝐡) 𝑔 (𝑒) βˆ’ (π΄β„Ž + π΅β„Ž ) 𝑔 (π‘’β„Ž ))

Assumptions 4. (1): 𝑔(β‹…, 𝑒) is measurable and differentiable for 𝑒 ∈ 𝑅, and 𝑔(π‘₯, β‹…) is continuous for π‘₯ ∈ Ξ“. (2): (πœ•/πœ•π‘’)𝑔(π‘₯, 𝑒) is Borel measurable and satisfies the inequality 0 < 𝑙 < (πœ•/πœ•π‘’)𝑔(π‘₯, 𝑒) < 𝐿 < ∞.

= ((𝐴 + 𝐡 βˆ’ π΄β„Ž βˆ’ π΅β„Ž ) 𝑔 (𝑒)) + (π΄β„Ž + π΅β„Ž ) (𝑔 (𝑒) βˆ’ 𝑔 (π‘’β„Ž ))

a.c

βˆ’1

(𝐼 + πΆβ„Ž ) (π‘€β„Ž + (π΄β„Ž + π΅β„Ž ) 𝑔󸀠 (𝑒))

(34)

a.c

󳨀→ (𝐼 + 𝐢)βˆ’1 (𝑀 + (𝐴 + 𝐡) 𝑔󸀠 (𝑒)) .

In this section, we derive the asymptotic expansion of errors for the solution and construct the extrapolation algorithm to obtain higher accuracy order solutions.

where π‘’β„Ž = π‘’β„Ž + 𝑑(𝑒 βˆ’ π‘’β„Ž ) (0 ≀ 𝑑 ≀ 1). Note that 𝑀 and 𝐡 are continuous operators, A is a logarithmic operator, and C is a Cauchy operator, so the approximate errors will be (𝐢 βˆ’ πΆβ„Ž )𝑒 = 𝑂(β„Ž2π‘š ), (𝑀 βˆ’ π‘€β„Ž )𝑒 = 𝑂(β„Ž2π‘š ), 𝑓 βˆ’ π‘“β„Ž = β„Ž3 πœ”1 + 𝑂(β„Ž5 ), and at 𝑑 = 𝑑𝑖 . (40)

Equation (37) can be simplified as 1 ( 𝐼 + πΆβ„Ž + π‘€β„Ž βˆ’ (π΄β„Ž + π΅β„Ž ) 𝑔󸀠 (π‘’β„Ž )) (𝑒 βˆ’ π‘’β„Ž ) 2 3

Theorem 5. Considering the asymptotic property and π‘₯(𝑑), 𝑓(𝑑) ∈ 𝑉2π‘š [0, 2πœ‹], there exists a function πœ”1 ∈ 𝑉2π‘šβˆ’2 [0, 2πœ‹] independent of β„Ž, such that

= β„Ž πœ” + 𝑂 (β„Ž ) , where πœ” = πœ”1 + πœ”2 . Since ((1/2)𝐼 + πΆβ„Ž ) uniformly bounded, (41) is rewritten as

(𝐼 + πΏβ„Ž ) (𝑒 βˆ’ π‘’β„Ž ) = β„Ž3 πœ™ + 𝑂 (β„Ž5 ) , βˆ’1

(35)

(41)

5

βˆ’1

4.1. Asymptotic Expansions

󡄨 󡄨 (𝑒 βˆ’ π‘’β„Ž )󡄨󡄨󡄨𝑑=𝑑𝑗 = β„Ž3 πœ”1 󡄨󡄨󡄨󡄨𝑑=𝑑 + 𝑂 (β„Ž5 ) 𝑗

+ (π΄β„Ž + π΅β„Ž ) 𝑔󸀠 (π‘’β„Ž ) (𝑒 βˆ’ π‘’β„Ž )

(𝐴 + 𝐡 βˆ’ π΄β„Ž βˆ’ π΅β„Ž ) 𝑔 (𝑒) = β„Ž3 πœ”2 + 𝑂 (β„Ž5 )

4. Asymptotic Expansions and Extrapolation

(39)

= (𝐴 + 𝐡 βˆ’ π΄β„Ž βˆ’ π΅β„Ž ) 𝑔 (𝑒)

The asymptotically compact convergence will be drawn according to the assumption about the function 𝑔(𝑒): (π΄β„Ž + π΅β„Ž ) 𝑔󸀠 (𝑒) 󳨀→ (𝐴 + 𝐡) 𝑔󸀠 (𝑒) ,

(38)

exists and is (42)

where πΏβ„Ž = ((1/2)𝐼 + πΆβ„Ž ) (π‘€β„Ž βˆ’ (π΄β„Ž + π΅β„Ž )𝑔󸀠 (π‘’β„Ž )) and πœ™ = βˆ’1 ((1/2)𝐼 + πΆβ„Ž ) πœ”.

6

Mathematical Problems in Engineering y 0.5

An auxiliary equation will be introduced with the solution πœ”1 (𝐼 + 𝐿) πœ”1 = πœ™,

(43)

and the corresponding approximate equations will be β„Ž

(𝐼 + 𝐿 ) πœ”1β„Ž = πœ™β„Ž .

0.3 x

(44)

Substituting (44) into (42) yields the equations: 󡄨 (𝐼 + πΏβ„Ž ) (π‘’β„Ž βˆ’ 𝑒 βˆ’ β„Ž3 πœ”1β„Ž )󡄨󡄨󡄨󡄨𝑑=𝑑 = 𝑂 (β„Ž5 ) . 𝑗

(45)

Noticing that πœ”1β„Ž ∈ 𝑉2π‘šβˆ’2 [0, 2πœ‹] and using the results of (42) yield (𝐼 + πΏβ„Ž ) (πœ”1 βˆ’ πœ”1β„Ž ) (𝑑𝑖 ) = 𝑂 (β„Ž3 ) .

(46)

Replace πœ”β„Žl by πœ”π‘™ and consider πΏβ„Ž is an asymptotic compact operator, 󡄨 (π‘’β„Ž βˆ’ 𝑒 βˆ’ β„Ž3 πœ”1 )󡄨󡄨󡄨󡄨𝑑=𝑑 = 𝑂 (β„Ž5 ) ; 𝑗

(47)

we complete the proof. 4.2. Extrapolation Algorithms. An asymptotic expansion about the errors in (35) implies that the extrapolation algorithms [24] can be applied to the solution of (2) to improve the approximate order. The high accuracy order 𝑂(β„Ž5 ) can be obtained by computing some coarse grids and fine grids on Ξ“ in parallel. The EAs are described as follows. Step 1. Take the mesh widths β„Ž and β„Ž/2 to obtain the solution of (26) in parallel, where π‘’β„Ž (𝑑𝑖 ), π‘’β„Ž/2 (𝑑𝑖 ) are the solutions on Ξ“ corresponding to β„Ž and β„Ž/2, respectively. Step 2. Use the values at coarse grids and fine grids to calculate the approximate values at 𝑑𝑖 π‘’β„Žβˆ—

1 (𝑑𝑖 ) = (8π‘’β„Ž/2 (𝑑𝑖 ) βˆ’ π‘’β„Ž (𝑑𝑖 )) , 7

(48)

and the error is |π‘’β„Žβˆ— (𝑑𝑖 ) βˆ’ 𝑒(𝑑𝑖 )| = 𝑂(β„Ž5 ). Step 3. So we can construct a posteriori error estimate: 󡄨󡄨 1 󡄨󡄨 󡄨 󡄨󡄨 󡄨󡄨𝑒 (𝑑𝑖 ) βˆ’ π‘’β„Ž/2 (𝑑𝑖 )󡄨󡄨󡄨 ≀ 󡄨󡄨󡄨󡄨𝑒 (𝑑𝑖 ) βˆ’ (8π‘’β„Ž/2 (𝑑𝑖 ) βˆ’ π‘’β„Ž (𝑑𝑖 ))󡄨󡄨󡄨󡄨 7 󡄨 󡄨 1 󡄨󡄨 󡄨 󡄨𝑒 (𝑑 ) βˆ’ π‘’β„Ž (𝑑𝑖 )󡄨󡄨󡄨 7 󡄨 β„Ž/2 𝑖 󡄨 󡄨 = 󡄨󡄨󡄨𝑒 (𝑑𝑖 ) βˆ’ π‘’β„Žβˆ— (𝑑𝑖 )󡄨󡄨󡄨 +

+ ≀

(49)

1 󡄨󡄨 󡄨 󡄨𝑒 (𝑑 ) βˆ’ π‘’β„Ž (𝑑𝑖 )󡄨󡄨󡄨 7 󡄨 β„Ž/2 𝑖

1 󡄨󡄨 󡄨 󡄨𝑒 (𝑑 ) βˆ’ π‘’β„Ž (𝑑𝑖 )󡄨󡄨󡄨 , 7 󡄨 β„Ž/2 𝑖

which is useful for constructing self-adaptive algorithms.

Figure 1: The shape of the elliptical body.

5. Numerical Example Example 1. We first introduce some notations for 𝑖 = 1, 2: π‘’π‘–β„Ž (𝑃) = |π‘’π‘–β„Ž (𝑃) βˆ’ 𝑒𝑖 (𝑃)| is the error of the displacement; π‘Ÿπ‘–β„Ž (𝑃) = π‘’π‘–β„Ž (𝑃)/π‘’π‘–β„Ž/2 (𝑃) is the error ratio; π‘’β„Žπ‘– (𝑃) = βˆ— |π‘’π‘–β„Ž (𝑃) βˆ’ π‘’π‘–βˆ— (𝑃)| is the error after one-step EAs; and π‘π‘–β„Ž (𝑃) = (1/7)|π‘’π‘–β„Ž/2 (𝑃) βˆ’ π‘’π‘–β„Ž (𝑃)| is a posteriori error estimate. Suppose Ξ© is an isotropic elliptical body with the axis π‘Ž = 0.3, 𝑏 = 0.5 in the plane domain shown in Figure 1. The parameter formulae for the boundary Ξ“ will be described as π‘₯ = 0.3 cos(𝑑), 𝑦 = 0.5 sin(𝑑), 𝑑 ∈ [0, 2πœ‹]. And the nonlinear boundary values 𝑝 = (𝑔1 , 𝑔2 )𝑇 are set as 𝑔1 = βˆ’π‘’12 + cos(𝑑), 𝑔2 = sin(𝑒2 ) + 2 sin(𝑑), 𝑑 ∈ [0, 2πœ‹]. We firstly calculate the numerical solutions π‘’β„Ž = (𝑒1β„Ž , 𝑒2β„Ž )𝑇 on the boundary Ξ“ following (26). Table 1 lists the approximate values of 𝑒1β„Ž (𝑃) at points 𝑃1 = (π‘Ž cos(πœ‹/8), 𝑏 sin(πœ‹/8)) and 𝑃2 = (π‘Ž cos(πœ‹/4), 𝑏 sin(πœ‹/4)). Table 2 lists the approximate values of 𝑒2β„Ž (𝑃) at points 𝑃1 = (π‘Ž cos(πœ‹/8), 𝑏 sin(πœ‹/8)) and 𝑃2 = (π‘Ž cos(πœ‹/4), 𝑏 sin(πœ‹/4)). From Tables 1-2, we can numerically see log2 π‘Ÿπ‘–β„Ž (𝑃) β‰ˆ 3, log2 π‘Ÿβ„Žπ‘– (𝑃) β‰ˆ 5,

(50)

which shows that the convergent orders are three for the approximation solutions π‘’π‘–β„Ž and will be improved to five orders after EAs. The normal derivative π‘β„Ž = (𝑝1β„Ž , 𝑝2β„Ž )𝑇 on Ξ“ can be obtained when the displacement π‘’β„Ž = (𝑒1β„Ž , 𝑒2β„Ž )𝑇 on Ξ“ is substituted into the boundary condition of (1). So following (4), the displacement π‘’β„Ž = (𝑒1β„Ž , 𝑒2β„Ž )𝑇 in Ξ© can be calculated. Table 3 shows the approximate values of the displacement (𝑒1β„Ž (𝑃0 ), 𝑒2β„Ž (𝑃0 ))𝑇 at an inner point 𝑃0 = (1/8)(cos(πœ‹/3), sin(πœ‹/3)) in Ξ©. From Table 3, we can numerically see that the β„Ž3 Richardson extrapolation algorithms are effective in obtaining high accuracy approximate solutions in which just the values at coarse grids and fine grids are applied. Furthermore, the a posteriori error estimate can be used to reckon the errors of the numerical solutions.

Mathematical Problems in Engineering

7

Table 1: The errors and errors ratio of 𝑒1β„Ž (𝑃) at points 𝑃 = 𝑃1 , 𝑃2 . 𝑛 𝑒1β„Ž (𝑃1 ) π‘Ÿ1β„Ž (𝑃1 ) π‘’β„Ž1 (𝑃1 ) π‘Ÿβ„Ž1 (𝑃1 ) 𝑒1β„Ž (𝑃2 ) π‘Ÿ1β„Ž (𝑃2 ) π‘’β„Ž1 (𝑃2 ) π‘Ÿβ„Ž1 (𝑃2 )

16 8.466𝐸 βˆ’ 4

32 1.040𝐸 βˆ’ 4 8.137 4.12𝐸 βˆ’ 07

6.085𝐸 βˆ’ 4

7.501𝐸 βˆ’ 5 8.112 3.91𝐸 βˆ’ 07

64 1.296𝐸 βˆ’ 5 8.028 1.29𝐸 βˆ’ 08 31.92 9.292𝐸 βˆ’ 6 8.073 1.24𝐸 βˆ’ 8 31.47

128 1.618𝐸 βˆ’ 6 8.011 4.03𝐸 βˆ’ 10 31.99 1.160𝐸 βˆ’ 6 8.008 4.00𝐸 βˆ’ 10 31.09

256 2.022𝐸 βˆ’ 7 8.002 1.24𝐸 βˆ’ 11 32.41 1.450𝐸 βˆ’ 7 8.000 1.30𝐸 βˆ’ 11 30.85

512 2.527𝐸 βˆ’ 8 8.000 3.93𝐸 βˆ’ 13 31.68 1.813𝐸 βˆ’ 8 8.000 4.03𝐸 βˆ’ 13 32.16

256 9.753𝐸 βˆ’ 8 8.003 2.45𝐸 βˆ’ 10 31.77 2.217𝐸 βˆ’ 7 8.001 2.79𝐸 βˆ’ 10 30.69

512 1.219𝐸 βˆ’ 8 8.000 7.75𝐸 βˆ’ 12 31.65 2.771𝐸 βˆ’ 8 8.000 8.83𝐸 βˆ’ 12 31.56

128 1.808𝐸 βˆ’ 7 8.002 7.04𝐸 βˆ’ 10 1.809𝐸 βˆ’ 7 9.599𝐸 βˆ’ 8 8.005 1.62𝐸 βˆ’ 9 9.604𝐸 βˆ’ 8

256 2.260𝐸 βˆ’ 8 8.000 2.22𝐸 βˆ’ 11 2.260𝐸 βˆ’ 8 1.200𝐸 βˆ’ 8 8.000 5.11𝐸 βˆ’ 11 1.200𝐸 βˆ’ 8

Table 2: The errors and errors ratio of 𝑒2β„Ž (𝑃) at points 𝑃 = 𝑃1 , 𝑃2 . 𝑛 𝑒2β„Ž (𝑃1 ) π‘Ÿ2β„Ž (𝑃1 ) π‘’β„Ž2 (𝑃1 ) π‘Ÿβ„Ž2 (𝑃1 ) 𝑒2β„Ž (𝑃2 ) π‘Ÿ2β„Ž (𝑃2 ) π‘’β„Ž2 (𝑃2 ) π‘Ÿβ„Ž2 (𝑃2 )

16 4.169𝐸 βˆ’ 4

32 5.100𝐸 βˆ’ 5 8.175 7.67𝐸 βˆ’ 06

9.338𝐸 βˆ’ 4

1.147𝐸 βˆ’ 4 8.138 8.30𝐸 βˆ’ 06

64 6.309𝐸 βˆ’ 6 8.083 2.44𝐸 βˆ’ 07 31.44 1.420𝐸 βˆ’ 5 8.080 2.75𝐸 βˆ’ 7 30.21

128 7.805𝐸 βˆ’ 7 8.018 7.80𝐸 βˆ’ 9 31.29 1.774𝐸 βˆ’ 6 8.005 8.75𝐸 βˆ’ 9 31.40

Table 3: The errors and errors ratio of (𝑒1β„Ž , 𝑒2β„Ž )𝑇 at inner points 𝑃 = 𝑃0 . 𝑛 𝑒1β„Ž (𝑃0 ) π‘Ÿ1β„Ž (𝑃0 ) π‘’β„Ž1 (𝑃0 ) 𝑝1β„Ž (𝑃0 ) 𝑒2β„Ž (𝑃0 ) π‘Ÿ2β„Ž (𝑃0 ) π‘’β„Ž2 (𝑃0 ) 𝑝2β„Ž (𝑃0 )

8 8.156𝐸 βˆ’ 4

4.372𝐸 βˆ’ 4

16 9.700𝐸 βˆ’ 5 8.409 2.13𝐸 βˆ’ 05 9.727𝐸 βˆ’ 5 5.145𝐸 βˆ’ 5 8.498 4.82𝐸 βˆ’ 05 5.154𝐸 βˆ’ 5

32 1.172𝐸 βˆ’ 5 8.276 6.81𝐸 βˆ’ 07 1.183𝐸 βˆ’ 5 6.227𝐸 βˆ’ 6 8.262 1.58𝐸 βˆ’ 6 6.249𝐸 βˆ’ 6

64 1.447𝐸 βˆ’ 6 8.099 2.21𝐸 βˆ’ 08 1.452𝐸 βˆ’ 6 7.684𝐸 βˆ’ 7 8.104 5.09𝐸 βˆ’ 8 7.688𝐸 βˆ’ 7

6. Conclusion

Acknowledgments

The mechanical quadrature methods provide a way to approximate the solutions of the nonlinear boundary value problems with high accuracy. The following conclusions can be drawn. These methods show that accuracy orders can be three and can be applied with the EAs to obtain higher accuracy. Furthermore, the larger scale of the problems, the more precision of the methods. To obtain the entry of discrete matrixes is very simple and straightforward, without any singular integrals. The problems, about Neumann boundary condition or nonlinear boundary condition in polygon elasticity, will be researched in our following work.

This project is supported by Natural Science Foundation of Chongqing (CSTC2013JCYJA00017) and supported by the Educational Council Foundation of Chongqing (KJ1500517 and KJ1600512).

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

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