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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 891519, 18 pages
http://dx.doi.org/10.1155/2012/891519
Research Article

Iterative Method for Solving the Second Boundary Value Problem for Biharmonic-Type Equation

1Institute of Information Technology, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi 10000, Vietnam
2Hanoi University of Industry, Minh Khai, Tu Liem, Hanoi 10000, Vietnam

Received 8 April 2012; Revised 11 June 2012; Accepted 11 June 2012

Academic Editor: Carla Roque

Copyright © 2012 Dang Quang A. and Nguyen Van Thien. 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.

Abstract

Solving boundary value problems (BVPs) for the fourth-order differential equations by the reduction of them to BVPs for the second-order equations with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by ourselves in recent works, we construct iterative method for the second BVP for biharmonic-type equation, which describes the deflection of a plate resting on a biparametric elastic foundation. The convergence rate of the method is established. The optimal value of the iterative parameter is found. Several numerical examples confirm the efficiency of the proposed method.

1. Introduction

Solving BVPs for the fourth-order differential equations by the reduction of them to BVPs for the second-order equations with the aim to use a lot of efficient algorithms for the latter ones attracts attention from many researchers. Namely, for the biharmonic equation with the Dirichlet boundary condition, there is intensively developed the iterative method, which leads the problem to two problems for the Poisson equation at each iteration (see e.g., [13]). Recently, Abramov and Ul’yanova [4] proposed an iterative method for the Dirichlet problem for the biharmonic-type equation, but the convergence of the method is not proved. In our previous works [5, 6] with the help of boundary or mixed boundary-domain operators appropriately introduced, we constructed iterative methods for biharmonic and biharmonic-type equations associated with the Dirichlet boundary condition. For the biharmonic-type equation with Neumann boundary conditions in [7] an iterative method also was proposed. It is proved that the iterative methods are convergent with the rate of geometric progression. In this paper we develop our technique in the above-mentioned papers for the second BVP for the biharmonic-type equation. Namely, we consider the following problem Δ2𝑢𝑎Δ𝑢+𝑏𝑢=𝑓inΩ,(1.1)𝑢=𝑔1onΓ,(1.2)Δ𝑢=𝑔2,onΓ,(1.3)

where Δ is the Laplace operator, Ω is a bounded domain in 𝑅𝑛(𝑛2),Γ is the sufficiently smooth boundary of Ω, and 𝑎,𝑏 are nonnegative constants. This problem has not yet considered in [4].

It should be noticed that when 𝑎=0,𝑏=0 (1.1) is the equation for a thin plate, and problem (1.1)–(1.3) is decomposed into two consecutive problems for the Poisson equations.

In this paper we suppose that 𝑎0,𝑏0,𝑎+𝑏>0. Then (1.1) describes the deflection of a plate resting on biparametric elastic foundation. For solving this equation several methods such as the boundary element, the finite element methods [810], domain/boundary element technique [11], and boundary differential integral equation (BDIE) method [12] were used. It should be noticed that at present the boundary element method is intensively developed and is used for solving more complex problems of plates and shells (see e.g., [1315]).

In this paper we use completely different approach to (1.1). Two cases will be treated in dependence on the sign of 𝑎24𝑏. In the case if 𝑎24𝑏0 we can immediately decompose the problem into two problems for second-order equations. In the opposite case we propose an iterative method for reducing problem (1.1)–(1.3) to a sequence of second-order problems. The convergence of the method is established and verified on examples.

2. Case 𝑎24𝑏0

In this case we always can lead the original problem (1.1)–(1.3) to two problems for second-order equations. To do this, we put 1𝜇=2𝑎+𝑎2.4𝑏(2.1) Then problem (1.1)–(1.3) is reduced to the following problems: 1𝜇Δ𝑣𝑏𝑣=𝑓,inΩ,𝑣=𝜇𝑔2𝑔11,onΓ,𝜇Δ𝑢𝑢=𝑣,inΩ,𝑢=𝑔1,onΓ.(2.2) These Dirichlet problems can be solved by known methods such as finite element method, boundary element method, or finite difference method. Some fast Poisson solvers in [16, 17] can be applied for the above problems.

3. Case 𝑎24𝑏<0

This case is very important in mechanics because (1.1) describes the bending plate on elastic foundation (see [18]).

3.1. Construction of Iterative Method on Continuous Level

As in [6], we set Δ𝑢=𝑣,(3.1)𝜑=𝑏𝑢.(3.2)

Then problem (1.1)–(1.3) leads to the following second-order problems Δ𝑣𝑎𝑣=𝑓+𝜑,inΩ,𝑣=𝑔2,onΓ,Δ𝑢=𝑣,inΩ,𝑢=𝑔1,onΓ,(3.3) where all the functions 𝑢,𝑣, and 𝜑 are unknown but they are related with each other by (3.2).

Now consider the following iterative process for finding 𝜑 and simultaneously for finding 𝑣,𝑢. (1)Given 𝜑(0)𝐿2(Ω), for example, 𝜑(0)=0inΩ.(3.4)(2)Knowing 𝜑(𝑘)(𝑥) on Ω(𝑘=0,1,) solve consecutively two problems Δ𝑣(𝑘)𝑎𝑣(𝑘)=𝑓+𝜑(𝑘)𝑣inΩ,(𝑘)=𝑔2onΓ,(3.5)Δ𝑢(𝑘)=𝑣(𝑘)𝑢inΩ,(𝑘)=𝑔1onΓ.(3.6)(3)Compute the new approximation 𝜑(𝑘+1)=1𝜏𝑘+1𝜑(𝑘)𝑏𝜏𝑘+1𝑢(𝑘)inΩ,(3.7) where 𝜏𝑘+1 is an iterative parameter to be chosen later.

3.2. Investigation of Convergence

In order to investigate the convergence of the iterative process (3.4)–(3.7), firstly we rewrite (3.7) in the canonical form of two-layer iterative scheme [19] 𝜑(𝑘+1)𝜑(𝑘)𝜏𝑘+1+𝜑(𝑘)+𝑏𝑢(𝑘)=0.(3.8)

Now, we introduce the operator 𝐴 defined by the formula 𝐴𝜑=𝑢,(3.9) where 𝑢 is the function determined from the problems Δ𝑣𝑎𝑣=𝜑inΩ,𝑣=0onΓ,(3.10)Δ𝑢=𝑣inΩ,𝑢=0onΓ.(3.11)

The properties of the operator 𝐴 will be investigated in the sequel.

Now, let us return to the problem (3.3). We represent their solutions in the form 𝑢=𝑢1+𝑢2,𝑣=𝑣1+𝑣2,(3.12) where 𝑢1,𝑢2,𝑣1,𝑣2 are the solutions of the problems Δ𝑣1𝑎𝑣1𝑣=𝜑inΩ,1=0onΓ,Δ𝑢1=𝑣1𝑢inΩ,1=0onΓ,(3.13)Δ𝑣2𝑎𝑣2𝑣=𝑓inΩ,2=𝑔2onΓ,Δ𝑢2=𝑣2𝑢inΩ,2=𝑔1onΓ.(3.14) According to the definition of 𝐴 we have 𝐴𝜑=𝑢1.(3.15) Since 𝜑 should satisfy the relation (3.2), taking into account the representation (3.12) we obtain the equation (𝐼+𝑏𝐴)𝜑=𝐹,(3.16) where 𝐼 is the identity operator, and 𝐹=𝑏𝑢2.(3.17)

Thus, we have reduced the original problem (1.1)–(1.3) to the operator equation (3.16), whose right-hand side is completely defined by the data functions 𝑓,𝑔, and , and coefficients 𝑎,𝑏.

Proposition 3.1 3.1. The iterative process (3.4)–(3.7) is a realization of the two-layer iterative scheme 𝜑(𝑘+1)𝜑(𝑘)𝜏𝑘+1+(𝐼+𝑏𝐴)𝜑(𝑘)=𝐹,𝑘=0,1,2,(3.18) for the operator equation (3.16).

Proof. Indeed, if in (3.5), (3.6) we put 𝑢(𝑘)=𝑢1(𝑘)+𝑢2,𝑣(𝑘)=𝑣1(𝑘)+𝑣2,(3.19) where 𝑣2,𝑢2 are the solutions of problem (3.14), then we get Δ𝑣1(𝑘)𝑎𝑣1(𝑘)=𝜑(𝑘)𝑣inΩ,1(𝑘)=0onΓ,Δ𝑢1(𝑘)=𝑣1(𝑘)𝑢inΩ,1(𝑘)=0onΓ.(3.20) From here it is easy to see that 𝐴𝜑(𝑘)=𝑢1(𝑘).(3.21) Therefore, taking into account the first relation of (3.19), the above equality, and (3.17), from (3.8) we obtain (3.18). Thus, the proposition is proved.

Proposition 3.1 enables us to lead the investigation of convergence of processs (3.4)–(3.7) to the study of scheme (3.18). For this reason we need some properties of the operator 𝐴.

Proposition 3.2. The operator 𝐴 defined by (3.9)–(3.11) is linear, symmetric, positive, and compact operator in the space 𝐿2(Ω).

Proof. The linearity of 𝐴 is obvious. To establish the other properties of 𝐴 let us consider the inner product (𝐴𝜑,𝜑) for two arbitrary functions 𝜑 and 𝜑 in 𝐿2(Ω). Recall that the operator 𝐴 is defined by (3.9)–(3.11). We denote by 𝑢 and 𝑣 the solutions of (3.10) and (3.11), where instead of 𝜑 there stands 𝜑. We have 𝐴𝜑,𝜑=Ω𝑢𝜑𝑑𝑥=Ω𝑢Δ𝑣𝑎𝑣=𝑑𝑥Ω𝑢Δ𝑣𝑑𝑥𝑎Ω𝑢𝑣𝑑𝑥.(3.22) Applying the Green formula for the functions 𝑢 and 𝑣, vanishing on the boundary Γ, we obtain Ω𝑢Δ𝑣𝑑𝑥=Ω𝑣Δ𝑢𝑑𝑥=Ω𝑣𝑣𝑑𝑥,Ω𝑢𝑣=Ω𝑢Δ𝑢=Ω𝑢𝑢𝑑𝑥.(3.23) Hence, 𝐴𝜑,𝜑=Ω𝑣𝑣𝑑𝑥+𝑎Ω𝑢𝑢𝑑𝑥.(3.24) Clearly, 𝐴𝜑,𝜑=𝐴,𝜑,𝜑(𝐴𝜑,𝜑)=Ω𝑣2𝑑𝑥+𝑎Ω||||𝑢2𝑑𝑥0(3.25) are due to 𝑎0. Moreover, it is easy seen that (𝐴𝜑,𝜑)=0 if and only if 𝜑=0. So, we have shown that the operator 𝐴 is symmetric and positive in 𝐿2(Ω).
It remains to show the compactness of 𝐴. As is well known that if 𝜑𝐿2(Ω) then problem (3.10) has a unique solution 𝑣𝐻2(Ω), therefore, problem (3.11) has a unique solution 𝑣𝐻4(Ω). Consequently, the operator 𝐴 maps 𝐿2(Ω) into 𝐻4(Ω). In view of the compact imbedding 𝐻4(Ω) into 𝐿2(Ω) it follows that 𝐴 is compact operator in 𝐿2(Ω).
Thus, the proof of Proposition 3.2 is complete.

Due to the above proposition the operator 𝐵=𝐼+𝑏𝐴(3.26) is linear, symmetric, positive definite, and bounded operator in the space 𝐿2(Ω). More precisely, we have 𝛾1𝐼<𝐵𝛾2𝐼,(3.27) where 𝛾1=1,𝛾2=1+𝑏𝐴.(3.28) Notice that since the operator 𝐴 is defined by (3.9)–(3.11) its norm 𝐴 depends on the value of 𝑎 but not on 𝑏 in (1.1).

From the theory of elliptic problems [20] we have the following estimates for the functions 𝑣,𝑢 given by (3.10), (3.11): 𝑣𝐻2(Ω)𝐶1𝜑𝐿2(Ω),𝑢𝐻4(Ω)𝐶2𝑣𝐻2(Ω),(3.29) where 𝐶1,𝐶2 are constants. Therefore, 𝑢𝐻4(Ω)𝐶1𝐶2𝜑𝐿2(Ω).(3.30) Before stating the result of convergence of the iterative process (3.5)–(3.7) we assume the following regularity of the data of the original problem (1.1)–(1.3): 𝑓𝐿2(Ω),𝑔1𝐻7/2(Γ),𝑔2𝐻5/2(Γ).(3.31) Then the problem (1.1)–(1.3) has a unique solution 𝑢𝐻4(Ω). For the function 𝑢2 determined by (3.14) we have also 𝑢2𝐻4(Ω).

Theorem 3.3. Let 𝑢 be the solution of problem (1.1)–(1.3) and 𝜑 be the solution of (3.16). Then, if {𝜏𝑘+1} is the Chebyshev collection of parameters, constructed by the bounds 𝛾1 and 𝛾2 of the operator 𝐵, namely 𝜏0=2𝛾1+𝛾2,𝜏𝑘=𝜏0𝜌0𝑡𝑘+1,𝑡𝑘=cos2𝑘1𝜌2𝑀𝜋,𝑘=1,,𝑀,0=1𝜉𝛾1+𝜉,𝜉=1𝛾2,(3.32) we have 𝑢(𝑀)𝑢𝐻4(Ω)𝐶𝑞𝑀,(3.33) where 𝐶=𝐶1𝐶2𝜑(0)𝜑𝐿2(Ω),(3.34) with 𝐶1,𝐶2 being the constant in (3.30), 𝑞𝑀=2𝜌𝑀11+𝜌12𝑀,𝜌1=1𝜉1+𝜉.(3.35) In the case of the stationary iterative process, 𝜏𝑘=𝜏0(𝑘=1,2,.) we have 𝑢(𝑘)𝑢𝐻4(Ω)𝐶𝜌𝑘0,𝑘=1,2,.(3.36)

Proof. According to the general theory of the two-layer iterative schemes (see [21]) for the operator equation (3.16) we have 𝜑(𝑀)𝜑𝐿2(Ω)𝑞𝑀𝜑(0)𝜑𝐿2(Ω),(3.37) if the parameter {𝜏𝑘+1} is chosen by the formulae (3.32) and 𝜑(𝑘)𝜑𝐻4(Ω)𝜌𝑘0𝜑(0)𝜑𝐿2(Ω),𝑘=1,2,(3.38) if 𝜏𝑘=𝜏0(𝑘=1,2,). In view of these estimates the corresponding estimates (3.33) and (3.36) follow from (3.30) applied to the problems Δ𝑣(𝑘)𝑣𝑣𝑎(𝑘)𝑣=𝜑(𝑘)𝑣𝜑,inΩ,(𝑘)Δ𝑢𝑣=0onΓ,(𝑘)𝑢=𝑣(𝑘)𝑢𝑣,inΩ,(𝑘)𝑢=0onΓ,(3.39) which are obtained in the result of the subtraction of (3.3) from (3.5) and (3.6), respectively. The theorem is proved.

Remark 3.4. From the above theorem it is easy to see that for each fixed value of 𝑎 the numbers 𝜌0 and 𝑞𝑀 characterizing the rate of convergence of the iterative method decrease with the decrease of 𝑏. So, the smaller 𝑏 is, the faster the iterative process converges. In the special case when 𝑏=0 the mentioned above numbers also are zero, hence the iterative process converges at once and the original problem (1.1)–(1.3) is decomposed into two second-order problems.

3.3. Computation of the Norm 𝐴

As we see in Theorem 3.3 for determining the iterative parameter 𝜏 we need the bounds 𝛾1 and 𝛾2 of the operator 𝐵, and in its turn the latter bound requires to compute 𝐴. Therefore, below we consider the problem of computation 𝐴.

Suppose the domain Ω=[0,𝑙1]×[0,𝑙2] in the plane 𝑥𝑂𝑦. In this case by Fourier method we found that the system of functions 𝑒𝑚𝑛2(𝑥,𝑦)=𝑙1𝑙2sin𝑚𝜋𝑥𝑙1sin𝑛𝜋𝑦𝑙2(𝑚,𝑛=1,2,)(3.40) is the eigenfunctions of the spectral problem Δ𝑢=𝜆𝑢inΩ,𝑢=0onΓ(3.41) corresponding to the eigenvalues 𝜆𝑚𝑛=𝜋2𝑚2𝑙21+𝑛2𝑙22.(3.42) Moreover, this system is orthogonal and complete in 𝐿2(Ω).

Now let a function 𝜑𝐿2(Ω) have the expansion 𝜑(𝑥,𝑦)=𝑚,𝑛=1𝜑𝑚𝑛𝑒𝑚𝑛(𝑥,𝑦).(3.43) Then we have 𝜑2=(𝜑,𝜑)=𝑚,𝑛=1||𝜑𝑚𝑛||2.(3.44) Representing the solution 𝑣 of (3.10) in the form of the double series 𝑣(𝑥,𝑦)=𝑚,𝑛=1𝑣𝑚𝑛𝑒𝑚𝑛(𝑥,𝑦)(3.45) we found 𝑣𝑚𝑛=𝜑𝑚𝑛𝜆𝑚𝑛.𝑎(3.46) Next, we seek the solution of (3.11) in the form 𝑢(𝑥,𝑦)=𝑚,𝑛=1𝑢𝑚𝑛𝑒𝑚𝑛(𝑥,𝑦).(3.47) Then from (3.45) we find 𝑢𝑚𝑛=𝑣𝑚𝑛𝜆𝑚𝑛.(3.48) From the definition of the operator 𝐴 by (3.9)–(3.11) we have (𝐴𝜑,𝜑)=(𝑢,𝜑)=𝑚,𝑛=1𝑢𝑚𝑛𝑒𝑚𝑛,𝑚,𝑛=1𝜑𝑚𝑛𝑒𝑚𝑛=𝑚,𝑛=1𝑢𝑚𝑛𝜑𝑚𝑛=𝑚,𝑛=1||𝜑𝑚𝑛||2𝜆𝑚𝑛𝜆𝑚𝑛=𝑎𝑚,𝑛=1||𝜑𝑚𝑛||2𝜋2𝑚2𝑙12+𝑛2𝑙22𝜋2𝑚2𝑙12+𝑛2𝑙22+𝑎𝑚,𝑛=1||𝜑𝑚𝑛||2𝜋2𝑙12+𝑙22𝜋2𝑙12+𝑙22+𝑎(3.49) due to the orthogonality of the system {𝑒𝑚𝑛} and (3.46), (3.48), and (3.42). Thus, there holds the estimate 1(𝐴𝜑,𝜑)𝜋2𝑙12+𝑙22𝜋2𝑙12+𝑙22+𝑎(𝜑,𝜑).(3.50) The sign of equality occurs for 𝜑=𝑒11(𝑥,𝑦). Since 𝐴 is shown to be symmetric and positive in 𝐿2(Ω) it follows: 𝐴=sup𝜑0(𝐴𝜑,𝜑)=1(𝜑,𝜑)𝜋2𝑙12+𝑙22𝜋2𝑙12+𝑙22.+𝑎(3.51)

4. Numerical Realization of the Iterative Method

In the previous section we proposed and investigated an iterative method for problem (1.1)–(1.3) in the case if 𝑎24𝑏<0. Now we study numerical realization of the method.

For simplicity we consider the case, where the domain Ω is a rectangle, Ω=[0,𝑙1]×[0,𝑙2] in the plane 𝑥𝑂𝑦. In this domain we construct an uniform grid Ω=(𝑥,𝑦),𝑥=𝑖1,𝑦=𝑗2,0𝑖𝑁1,0𝑗𝑁2,(4.1) where 1=𝑙1/𝑁1,2=𝑙2/𝑁2.

Denote by Ω the set of inner nodes, by Γ the set of boundary nodes of the grid, and by 𝑣,𝑢, the grid functions defined on Ω.

Now consider a discrete version of the iterative method (3.4)–(3.7) when 𝜏𝑘𝜏. (1)Given a starting 𝜑(0), for example, 𝜑(0)=0inΩ.(4.2)(2)Knowing 𝜑(𝑘) on Ω(𝑘=0,1,) solve consecutively two difference problems Λ𝑣(𝑘)𝑎𝑣(𝑘)=𝑓+𝜑(𝑘)inΩ,𝑣(𝑘)=𝑔2onΓ,(4.3)Λ𝑢(𝑘)=𝑣(𝑘)inΩ,𝑢(𝑘)=𝑔1onΓ,(4.4) where Λ is the discrete Laplace operator, (Λ𝑣)𝑖𝑗=𝑣𝑖1,𝑗2𝑣𝑖𝑗+𝑣𝑖+1,𝑗21+𝑣𝑖,𝑗12𝑣𝑖𝑗+𝑣𝑖,𝑗+122.(4.5)(3)Compute the new approximation 𝜑(𝑘+1)=1𝜏𝜑(𝑘)𝑏𝜏𝑢(𝑘)inΩ.(4.6)It is easy to see that the convergence of the above iterative method is related to the discrete version 𝐴 of the operator 𝐴, defined by the formula 𝐴𝜑=𝑢,(4.7) where 𝑢 is determined from the difference problems Λ𝑣𝑎𝑣=𝜑inΩ,𝑣=0onΓ,Λ𝑢=𝑣inΩ,𝑢=0onΓ.(4.8) Using the results of the spectral problem for the discrete Laplace operator Λ (see [19]) we find the bounds of 𝐴: 1𝛽2𝛽2+𝑎𝐼𝐴1𝛽1𝛽1+𝑎𝐼,(4.9) where 𝛽1=421sin2𝜋12𝑙1+422sin2𝜋22𝑙2,𝛽2=421cos2𝜋12𝑙1+422cos2𝜋22𝑙2.(4.10) Therefore, for the operator 𝐵, the discrete version of 𝐵, we obtain the estimate 𝛾1𝐼𝐵𝛾2𝐼,(4.11) where 𝛾1𝑏=1+𝛽2𝛽2+𝑎,𝛾2𝑏=1+𝛽1𝛽1.+𝑎(4.12) Hence, we choose 𝜏=2𝛾1+𝛾2,(4.13) which is the optimal parameter of the iterative process (4.2)–(4.6).

Now we study the deviation of 𝑢(𝑘) from 𝑢(𝑘) obtained by the iterative process (3.4)–(3.7). In the future for short we will write instead of .

Proposition 4.1. For any 𝑘=0,1, there holds the estimate 𝜑(𝑘)𝜑(𝑘)=𝑂2,𝑢(𝑘)𝑢(𝑘)=𝑂2,(4.14) where 2=21+22,  𝑢(𝑘),𝜑(𝑘) are computed by the process (3.4)–(3.7) and 𝑢(𝑘),𝜑(𝑘) are computed by (4.2)–(4.6).

Proof. We shall prove this proposition by induction in 𝑘.
For 𝑘=0 we have ||𝜑(0)𝜑(0)||=0 and the second estimate in (4.14) is valid due to the the second-order accuracy of the iterative schemes (4.3) and (4.4) for the problems (3.5) and (3.6) (see [21]).
Now suppose (4.14) is valid for 𝑘10. We shall show that it also is valid for 𝑘. For this purpose we recall that 𝜑(𝑘) is computed by the formula 𝜑(𝑘)=(1𝜏)𝜑(𝑘1)𝑏𝜏𝑢(𝑘1)onΩ,(4.15) where 2𝜏=)(2+𝑏𝐴(4.16) and 𝜑(𝑘) is computed by the formula 𝜑(𝑘)=1𝜏𝜑(𝑘1)𝑏𝜏𝑢(𝑘1)onΩ,(4.17)𝜏 being given by (4.13) and (4.12).
From (4.10)–(4.13), (4.16), and (3.51) it is possible to obtain the estimate 𝜏=𝜏+𝑂2.(4.18) Next, subtracting (4.15) from (4.17) and taking into account the above formula we get 𝜑(𝑘)𝜑(𝑘)𝜑=(1𝜏)(𝑘1)𝜑(𝑘1)𝑢+𝜏𝑏(𝑘1)𝑢(𝑘1)+𝑂2.(4.19) By the assumptions of the induction 𝑢(𝑘1)𝑢(𝑘1)=𝑂2,𝜑(𝑘1)𝜑(𝑘1)=𝑂2(4.20) from (4.19) it follows 𝜑(𝑘)𝜑(𝑘)=𝑂(2). Now, having in mind this estimate due to the second-order approximation of the difference operators in (4.3) and (4.4) we get the second estimate in (4.14). Thus, the proof of the proposition is complete.

In realization of the discrete iterative process (4.2)–(4.6) we shall stop iterations when 𝑢(𝑘)𝑢(𝑘1)<TOL, where TOL is a some given accuracy. Then for the total error of the discrete solution 𝑢(𝑘) there there holds the following theorem.

Theorem 4.2. For the total error of the discrete solution 𝑢(𝑘) from the exact solution 𝑢 of the original problem (1.1)–(1.3) there holds the estimate 𝑢(𝑘)𝑢TOL+𝐶22+𝐶1𝜌0𝑘1,(4.21) where 𝐶1,𝐶2 are some constants and 𝜌0 is the number in (3.36).

Proof. We have the following estimate 𝑢(𝑘)𝑢𝑢(𝑘)𝑢(𝑘1)+𝑢(𝑘1)𝑢(𝑘1)+𝑢(𝑘1).𝑢(4.22) Since the space 𝐻4(Ω) is continuously embedded to the space 𝐶(Ω) (see [20]) from (3.36) we have 𝑢(𝑘1)𝑢𝐶1𝜌0𝑘1(4.23) for some constant 𝐶1. Now using Proposition 4.1 and the above estimate, from (4.22) we obtain (4.21). Thus, the theorem is proved.

Below we report the results of some numerical examples for testing the convergence of the iterative method. In all examples we test the iterative method for some values of 𝑎 and 𝑏 with TOL=105. The results of convergence of the method are given in tables, where 𝑘 is the number of iterations, 𝐸 is the error of approximate solution 𝑢(𝑘), 𝐸=||𝑢(𝑘)𝑢||.

Example 4.3. We take an exact solution 𝑢=sin𝑥sin𝑦 in the rectangle [0,𝜋]×[0,𝜋] and corresponding boundary conditions. The right-hand side of (1.1) in this case is 𝑓=(4+2𝑎+𝑏)sin𝑥sin𝑦.
The results of convergence in the case of the uniform grids including 65×65,129×129, and 257×257 nodes for 𝑎=0,𝑎=0.5 and 𝑎=1 are presented in Tables 1, 2, and 3, respectively.

tab1
Table 1: Convergence of the method in Example 4.3 for 𝑎=0.
tab2
Table 2: Convergence of the method in Example 4.3 for 𝑎=0.5.
tab3
Table 3: Convergence of the method in Example 4.3 for 𝑎=1.

Example 4.4. We take an exact solution 𝑢=𝑥sin𝑦+𝑦sin𝑥 in the rectangle [𝜋,𝜋]×[𝜋,𝜋] with corresponding boundary conditions. The the right-hand side of (1.1) is 𝑓=(1+𝑎+𝑏)(𝑥sin𝑦+𝑦sin𝑥).
The results of convergence in the case of the grids including 65×65,129×129, and 257×257 nodes for 𝑎=0,𝑎=0.5, and 𝑎=1 are presented in Tables 4, 5, and 6, respectively.
In the two above examples the grid step sizes are =𝜋/64,𝜋/128,𝜋/256. Therefore, 2=0.0024,6.0239𝑒4,  1.5060𝑒4. According to the estimate (4.21) the total error of the discrete approximate solution depends on 2. The columns 𝐸1,𝐸2,𝐸3 in Tables 16 show this fact. It is interesting to notice that in these tables 𝐸1/𝐸2,𝐸2/𝐸34 and (64×64)/(128×128)=(128×128)/(256×256)=4. It means that if the number of grid nodes increases in 4 times then it is expected that the accuracy of the approximate solution increases in the same times. From the tables we also observe that the number of iterations increases with the increase of the parameter 𝑏 for fixed values of parameter 𝑎. This confirms Remark 3.4. We also remark that for each pair of the parameters 𝑎and𝑏 the number of iterations for achieving an accuracy corresponding to the grid step size (or density of grid) does not depend on the grid step size.
In general the error of discrete approximate solution strongly depends on the step size of grid. So, it is not expected to get an approximate solution of higher accuracy on a grid of low density. However, in some exceptional cases we can obtain very accurate approximate solution on sparse grid. Below is an example showing this fact.

tab4
Table 4: Convergence of the method in Example 4.4 for 𝑎=0.
tab5
Table 5: Convergence of the method in Example 4.4 for 𝑎=0.5.
tab6
Table 6: Convergence of the method in Example 4.4 for 𝑎=1.

Example 4.5. We take an exact solution 𝑢=(𝑥24)(𝑦21) in the rectangle [2,2]×[1,1]. The the right-hand side of (1.1) is 𝑓=82𝑎(𝑥2+𝑦25)+𝑏(𝑥24)(𝑦21).
The results of convergence in the case of the grids including 65×65 and 129×129 nodes are presented in Tables 7 and 8, respectively.
From Tables 7 and 8 we see a high accuracy even on the grid 65×65. The reason of this fact is that for quadratic function the approximation error of the central difference scheme is zero for any grid step size.
The above three numerical examples demonstrate the fast convergence of the iterative method (3.4)–(3.7) for problem (1.1)–(1.3).
Below, we consider an example for examining the variation of the solution of Prob. (1.1)–(1.3) in dependence of the parameters 𝑎 and 𝑏.

tab7
Table 7: Convergence of the method in Example 4.5 for grid 65×65.
tab8
Table 8: Convergence of the method in Example 4.5 for grid 129×129.

Example 4.6. We take the the right-hand side of (1.1) 𝑓=1 and the boundary conditions 𝑔1=𝑔2=0 and use the proposed method for finding approximate solution for different values of 𝑎 and 𝑏. In all experiments we use the grid of 65×65 nodes in the domain Ω=[2,2]×[1,1] and TOL=105. It turns out that the number of iterations in all cases of 𝑎 and 𝑏 does not exceed 7 and the value of the solution at any fixed point is decreasing with the growth of 𝑎 and 𝑏. This fact is obvious from Figure 1. The graph of the solution in the case of 𝑎=0.5 and 𝑏=1 is given in Figure 2.

891519.fig.001
Figure 1: Variation of the value of 𝑢 in the middle point with 𝑏 for several values of 𝑎.
891519.fig.002
Figure 2: Solution in the case of 𝑎=0.5 and 𝑏=1.

5. Concluding Remark

In the paper an iterative method was proposed for reducing the second problem for biharmonic-type equation to a sequence of Dirichlet problems for second-order equations. The convergence of the method was proved. In the case when the computational domain is a rectangle the optimal iterative parameter was given. Several numerical examples in this case show fast convergence of the method. When the computational domain consists of rectangles the proposed iterative method can be applied successfully if combining with the domain decomposition method.

Acknowledgments

This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the Grant 102.99–2011.24. The authors would like to thank the anonymous reviewers sincerely for their helpful comments and remarks to improve the original manuscript.

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