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

Authors: Dang Quang A.; Nguyen Van Thien
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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., [1–3]). 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
(1.1)Δ2u-aΔu+bu=finΩ,(1.2)u=g1on⁡Γ,(1.3)Δu=g2,on⁡Γ,whereΔ is the Laplace operator, Ω is a bounded domain in Rn(n≥2),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 whena=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 thata≥0,b≥0,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 [8–10], 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., [13–15]).In this paper we use completely different approach to (1.1). Two cases will be treated in dependence on the sign of a2-4b. In the case if a2-4b≥0 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. Casea2-4b≥0

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
(2.1)μ=12(a+a2-4b).
Then problem (1.1)–(1.3) is reduced to the following problems:
(2.2)μΔv-bv=f,inΩ,v=1μg2-g1,on⁡Γ,1μΔu-u=v,inΩ,u=g1,on⁡Γ.
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. Casea2-4b<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
(3.1)Δu=v,(3.2)φ=-bu.Then problem (1.1)–(1.3) leads to the following second-order problems
(3.3)Δv-av=f+φ,inΩ,v=g2,on⁡Γ,Δu=v,inΩ,u=g1,on⁡Γ,
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 findingφ and simultaneously for finding v,u. (1)
Givenφ(0)∈L2(Ω), for example,
(3.4)φ(0)=0inΩ.(2)
Knowingφ(k)(x) on Ω(k=0,1,…) solve consecutively two problems
(3.5)Δv(k)-av(k)=f+φ(k)inΩ,v(k)=g2on⁡Γ,(3.6)Δu(k)=v(k)inΩ,u(k)=g1on⁡Γ.(3)
Compute the new approximation(3.7)φ(k+1)=(1-τk+1)φ(k)-bτk+1u(k)inΩ,
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]
(3.8)φ(k+1)-φ(k)τk+1+(φ(k)+bu(k))=0.Now, we introduce the operatorA defined by the formula
(3.9)Aφ=u,
where u is the function determined from the problems
(3.10)Δv-av=φinΩ,v=0on⁡Γ,(3.11)Δu=vinΩ,u=0on⁡Γ.The properties of the operatorA will be investigated in the sequel.Now, let us return to the problem (3.3). We represent their solutions in the form
(3.12)u=u1+u2,v=v1+v2,
where u1,u2,v1,v2 are the solutions of the problems
(3.13)Δv1-av1=φinΩ,v1=0on⁡Γ,Δu1=v1inΩ,u1=0on⁡Γ,(3.14)Δv2-av2=finΩ,v2=g2on⁡Γ,Δu2=v2inΩ,u2=g1on⁡Γ.
According to the definition of A we have
(3.15)Aφ=u1.
Since φ should satisfy the relation (3.2), taking into account the representation (3.12) we obtain the equation
(3.16)(I+bA)φ=F,
where I is the identity operator, and
(3.17)F=-bu2.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 h, and coefficients a,b.Proposition 3.13.1.
The iterative process (3.4)–(3.7) is a realization of the two-layer iterative scheme
(3.18)φ(k+1)-φ(k)τk+1+(I+bA)φ(k)=F,k=0,1,2,…
for the operator equation (3.16).Proof.
Indeed, if in (3.5), (3.6) we put
(3.19)u(k)=u1(k)+u2,v(k)=v1(k)+v2,
where v2,u2 are the solutions of problem (3.14), then we get
(3.20)Δv1(k)-av1(k)=φ(k)inΩ,v1(k)=0on⁡Γ,Δu1(k)=v1(k)inΩ,u1(k)=0on⁡Γ.
From here it is easy to see that
(3.21)Aφ(k)=u1(k).
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.Proposition3.1 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 operatorA defined by (3.9)–(3.11) is linear, symmetric, positive, and compact operator in the space L2(Ω).Proof.
The linearity ofA 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
(3.22)(Aφ,φ-)=∫Ωuφ-dx=∫Ωu(Δv--av-)dx=∫ΩuΔv-dx-a∫Ωuv-dx.
Applying the Green formula for the functions u and v-, vanishing on the boundary Γ, we obtain
(3.23)∫ΩuΔv-dx=∫Ωv-Δudx=∫Ωv-vdx,∫Ωuv-=∫ΩuΔu-=-∫Ω∇u⋅∇u-dx.
Hence,
(3.24)(Aφ,φ-)=∫Ωvv-dx+a∫Ω∇u⋅∇u-dx.
Clearly,
(3.25)(Aφ,φ-)=(Aφ-,φ),(Aφ,φ)=∫Ωv2dx+a∫Ω|∇u|2dx≥0
are due to a≥0. 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 ofA. As is well known that if φ∈L2(Ω) then problem (3.10) has a unique solution v∈H2(Ω), therefore, problem (3.11) has a unique solution v∈H4(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 Proposition3.2 is complete.Due to the above proposition the operator(3.26)B=I+bA
is linear, symmetric, positive definite, and bounded operator in the space L2(Ω). More precisely, we have
(3.27)γ1I<B≤γ2I,
where
(3.28)γ1=1,γ2=1+b‖A‖.
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):
(3.29)‖v‖H2(Ω)≤C1‖φ‖L2(Ω),‖u‖H4(Ω)≤C2‖v‖H2(Ω),
where C1,C2 are constants. Therefore,
(3.30)‖u‖H4(Ω)≤C1C2‖φ‖L2(Ω).
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):
(3.31)f∈L2(Ω),g1∈H7/2(Γ),g2∈H5/2(Γ).
Then the problem (1.1)–(1.3) has a unique solution u∈H4(Ω). For the function u2 determined by (3.14) we have also u2∈H4(Ω).Theorem 3.3.
Letu 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
(3.32)τ0=2γ1+γ2,τk=τ0ρ0tk+1,tk=cos⁡2k-12Mπ,k=1,…,M,ρ0=1-ξ1+ξ,ξ=γ1γ2,
we have
(3.33)‖u(M)-u‖H4(Ω)≤CqM,
where
(3.34)C=C1C2‖φ(0)-φ‖L2(Ω),
with C1,C2 being the constant in (3.30),
(3.35)qM=2ρ1M1+ρ12M,ρ1=1-ξ1+ξ.
In the case of the stationary iterative process, τk=τ0(k=1,2,….) we have
(3.36)‖u(k)-u‖H4(Ω)≤Cρ0k,k=1,2,….Proof.
According to the general theory of the two-layer iterative schemes (see [21]) for the operator equation (3.16) we have
(3.37)‖φ(M)-φ‖L2(Ω)≤qM‖φ(0)-φ‖L2(Ω),
if the parameter {τk+1} is chosen by the formulae (3.32) and
(3.38)‖φ(k)-φ*‖H4(Ω)≤ρ0k‖φ(0)-φ*‖L2(Ω),k=1,2,…
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
(3.39)Δ(v(k)-v)-a(v(k)-v)=φ(k)-φ,inΩ,v(k)-v=0on⁡Γ,Δ(u(k)-u)=v(k)-v,inΩ,u(k)-u=0on⁡Γ,
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 ofa 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 Theorem3.3 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 domainΩ=[0,l1]×[0,l2] in the plane xOy. In this case by Fourier method we found that the system of functions
(3.40)emn(x,y)=2l1l2sin⁡mπxl1sin⁡nπyl2(m,n=1,2,…)
is the eigenfunctions of the spectral problem
(3.41)Δu=λuinΩ,u=0on⁡Γ
corresponding to the eigenvalues
(3.42)λmn=-π2(m2l12+n2l22).
Moreover, this system is orthogonal and complete in L2(Ω).Now let a functionφ∈L2(Ω) have the expansion
(3.43)φ(x,y)=∑m,n=1∞φmnemn(x,y).
Then we have
(3.44)‖φ‖2=(φ,φ)=∑m,n=1∞|φmn|2.
Representing the solution v of (3.10) in the form of the double series
(3.45)v(x,y)=∑m,n=1∞vmnemn(x,y)
we found
(3.46)vmn=φmnλmn-a.
Next, we seek the solution of (3.11) in the form
(3.47)u(x,y)=∑m,n=1∞umnemn(x,y).
Then from (3.45) we find
(3.48)umn=vmnλmn.
From the definition of the operator ∥A∥ by (3.9)–(3.11) we have
(3.49)(Aφ,φ)=(u,φ)=(∑m,n=1∞umnemn,∑m,n=1∞φmnemn)=∑m,n=1∞umnφmn=∑m,n=1∞|φmn|2λmn(λmn-a)=∑m,n=1∞|φmn|2[π2(m2l1-2+n2l2-2)][π2(m2l1-2+n2l2-2)+a]≤∑m,n=1∞|φmn|2π2(l1-2+l2-2)[π2(l1-2+l2-2)+a]
due to the orthogonality of the system {emn} and (3.46), (3.48), and (3.42). Thus, there holds the estimate
(3.50)(Aφ,φ)≤1π2(l1-2+l2-2)[π2(l1-2+l2-2)+a](φ,φ).
The sign of equality occurs for φ=e11(x,y). Since A is shown to be symmetric and positive in L2(Ω) it follows:
(3.51)‖A‖=sup⁡φ≠0(Aφ,φ)(φ,φ)=1π2(l1-2+l2-2)[π2(l1-2+l2-2)+a].


## 3.1. Construction of Iterative Method on Continuous Level

As in [6], we set
(3.1)Δu=v,(3.2)φ=-bu.Then problem (1.1)–(1.3) leads to the following second-order problems
(3.3)Δv-av=f+φ,inΩ,v=g2,on⁡Γ,Δu=v,inΩ,u=g1,on⁡Γ,
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 findingφ and simultaneously for finding v,u. (1)
Givenφ(0)∈L2(Ω), for example,
(3.4)φ(0)=0inΩ.(2)
Knowingφ(k)(x) on Ω(k=0,1,…) solve consecutively two problems
(3.5)Δv(k)-av(k)=f+φ(k)inΩ,v(k)=g2on⁡Γ,(3.6)Δu(k)=v(k)inΩ,u(k)=g1on⁡Γ.(3)
Compute the new approximation(3.7)φ(k+1)=(1-τk+1)φ(k)-bτk+1u(k)inΩ,
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]
(3.8)φ(k+1)-φ(k)τk+1+(φ(k)+bu(k))=0.Now, we introduce the operatorA defined by the formula
(3.9)Aφ=u,
where u is the function determined from the problems
(3.10)Δv-av=φinΩ,v=0on⁡Γ,(3.11)Δu=vinΩ,u=0on⁡Γ.The properties of the operatorA will be investigated in the sequel.Now, let us return to the problem (3.3). We represent their solutions in the form
(3.12)u=u1+u2,v=v1+v2,
where u1,u2,v1,v2 are the solutions of the problems
(3.13)Δv1-av1=φinΩ,v1=0on⁡Γ,Δu1=v1inΩ,u1=0on⁡Γ,(3.14)Δv2-av2=finΩ,v2=g2on⁡Γ,Δu2=v2inΩ,u2=g1on⁡Γ.
According to the definition of A we have
(3.15)Aφ=u1.
Since φ should satisfy the relation (3.2), taking into account the representation (3.12) we obtain the equation
(3.16)(I+bA)φ=F,
where I is the identity operator, and
(3.17)F=-bu2.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 h, and coefficients a,b.Proposition 3.13.1.
The iterative process (3.4)–(3.7) is a realization of the two-layer iterative scheme
(3.18)φ(k+1)-φ(k)τk+1+(I+bA)φ(k)=F,k=0,1,2,…
for the operator equation (3.16).Proof.
Indeed, if in (3.5), (3.6) we put
(3.19)u(k)=u1(k)+u2,v(k)=v1(k)+v2,
where v2,u2 are the solutions of problem (3.14), then we get
(3.20)Δv1(k)-av1(k)=φ(k)inΩ,v1(k)=0on⁡Γ,Δu1(k)=v1(k)inΩ,u1(k)=0on⁡Γ.
From here it is easy to see that
(3.21)Aφ(k)=u1(k).
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.Proposition3.1 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 operatorA defined by (3.9)–(3.11) is linear, symmetric, positive, and compact operator in the space L2(Ω).Proof.
The linearity ofA 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
(3.22)(Aφ,φ-)=∫Ωuφ-dx=∫Ωu(Δv--av-)dx=∫ΩuΔv-dx-a∫Ωuv-dx.
Applying the Green formula for the functions u and v-, vanishing on the boundary Γ, we obtain
(3.23)∫ΩuΔv-dx=∫Ωv-Δudx=∫Ωv-vdx,∫Ωuv-=∫ΩuΔu-=-∫Ω∇u⋅∇u-dx.
Hence,
(3.24)(Aφ,φ-)=∫Ωvv-dx+a∫Ω∇u⋅∇u-dx.
Clearly,
(3.25)(Aφ,φ-)=(Aφ-,φ),(Aφ,φ)=∫Ωv2dx+a∫Ω|∇u|2dx≥0
are due to a≥0. 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 ofA. As is well known that if φ∈L2(Ω) then problem (3.10) has a unique solution v∈H2(Ω), therefore, problem (3.11) has a unique solution v∈H4(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 Proposition3.2 is complete.Due to the above proposition the operator(3.26)B=I+bA
is linear, symmetric, positive definite, and bounded operator in the space L2(Ω). More precisely, we have
(3.27)γ1I<B≤γ2I,
where
(3.28)γ1=1,γ2=1+b‖A‖.
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):
(3.29)‖v‖H2(Ω)≤C1‖φ‖L2(Ω),‖u‖H4(Ω)≤C2‖v‖H2(Ω),
where C1,C2 are constants. Therefore,
(3.30)‖u‖H4(Ω)≤C1C2‖φ‖L2(Ω).
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):
(3.31)f∈L2(Ω),g1∈H7/2(Γ),g2∈H5/2(Γ).
Then the problem (1.1)–(1.3) has a unique solution u∈H4(Ω). For the function u2 determined by (3.14) we have also u2∈H4(Ω).Theorem 3.3.
Letu 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
(3.32)τ0=2γ1+γ2,τk=τ0ρ0tk+1,tk=cos⁡2k-12Mπ,k=1,…,M,ρ0=1-ξ1+ξ,ξ=γ1γ2,
we have
(3.33)‖u(M)-u‖H4(Ω)≤CqM,
where
(3.34)C=C1C2‖φ(0)-φ‖L2(Ω),
with C1,C2 being the constant in (3.30),
(3.35)qM=2ρ1M1+ρ12M,ρ1=1-ξ1+ξ.
In the case of the stationary iterative process, τk=τ0(k=1,2,….) we have
(3.36)‖u(k)-u‖H4(Ω)≤Cρ0k,k=1,2,….Proof.
According to the general theory of the two-layer iterative schemes (see [21]) for the operator equation (3.16) we have
(3.37)‖φ(M)-φ‖L2(Ω)≤qM‖φ(0)-φ‖L2(Ω),
if the parameter {τk+1} is chosen by the formulae (3.32) and
(3.38)‖φ(k)-φ*‖H4(Ω)≤ρ0k‖φ(0)-φ*‖L2(Ω),k=1,2,…
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
(3.39)Δ(v(k)-v)-a(v(k)-v)=φ(k)-φ,inΩ,v(k)-v=0on⁡Γ,Δ(u(k)-u)=v(k)-v,inΩ,u(k)-u=0on⁡Γ,
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 ofa 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 Theorem3.3 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 domainΩ=[0,l1]×[0,l2] in the plane xOy. In this case by Fourier method we found that the system of functions
(3.40)emn(x,y)=2l1l2sin⁡mπxl1sin⁡nπyl2(m,n=1,2,…)
is the eigenfunctions of the spectral problem
(3.41)Δu=λuinΩ,u=0on⁡Γ
corresponding to the eigenvalues
(3.42)λmn=-π2(m2l12+n2l22).
Moreover, this system is orthogonal and complete in L2(Ω).Now let a functionφ∈L2(Ω) have the expansion
(3.43)φ(x,y)=∑m,n=1∞φmnemn(x,y).
Then we have
(3.44)‖φ‖2=(φ,φ)=∑m,n=1∞|φmn|2.
Representing the solution v of (3.10) in the form of the double series
(3.45)v(x,y)=∑m,n=1∞vmnemn(x,y)
we found
(3.46)vmn=φmnλmn-a.
Next, we seek the solution of (3.11) in the form
(3.47)u(x,y)=∑m,n=1∞umnemn(x,y).
Then from (3.45) we find
(3.48)umn=vmnλmn.
From the definition of the operator ∥A∥ by (3.9)–(3.11) we have
(3.49)(Aφ,φ)=(u,φ)=(∑m,n=1∞umnemn,∑m,n=1∞φmnemn)=∑m,n=1∞umnφmn=∑m,n=1∞|φmn|2λmn(λmn-a)=∑m,n=1∞|φmn|2[π2(m2l1-2+n2l2-2)][π2(m2l1-2+n2l2-2)+a]≤∑m,n=1∞|φmn|2π2(l1-2+l2-2)[π2(l1-2+l2-2)+a]
due to the orthogonality of the system {emn} and (3.46), (3.48), and (3.42). Thus, there holds the estimate
(3.50)(Aφ,φ)≤1π2(l1-2+l2-2)[π2(l1-2+l2-2)+a](φ,φ).
The sign of equality occurs for φ=e11(x,y). Since A is shown to be symmetric and positive in L2(Ω) it follows:
(3.51)‖A‖=sup⁡φ≠0(Aφ,φ)(φ,φ)=1π2(l1-2+l2-2)[π2(l1-2+l2-2)+a].


## 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 a2-4b<0. Now we study numerical realization of the method.For simplicity we consider the case, where the domainΩ is a rectangle, Ω=[0,l1]×[0,l2] in the plane xOy. In this domain we construct an uniform grid
(4.1)Ω-h={(x,y),x=ih1,y=jh2,0≤i≤N1,0≤j≤N2},
where h1=l1/N1,h2=l2/N2.Denote byΩh the set of inner nodes, by Γh the set of boundary nodes of the grid, and by vh,uh,… the grid functions defined on Ωh.Now consider a discrete version of the iterative method (3.4)–(3.7) when τk≡τ. (1)
Given a startingφh(0), for example,
(4.2)φh(0)=0inΩh.(2)
Knowingφh(k) on Ωh(k=0,1,…) solve consecutively two difference problems
(4.3)Λvh(k)-avh(k)=fh+φh(k)inΩh,vh(k)=g2hon⁡Γh,(4.4)Λuh(k)=vh(k)inΩh,uh(k)=g1hon⁡Γh,
where Λ is the discrete Laplace operator,
(4.5)(Λv)ij=vi-1,j-2vij+vi+1,jh12+vi,j-1-2vij+vi,j+1h22.(3)
Compute the new approximation(4.6)φh(k+1)=(1-τh)φh(k)-bτhuh(k)inΩh.It is easy to see that the convergence of the above iterative method is related to the discrete version Ah of the operator A, defined by the formula
(4.7)Ahφh=uh,
where uh is determined from the difference problems
(4.8)Λvh-avh=φhinΩh,vh=0on⁡Γh,Λuh=vhinΩh,uh=0on⁡Γh.
Using the results of the spectral problem for the discrete Laplace operator Λ (see [19]) we find the bounds of Ah:
(4.9)1β2(β2+a)I≤Ah≤1β1(β1+a)I,
where
(4.10)β1=4h12sin⁡2πh12l1+4h22sin⁡2πh22l2,β2=4h12cos⁡2πh12l1+4h22cos⁡2πh22l2.
Therefore, for the operator Bh, the discrete version of B, we obtain the estimate
(4.11)γ1hI≤Bh≤γ2hI,
where
(4.12)γ1h=1+bβ2(β2+a),γ2h=1+bβ1(β1+a).
Hence, we choose
(4.13)τh=2γ1h+γ2h,
which is the optimal parameter of the iterative process (4.2)–(4.6).Now we study the deviation ofuh(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 anyk=0,1,… there holds the estimate
(4.14)‖φh(k)-φ(k)‖=O(h2),‖uh(k)-u(k)‖=O(h2),
where h2=h12+h22,  u(k),φ(k) are computed by the process (3.4)–(3.7) and uh(k),φh(k) are computed by (4.2)–(4.6).Proof.
We shall prove this proposition by induction ink.
Fork=0 we have ||φh(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 k-1≥0. We shall show that it also is valid for k. For this purpose we recall that φ(k) is computed by the formula
(4.15)φ(k)=(1-τ)φ(k-1)-bτu(k-1)on⁡Ω,
where
(4.16)τ=2(2+b‖A‖)
and φh(k) is computed by the formula
(4.17)φh(k)=(1-τh)φh(k-1)-bτhuh(k-1)on⁡Ωh,τh 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
(4.18)τh=τ+O(h2).
Next, subtracting (4.15) from (4.17) and taking into account the above formula we get
(4.19)φh(k)-φ(k)=(1-τ)(φh(k-1)-φ(k-1))+τb(uh(k-1)-u(k-1))+O(h2).
By the assumptions of the induction
(4.20)‖uh(k-1)-u(k-1)‖=O(h2),‖φh(k-1)-φ(k-1)‖=O(h2)
from (4.19) it follows ∥φh(k)-φ(k)∥=O(h2). 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 ∥uh(k)-uh(k-1)∥<TOL, where TOL is a some given accuracy. Then for the total error of the discrete solution uh(k) there there holds the following theorem.Theorem 4.2.
For the total error of the discrete solutionuh(k) from the exact solution u of the original problem (1.1)–(1.3) there holds the estimate
(4.21)‖uh(k)-u‖≤TOL+C2h2+C1ρ0k-1,
where C1,C2 are some constants and ρ0 is the number in (3.36).Proof.
We have the following estimate(4.22)‖uh(k)-u‖≤‖uh(k)-uh(k-1)‖+‖uh(k-1)-u(k-1)‖+‖u(k-1)-u‖.
Since the space H4(Ω) is continuously embedded to the space C(Ω) (see [20]) from (3.36) we have
(4.23)‖u(k-1)-u‖≤C1ρ0k-1
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 ofa and b with TOL=10-5. 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 uh(k), E=||uh(k)-u||.Example 4.3.
We take an exact solutionu=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 including65×65,129×129, and 257×257 nodes for a=0,a=0.5 and a=1 are presented in Tables 1, 2, and 3, respectively.Table 1
Convergence of the method in Example4.3 for a=0.
b
Grid65×65
Grid129×129
Grid257×257
E
1
/
E
2
E
2
/
E
3
k
E
1
k
E
2
k
E
3
0.3
5
3.7378
e
-
4
5
9.3527
e
-
5
5
2.3477
e
-
5
3.9965
3.9838
0.7
6
3.4125
e
-
4
6
8.4859
e
-
5
6
2.0771
e
-
5
4.0214
4.0855
1.0
7
3.2179
e
-
4
7
8.0793
e
-
5
7
2.0550
e
-
5
3.9829
3.9315
1.5
8
2.9118
e
-
4
8
7.2101
e
-
5
8
1.7337
e
-
5
4.0385
4.1588
2.0
9
2.6904
e
-
4
9
6.8215
e
-
5
9
1.8013
e
-
5
3.9440
3.7870
2.5
10
2.4561
e
-
4
10
6.0247
e
-
5
10
1.3909
e
-
5
4.0767
4.3315Table 2
Convergence of the method in Example4.3 for a=0.5.
b
Grid65×65
Grid129×129
Grid257×257
E
1
/
E
2
E
2
/
E
3
k
E
1
k
E
2
k
E
3
0.3
5
3.7378
e
-
4
5
8.5291
e
-
5
5
2.1354
e
-
5
3.9990
3.9941
0.7
6
3.4125
e
-
4
6
7.9097
e
-
5
6
1.9648
e
-
5
4.0069
4.0257
1.0
7
3.2179
e
-
4
7
7.5415
e
-
5
7
1.8938
e
-
5
3.9959
3.9822
1.5
8
2.9118
e
-
4
8
6.9316
e
-
5
8
1.7184
e
-
5
4.0086
4.0338
2.0
9
2.6904
e
-
4
9
6.4782
e
-
5
9
1.6374
e
-
5
3.9892
3.9564
2.5
9
2.4561
e
-
4
9
6.1521
e
-
5
9
1.6340
e
-
5
3.9378
3.7651Table 3
Convergence of the method in Example4.3 for a=1.
b
Grid65×65
Grid129×129
Grid257×257
E
1
/
E
2
E
2
/
E
3
k
E
1
k
E
2
k
E
3
0.3
5
3.1879
e
-
4
5
7.9702
e
-
5
5
1.9938
e
-
5
3.9998
3.9975
0.7
6
2.9968
e
-
4
6
7.4868
e
-
5
6
1.8672
e
-
5
4.0028
4.0096
1.0
6
2.8644
e
-
4
6
7.1267
e
-
5
6
1.7480
e
-
5
4.0193
4.0771
1.5
7
2.6823
e
-
4
7
6.7405
e
-
5
7
1.7204
e
-
5
3.9794
3.9180
2.0
8
2.5061
e
-
4
8
6.2346
e
-
5
8
1.5283
e
-
5
4.0197
4.0794
2.5
9
2.3658
e
-
4
9
5.9385
e
-
5
9
1.5090
e
-
5
3.9838
3.9354Example 4.4.
We take an exact solutionu=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 including65×65,129×129, 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 areh=π/64,π/128,π/256. Therefore, h2=0.0024,6.0239e-4,  1.5060e-4. According to the estimate (4.21) the total error of the discrete approximate solution depends on h2. The columns E1,E2,E3 in Tables 1–6 show this fact. It is interesting to notice that in these tables E1/E2,E2/E3≈4 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.Table 4
Convergence of the method in Example4.4 for a=0.
b
Grid65×65
Grid129×129
Grid257×257
E
1
/
E
2
E
2
/
E
3
k
E
1
k
E
2
k
E
3
0.3
6
5.4815
e
-
4
6
1.3703
e
-
4
6
3.4247
e
-
5
4.0002
4.0012
0.7
7
5.0136
e
-
4
7
1.2555
e
-
4
7
3.1525
e
-
5
3.9933
3.9826
1.0
8
4.7101
e
-
4
8
1.1763
e
-
4
8
2.9257
e
-
5
4.0042
4.0206
1.5
9
4.2909
e
-
4
9
1.0770
e
-
4
9
2.7341
e
-
5
3.9841
3.9391
2.0
10
3.9191
e
-
4
10
9.7264
e
-
5
10
2.3582
e
-
5
4.0293
4.1245
2.5
11
3.6420
e
-
4
11
9.2141
e
-
5
11
2.4078
e
-
5
3.9526
3.8268Table 5
Convergence of the method in Example4.4 for a=0.5.
b
Grid65×65
Grid129×129
Grid257×257
E
1
/
E
2
E
2
/
E
3
k
E
1
k
E
2
k
E
3
0.3
5
5.1875
e
-
4
5
1.2985
e
-
4
5
3.2589
e
-
5
3.9950
3.9845
0.7
7
4.8209
e
-
4
7
1.2061
e
-
4
7
3.0184
e
-
5
3.9971
3.9958
1.0
7
4.5846
e
-
4
7
1.1495
e
-
4
7
2.9067
e
-
5
3.9883
3.9547
1.5
8
4.2213
e
-
4
8
1.0498
e
-
4
8
2.5687
e
-
5
4.0211
4.0869
2.0
9
3.9362
e
-
4
9
9.9099
e
-
5
9
2.5458
e
-
5
3.9720
3.8926
2.5
10
3.6555
e
-
4
10
9.0709
e
-
5
10
2.1945
e
-
5
4.0299
4.1335Table 6
Convergence of the method in Example4.4 for a=1.
b
Grid65×65
Grid129×129
Grid257×257
E
1
/
E
2
E
2
/
E
3
k
E
1
k
E
2
k
E
3
0.3
5
4.9843
e
-
4
5
1.2472
e
-
4
5
3.1232
e
-
5
3.9964
3.9933
0.7
6
4.6839
e
-
4
6
1.1696
e
-
4
6
2.9067
e
-
5
4.0047
4.0238
1.0
7
4.4871
e
-
4
7
1.1229
e
-
4
7
2.8174
e
-
5
3.9960
3.9856
1.5
8
4.1852
e
-
4
8
1.0449
e
-
4
8
2.5970
e
-
5
4.0054
4.0235
2.0
9
3.9278
e
-
4
9
9.8369
e
-
5
9
2.4760
e
-
5
3.9929
3.9729
2.5
9
3.7068
e
-
4
9
9.3636
e
-
5
9
2.4338
e
-
5
3.9587
3.8473Example 4.5.
We take an exact solutionu=(x2-4)(y2-1) in the rectangle [-2,2]×[-1,1]. The the right-hand side of (1.1) is f=8-2a(x2+y2-5)+b(x2-4)(y2-1).
The results of convergence in the case of the grids including65×65 and 129×129 nodes are presented in Tables 7 and 8, respectively.
From Tables7 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.Table 7
Convergence of the method in Example4.5 for grid 65×65.
b
a
=
0
a
=
0.5
a
=
1
k
error
k
error
k
error
0.3
5
7.8052
e
-
9
5
3.7143
e
-
9
5
1.9460
e
-
9
0.7
6
1.7710
e
-
8
5
2.3933
e
-
7
5
1.2646
e
-
7
1.0
6
1.3959
e
-
7
6
5.8675
e
-
8
6
2.7519
e
-
8
1.5
7
1.0251
e
-
7
7
3.8079
e
-
8
6
2.8517
e
-
7
2.0
7
6.6466
e
-
7
7
2.5155
e
-
7
7
1.0720
e
-
7
2.5
8
3.2038
e
-
7
8
1.0795
e
-
7
7
4.5822
e
-
7Table 8
Convergence of the method in Example4.5 for grid 129×129.
b
a
=
0
a
=
0.5
a
=
1
k
error
k
error
k
error
0.3
5
7.7939
e
-
9
5
3.7094
e
-
9
5
1.9437
e
-
9
0.7
6
1.7679
e
-
8
5
2.3901
e
-
7
5
1.2630
e
-
7
1.0
6
1.3935
e
-
7
6
5.8579
e
-
8
6
2.7476
e
-
8
1.5
7
1.0231
e
-
7
7
3.8009
e
-
8
6
2.8474
e
-
7
2.0
7
6.6338
e
-
7
7
2.5109
e
-
7
7
1.0701
e
-
7
2.5
8
3.1968
e
-
7
8
1.0773
e
-
7
7
4.5744
e
-
7Example 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=10-5. 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.Figure 1
Variation of the value ofu in the middle point with b for several values of a.Figure 2
Solution in the case ofa=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.



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