MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 828615 10.1155/2013/828615 828615 Research Article A Nonoverlapping Domain Decomposition Method for an Exterior Anisotropic Quasilinear Elliptic Problem in Elongated Domains Liu Baoqing Du Qikui Chang Yong-Kui Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences Nanjing Normal University, Nanjing, Jiangsu 210023 China nnu.cn 2013 14 2 2013 2013 02 09 2012 17 12 2012 2013 Copyright © 2013 Baoqing Liu and Qikui Du. 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.

Based on the Kirchhoff transformation, a nonoverlapping domain decomposition method is discussed for solving exterior anisotropic quasilinear problems with elliptic artificial boundary. By the principle of the natural boundary reduction, we obtain the natural integral equation for the anisotropic quasilinear problem on elliptic artificial boundaries and construct the algorithm and analyze its convergence. Moreover, we give the existence and uniqueness result for the original problem. Finally, some numerical examples are presented to illustrate the feasibility of the method.

1. Introduction

When solving a problem modelled by a linear or nonlinear partial differential equation in the bounded or unbounded domain, domain decomposition methods are one of the most efficient techniques. one can refer to  and references therein for more details. Based on natural boundary reduction [5, 6], the overlapping and nonoverlapping domain decomposition methods can be viewed as effective ways to solve the problems in the unbounded domains. These techniques have been used to solve many linear problems , and they have also been generalized to linear or nonlinear wave problems [5, 9, 10]. In this paper, we consider a nonoverlapping domain decomposition method for an exterior anisotropic quasilinear elliptic problem with elliptical artificial boundary. By the Kirchhoff transformation, we will discuss some exterior anisotropic quasilinear elliptic problems  by virtue of the nonoverlapping domain decomposition method.

Let Ω be an elongated, bounded, and simple connected domain in 2 with sufficiently smooth boundary Ω=Γ0, Ωc=2/Ω-. We consider the numerical solution to the exterior anisotropic quasilinear problem (1)-(x(αa(x,u)ux)+y(βa(x,u)uy))=f(x),-(x(αa(x,u)ux)+y(βa(x,u)uy))=in  Ωc,u=0,on  Γ0,(2)u(x)=𝒪(1),as  |x|, with β>α>0 or α=β=1, x=(x,y), a(·,·), and f(x) are the given functions which will be ranked as below.

Problem (1) has many physical applications in, for example, the field of heat transfer, where a is the thermal conductivity of the medium and u is the temperature field; the field of compressible flow, where a is the density and u is the velocity potential. Problem (1) can also describe a temperature distribution in large transformers whose magnetic cores (consisting of iron tins) are nonlinear anisotropic media where a is the heat conductivity. One can also refer to [1114, 16] for more details.

Following [11, 13], suppose that the given function a(·,·) satisfies (3)0<C0a(x,u)C1,u,and for almost all  xΩc, Where the two constants C0,C1. In the following, we suppose that the function f(x)L2(Ωc) has compact support, that is, there exists a constant γ0>0, such that (4)suppfΩγ0={x2|x|γ0}.

We also assume that (5)a(x,u)=˙a0(u),when  |x|γ0. Now, we introduce an elliptical artificial boundary Γ1 enclosing Γ0 such that (6)Γ1={(μ,φ)  μ=μ1>γ0,0φ2π},and dist(Γ1,Γ0)=δ0>0.

Then, Ωc is divided into two nonoverlapping subdomains Ω1 and Ω2 (see Figure 1), where Ω1 denotes the bounded domain between Γ0 and Γ1, and Ω2 refers to the unbounded domain outside Γ1. The original problem (1) is decomposed into two subproblems in domains Ω1 and Ω2 with Ω1Ω2=. We have the nonoverlapping domain decomposition algorithm.

The illustration of domains Ω1 and Ω2.

Step 1.

Choose an initial value λ0H1/2(Γ1), and put k=0.

Step 2.

Solve a Dirichlet boundary value problem in the exterior domain Ω2: (7)-(x(αa(x,u2k)u2kx)+y(βa(x,u2k)u2ky))=0,in  Ω2,u2k=λk,on  Γ1,u2k(x)=𝒪(1),as  |x|.

Step 3.

Solve a mixed boundary value problem in the interior domain Ω1: (8)-(x(αa(x,u1k)u1kx)+y(βa(x,u1k)u1ky))=f(x),-(x(αa(x,u1k)u1kx)+y(βa(x,u1k)u1ky))  in  Ω1,u1kn1=-u2kn2,on  Γ1,u1k=0,on  Γ0.

Step 4.

Update the boundary value 0<θk<1, (9)λk+1=θku1k+(1-θk)λk,on  Γ1.

Step 5.

Put k=k+1, turn to Step 2.

The relaxation factor θk is a suitably chosen real number. Notice that, in Step 3, we solve problem (8) by the standard finite element method and only need the normal derivative of the solution to problem (7) in Step 2. So, we need not to solve (7) directly. Based on the Kirchhoff transformation, the natural integral equation for the quasilinear problem can be obtained by the natural boundary reduction [6, 17, 18]. Particularly, when a(x,u)=c which is independent of x and u, [6, 8, 19] have discussed the corresponding problems by this technique. Now, we introduce the so-called Kirchhoff transformation  (10)w(x)=0u(x)a(ξ)dξ,xΩc. Then, we have (11)w=a(u)u,(12)  (αwx,βwy)=(αa(u)ux,βa(u)uy). From (7), we have that w satisfies the following problem (13)-(α2wkx2+β2wky2)=0,in  Ω2,wk=0λka0(ξ)dξ,on  Γ1. The rest of the paper is organized as follows. In Section 2, we obtain the natural integral equation for the quasilinear problem in the elliptical unbounded domain. In Section 3, we state the nonoverlapping domain decomposition method and discuss the convergence of the algorithm. We also show the existence and uniqueness of the original problem. At last, in Section 4, we present some numerical examples to illustrate the efficiency and feasibility of our method.

2. Natural Boundary Reduction

In this section, by virtue of the Poisson integral formula and natural integral equation for the linear problem, we will obtain the corresponding results for the quasilinear problem in Ω2. For this purpose, we need to discuss some properties between elliptic coordinates (μ,φ) and Cartesian coordinates (x,y) first. The relationship between the two coordinates can be expressed as below (14)x=f0coshμcosφ,y=f0sinhμsinφ, with f0=a2-b2, a=f0coshμ1, and b=f0sinhμ1. Following from , we have the following.

Theorem 1.

The transformation between elliptic coordinates and Cartesian coordinates in (14) possesses the following property.

The Jacobian determinant of (14) is (15)J=f02cosh2μsin2φ+f02sinh2μcos2φ=f02(cosh2μ-cos2φ),J=0 if and only if (x,y)=(±f0,0).

For uC2(2), the following holds (16)2uμ2+2uφ2=J(2ux2+2uy2).

For the exterior domain Ω2, we have (17)uν=-1Juμ,

where ν refers to the unit exterior normal vector on Γμ1(regarded as the inner boundary of Ω2).

Proof.

The conclusions 1 and 2 can be obtained by direct computation. And 3 follows from the property (18)ν=-1J(f0sinhμcosφ,f0coshμsinφ).

2.1. Natural Integral Equation for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M74"><mml:mi>α</mml:mi><mml:mo mathvariant="bold"> </mml:mo><mml:mo mathvariant="bold">=</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>β</mml:mi><mml:mo mathvariant="bold"> </mml:mo><mml:mo mathvariant="bold">=</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>

Assume that w(x) is the solution to problem (13) and the value w||μ|=μ1 is given, that is, (19)w||μ|=μ1=w0(φ). Then, based on the natural boundary reduction, there are the Poisson integral formulas (20)w(μ,φ)=e2μ-e2μ12π×02πw0(φ)e2μ+e2μ1-2eμ+μ1cos(φ-φ)dφ,μ>μ1, or (21)w(μ,φ)=1πj=1ej(μ1-μ)×02πcosj(φ-φ)w0(φ)+12π02πw0(φ)dφ,μ>μ1, and the natural integral equation (22)wn=1J0[-14πsin2(φ/2)*w0(φ)],μ=μ1, or (23)wn=1πJ0j=1j×02πcosj(φ-φ)w0(μ1,φ)dφ,μ=μ1, the definition of J0 can be found in the following. The Poisson integral formulas (20)-(21) and the natural integral equation (22)-(23) can also be expressed in the Fourier series forms (24)w(μ,φ)=j=-+aje|j|(μ1-μ)+ijφ,μ>μ1,(25)wn=1J0j=-+|j|ajeijφ,μ=μ1, with aj=(1/2π)02πw0(φ)e-ijφdφ, J0=f02(cosh2μ1sin2φ+sinh2μ1cos2φ)=f02(cosh2μ1-cos2φ), i=-1.

From (12), we obtain (26)wn=a0(u)un. Combining (11), (25), and (26), we get the exact artificial boundary condition of u on Γμ1(27)a0(u)u(μ,φ)n|μ=μ1=1J0j=-+|j|ajeijφ=1πJ0j=1+j02πcosj(φ-φ)(0u(μ1,φ)a0(y)dy)dφ=˙𝒦1(u(μ1,φ)), with aj=(1/2π)02π(0u(μ1,φ)a0(y)dy)e-ijφdφ, J0=f02(cosh2μ1-cos2φ), i=-1.

2.2. Natural Integral Equation for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M94"><mml:mi>β</mml:mi><mml:mo> </mml:mo><mml:mo mathvariant="bold">></mml:mo><mml:mo> </mml:mo><mml:mi>α</mml:mi><mml:mo> </mml:mo><mml:mo mathvariant="bold">></mml:mo><mml:mo> </mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>

Now, we assume that Γμ1 can be expressed in the form: Γμ1={(x,y)|px2+qy2=R2}, with βq>αp>0. We also assume that w(x) is the solution to problem (13) and the value w||μ|=μ1 is given, namely, (28)w||μ|=μ1=w0(φ). Let x=αξ and y=βη. Then, the boundary Γμ1 is changed by the elliptic boundary Γ~1={(ξ,η)|αpξ2+βqη2=R2}; the unit exterior normal vector on Γ~1 is (29)ν=-(αpcosθ,βqsinθ)αpcos2θ+βqsin2θ. By the above transformation, problem (13) changes into (30)-(2wξ2+2wη2)=0,in  Ω~2,(31)w=w0,on  Γ~1.

This is the right problem we talked in Section 2.1. Similar with (14), we let (32)ξ=f0coshμcosφ,η=f0sinhμsinφ, with (33)f0=βq-αpαpβqR,μ0=  ln(βq+αpβq-αp),Γ~1={(μ,φ)μ=μ0,φ[0,2π]},Ω~2={(μ,φ)μ>μ0,φ[0,2π]}. Then, just the same as the problem discussed in Section 2.1, we have the natural integral equation on Γμ1(34)αnxwx+βnywy=-αpβqpcos2φ+qsin2φ[14πRsin2(φ/2)*w0(φ)], where (nx,ny)=(x/R,y/R) is the unit exterior normal vector on Γμ1. From (13), we obtain (35)αnxwx+βnywy=αnxa0(u)ux+βnya0(u)uy. Combining (11), (34), and (35), we obtain the exact artificial boundary condition of u on Γμ1(36)(αnxa0(u)ux+βnya0(u)uy)|μ=μ1=-αpβqpcos2φ+qsin2φ×[14πRsin2(φ/2)*(0u(μ1,φ)a0(y)dy)]=αpβqpcos2φ+qsin2φj=-+j2πR×02πcosj(φ-φ)(0u(μ1,φ)a0(y)dy)dφ=˙𝒦1(u(μ1,φ)).

3. Variational Problem and Convergence Analysis of the Algorithm 3.1. The Equivalent Variational Problem

Now, we consider problem (8). We will use Wm,p denoting the standard Sobolev spaces. · and |·| refer to the corresponding norms and seminorms. Especially, we define Hm(Ω)=Wm,2(Ω), ·m,Ω=·m,2,Ω, and |·|m,Ω=|·|m,2,Ω. Let us introduce the space (37)V={vH1(Ω1)|v|Γ0=0}, and the corresponding norms (38)v0,Ω1=Ω1|v|2dx,v1,Ω1=Ω1(|v|2+|v|2)dx.

The boundary value problem (8) is equivalent to the following variational problem. (39)Find  uV,such thatD(u;u,v)+D^(u;u,v)=F(v),vV, with (40)D(t;u,v)=Ω1a(x,t)(αuxvx+βuyvy)dx,(41)D^(t;u,v)=j=1+αβjπ×02π02πa0(t(μ1,φ))×u(μ1,φ)φv(μ1,φ)φ×cosj(φ-φ)dφdφ, where D^(t;u,v) follows from Green’s formula, (27) with ds=J0dφ, and (36) with ds=(R/pq)pcos2φ+qsin2φdφ.

And, (42)F(v)=Ω1f(x)v(x)dx.

3.2. Convergence Analysis of the Method in Continuous Case

From (10)-(11), the original problem (1) can be changed to (43)-(α2wx2+β2wy2)=f(x),in  Ωc,w=0, on  Γ0,w(x)=𝒪(1), as  |x|. Then, we let x=αξ and y=βη. Problem (43) becomes (44)-Δw=-(2wξ2+2wη2)=f(x),in  Ω~c,w=0,on  Γ~0,w(x)=𝒪(1),as  |x|, where Ω~c and Γ~0 are the corresponding changes from Ωc and Γ0, respectively. Let g be extended to Ω~c, w=u-g, f=Δg. Then, problem (44) is equivalent to (45)-Δu=0,in  Ω~c,u=g,on  Γ~0,u(x)=𝒪(1),as  |x|.

Since it is difficult to estimate the convergence rate for a general unbounded domain Ωc, we here let Ωc be an exterior domain of an ellipse Γ0, with μ=μ0, and Γ1 is taken as stated in Section 1. We introduce the following conclusion first.

Lemma 2.

If u is the solution of (46)-Δu=0,in  Ω1,u=u0,on  Γ0,un=un,on  Γ1, where Ω1 is the elliptical ring domain between Γ0 and Γ1, (47)u0=j=-+cjeijϕH1/2(Γ0),un=1J0(-,j0+dj|j|eijϕ+d0),H-1/2(Γ1), then there exist a unique uH1(Ω1) and (48)u(μ,ϕ)=-,j0+({(e|j|(μ1-μ0)+e|j|(μ0-μ1))-1}(cj(e|j|(μ-μ1)+e|j|(μ1-μ))+dj×(e|j|(μ-μ0)-e|j|(μ0-μ)))×(e|j|(μ1-μ0)+e|j|(μ0-μ1))-1)×eijϕ+c0+d0(μ-μ0).

Proof.

The result can be obtained directly from (46) by the separation of variables.

Theorem 3.

If 0<θk<1, then the nonoverlapping domain decomposition method (7)–(9) is convergent.

Proof.

We assume that the exact solution to problem (1) is u, and we let λ=u|Γ1, uk=u|Ωk, and k=1,2. Then, following (7)–(8), we let e1k=λ-u1k and e1k|Γ1=λ-λk=˙e2k. From (24), we assume that e2k=j=-+ajeijϕH1/2(Γ1). By the natural integral equation (27), we have (49)e1kn=-𝒦1(e2k)=-1J0j=-+|j|ajeijϕ. So, e1k satisfies (50)-Δe1k=0,in  Ω1,e1k=0,on  Γ0,e1kn=1J0j=-+|j|ajeijϕ,on  Γ1. By Lemma 2, one obtains (51)e1k=-j=-+ajHj(μ)eijϕ, with Hj(μ)=(e|j|(μ-μ0)-e|j|(μ0-μ))/(e|j|(μ1-μ0)+e|j|(μ0-μ1)). From (51), e1k confines on Γ1 can be expressed as (52)e1kΓ1=-j=-+ajHj(μ1)eijϕ,𝒦1(e1k)=-1J0j=-+|j|ajHj(μ1)eijϕ.

Then, we have (53)e1k+1n=-𝒦1(λ-λk+1)=𝒦1(θku1k+(1-θk)λk-λ)=-θk𝒦1(e1k)-(1-θk)𝒦1(e2k)=1J0j=-+|j|aj(θkHj(μ1)-1+θk)eijϕ.

Let Em=˙e1k/n-1/2,Γ12, then Em=2πj=-+(j2/1+j2)|aj|2 and (54)Em+1=2πj=-+j21+j2|aj|2(θkHj(μ1)-1+θk)2=(1-θk)2Em+2πj=-+j21+j2|aj|2×θkHj(μ1)[θk(Hj(μ1)+2)-2].

Assuming that δ1=infj/{0}2/(2+Hj(μ1)), then 1>δ12/3.

If 0<θkδ1, k=0,1,2,, then (55)Em+1<(1-θk)2Em, or equally (56)Em+1<j=1m(1-θj)2E1rmE1,19r<1.

By the trace theorem, we have (57)e1k1,Ω12CEm0,m. From (54), one also has (58)Em+1=2πj=-+j21+j2|aj|2(θkHj(μ1)-1+θk)2=(1-2θk)2Em+2πj=-+j21+j2|aj|2×θkIj(μ1)[θk(Ij(μ1)-2)+1], with Ij(μ1)=(1-Hj(μ1))/2. Assuming that δ2=supj/{0}1/(2-Ij(μ1)), then 0<δ22/3.

For δ2θk<1, k=0,1,2,, the convergence result can be obtained similarly with (55)–(57). Therefore, for 0<θk<1, the nonoverlapping domain decomposition method is convergent.

3.3. Discrete Nonoverlapping Alternating Algorithm and Convergence Analysis

Divide the arc Γμ1 into M parts, and take a finite element subdivision in Ω1 such that their nodes on Γμ1 are coincident. That is, we make a regular and quasiuniform triangulation 𝒯h on Ω1, such that (59)Ω1=K𝒯hK, where K is a (curved) triangle and h is the maximal diameter of triangles. Let (60)Vh={vhV|v|K  is a linear polynomial,K𝒯h}. Then, the approximate problem of (39) can be written as follows (61)  FinduhVh,such thatD(uh;uh,vh)+D^(uh;uh,vh)=F(vh),vhVh, with (62)D(th;uh,vh)=Ω1a(x,th)(αuhxvhx+βuhyvhy)dx,(63)D^(th;uh,vh)=j=1+αβjπ02π02πa0(th(μ1,φ))  ×uh(μ1,φ)φvh(μ1,φ)φ·cos  j(φ-φ)dφdφ. In practice, the sum of (63) is truncated to a finite number of terms N. By the hypothesis of a(·,·), it is not difficult to know the following result.

Lemma 4.

There exists a constant C2>0 which has a different meaning in a different place and is related to α and β, such that (64)|D(u;u,v)+D^(u;u,v)|C2u1,Ω1v1,Ω1,D(u;u,u)+D^(u;u,u)C0u1,Ω12,u,vV.

Proof.

From (3) and (40), one can obtain that (65)|D(u;u,v)|C2u1,Ω1v1,Ω1,D(u;u,u)C0u1,Ω12,u,vV. For the natural integral equation, one can obtain the following result (66)|D^(u;u,v)|C2′′u1,Ω1v1,Ω1,D^(u;u,u)0,u,vV.

For this purpose, for u,vV, we assume that (67)u(μ1,φ)=j=-+ujeijφ,v(μ1,φ)=j=-+vjeijφ, with uj=(1/2π)02πu(μ1,φ)e-ijφdφ, vj=(1/2π)02πv(μ1,φ)e-ijφdφ. Then, we have (68)uφ(μ1,φ)=j=-+ijujeijφ,vφ(μ1,φ)=j=-+ijvjeijφ.

Combining property (3), Cauchy’s inequality, and the trace theorem, we have (69)|D^(u;u,v)|C2′′(j=-+|j|·|uj|2)1/2×(j=-+|j|·|vj|2)1/2C2′′(j=-+(1+j2)1/2·|uj|2)1/2×(j=-+(1+j2)1/2·|vj|2)1/2C2′′u1/2,Γμ1v1/2,Γμ1C2′′u1,Ω1v1,Ω1,u,vV.

Next, we show that D^(u;u,u)0, for any uV. For any given vV, let us consider the following auxiliary problem in Ω1: (70)-·(a(x,u)u)=0,in  Ω1,u=0,on  Γ0,u=v,on  Γμ1.

From the analysis in Section 2.1, we know that the solution u to problem (70) satisfies (71)a0(u)un|Γμ1=𝒦1(u(μ1,φ))=𝒦1(v(μ1,φ)). Multiplying (70) by u and integrating over Ω1, we have (72)D^(u;u,u)=Ω1a0(u)|u|2dx0.

Letting C2=C2+C2′′, we can get the desired result.

We are now in the position to show the existence and uniqueness result for this type of problem (1). We begin with the following estimate.

Lemma 5.

Any solution u to problem (39) satisfies (73)u1,Ω1C3fL2(Ω1), with C3 a positive constant.

Proof.

Taking v equal to u in (39), by Lemma 4, one has (74)C0u1,Ω12D(u;u,u)+D^(u;u,u)=L(u). Then, the desired result follows from Cauchy-Schwarz and Poincaré-Friedrichs inequalities.

Theorem 6.

Problem (39) is uniquely solvable.

Proof.

Since the space V is separable (indeed, it is a closed subspace of the space H1(Ω1) which is separable), there exist increasing sequences of finite-dimensional Hilbert subspaces Vm of V such that (75)V=m0Vm.

We define a mapping Φm:VmVm by (76)(Φm(u),v)=D(u;u,v)+D^(u;u,v)-L(v),u,vVm, where (·,·) stands for the scalar product on Vm×Vm. Since the function a(x,u) is bounded and the trace of the function in V on Γ1 belongs to H1/2(Γ1), hence at least to L4(Γ1) from the Sobolev embedding, each mapping Φm is well defined and continuous on Vm. What’s more, by Lemmas 4 and 5, we obtain (77)(Φm(u),u)=D(u;u,u)+D^(u;u,u)-L(u)C0u1,Ω12-C3fL2(Ω1)u1,Ω1. So, the right-hand side is nonnegative on the circle of radius μ which is defined by (78)μ=C3C0fL2(Ω1). Applying Brouwer’s fixed point theorem  yields the existence of um in Vm, with norm less than μ, such that (79)Φm(um)=0. Since the sequence {um}m is bounded by μ in V, there exists a subsequence which converges weakly to u in Vm. Using the compactness of the imbedding of H1/2(Γ1) into L3(Γ1), we obtain that u is a solution to problem (39).

Now, we show the uniqueness of the solution. Let u and u- be two solutions to this problem. Then, taking v equal to u and u- in (39), respectively, and combining with Lemma 4, one obtains (80)C2u1,Ω1u-1,Ω1D(u-;u-,u)+D^(u-;u-,u)=D(u;u,u)+D^(u;u,u)C0u1,Ω12,(81)C2u1,Ω1u-1,Ω1D(u;u,u-)+D^(u;u,u-)=D(u-;u-,u-)+D^(u-;u-,u-)C0u-1,Ω12. Since C2>C0, combining (80) with (81), we deduce the desired result.

From the discrete problem (61), similar to the discussion for the linear problem in [6, 8], we can get a system of algebraic equations for our quasilinear problem with the following form: (82)(A11+KhA12A21A22)(UV)=(0b), where U is a vector whose components are function values at nodes on Γ1, and V is a vector whose components are function values at interior nodes of Ω1. The matrix A=˙A(u)=(A11A12A21A22) is the stiffness matrix obtained from finite element in Ω1, while Kh=˙Kh(u|Γ1) follows from the natural boundary element method on Γ1.

Problem (82) can also be rewritten as follows: (83)(A11A12A21A22)(UV)=(-KhUb). Then, we have the iterative algorithm (84)(A11A12A21A22)(UkVk)=(-KhΛkb), with (85)Λk+1=θkUk+(1-θk)Λk,k=0,1,.

By condition (3), one obtains that A is a positive definite matrix, so we know that A22-1 exists. Now, we let Sh=Sh(1)+Kh be the discrete analogue of the Steklov-Poincaré operator on Γ1, with Sh(1)=A11-A12A22-1A21 and B=-A12A22-1b. Then, similar to the proof of Theorems 7.6 and  7.7 in , we have that the alternating algorithm (84)-(85) is equivalent to the preconditioned Richardson iteration: (86)Sh(1)(Λk+1-Λk)=θk(B-ShΛk).

And we also have the following convergence result.

Theorem 7.

If 0<minθkmaxθk<1, then the discrete nonoverlapping alternating method (84)-(85) is convergent, and both the convergence rate and the condition number of [Sh(1)]-1Sh are independent of the mesh size h.

4. Numerical Examples

In this section, we will give some examples to confirm our theoretical results. In the following, we choose the finite element space as given in (60). For simplicity, we let (87)Δμ=μ1-μ0m,Δφ=2πM,e0(k)=u-uh,NkL2(Ωi),e(k)=u-uh,NkL(Ωi). And we let eh(k) denote the maximal error between the iteration k-1 and k, that is, eh(k)=uh,Nk-uh,Nk-1L(Ωi) and qh(k)=eh(k-1)/eh(k) simulate the convergence rate.

We give the following four examples. Examples 8 and 11 show the relationship between meshes and convergence rate for the cases α=β=1 and β>α>0, respectively. Example 9 focuses on the effect of the relaxation factor θk to the convergence rate. And Example 10 wants to show the relationship between the coefficients α,β and the convergence rate.

Example 8.

We assume the exterior domain Ωc with elliptical boundary Γ0={(μ0,φ)|μ0=0.8,0φ2π}, Γμ1={(μ1,φ)|μ1>μ0,0φ2π}. Now, we consider the problem (88)-·(a(x,u)u)=f,in  Ω1,u=0,on  Γ0,a0(u)un=𝒦1(u(μ1,φ)),on  Γμ1, when a(x,u)=1/(1+u2), f=0, and f0=1.35.

The exact solution of Example 8 is u=tan(2sinhμsinφ/f0(cosh2μ+cos2φ)). For different meshes and μ1, we have the following results ranked in Tables 1 and 2.

The relationship between meshes and convergence rate (N=10, θk=0.58, and μ1=1.5).

( m , M ) Error Iteration number
0 1 2 3 4 5 6 9
(2,8) e 4.9212 E - 01 3.7241 E - 01 3.0583 E - 01 2.7007 E - 01 2.5029 E - 01 2.3880 E - 01 2.3175 E - 01 2.2183 E - 01
e h 1.1971 E - 01 6.6581 E - 02 3.5758 E - 02 1.9781 E - 02 1.1487 E - 02 7.0490 E - 03 2.2318 E - 03
q h 1.7980 1.8620 1.8077 1.7220 1.6296 1.3992

(4,16) e 3.4870 E - 01 1.9037 E - 01 1.1902 E - 01 8.8599 E - 02 7.5058 E - 02 7.0101 E - 02 6.9010 E - 02 6.8244 E - 02
e h 1.5833 E - 01 7.1346 E - 02 3.0423 E - 02 1.3541 E - 02 6.3774 E - 03 3.2007 E - 03 6.2995 E - 04
q h 2.2192 2.3452 2.2467 2.1233 1.9925 1.5865

The relationship between meshes and convergence rate (N=10, θk=0.58, and μ1=1.0).

( m , M) Error Iteration number
0 1 2 3 4 5 6 9
(2,8) e 3.7701 E - 01 3.2871 E - 01 2.9909 E - 01 2.8081 E - 01 2.6906 E - 01 2.6116 E - 01 2.5562 E - 01 2.4614 E - 01
e h 4.8299 E - 02 2.9614 E - 02 1.8287 E - 02 1.1749 E - 02 7.8958 E - 03 5.5417 E - 03 2.3769 E - 03
q h 1.6309 1.6194 1.5564 1.4880 1.4248 1.2857

(4,16) e 3.0912 E - 01 2.0526 E - 01 1.4917 E - 01 1.1931 E - 01 1.0260 E - 01 9.2674 E - 02 8.6414 E - 02 7.7207 E - 02
e h 1.0386 E - 01 5.6091 E - 02 2.9857 E - 02 1.6710 E - 02 9.9296 E - 03 6.2607 E - 03 2.1229 E - 03
q h 1.8516 1.8787 1.7868 1.6828 1.5860 1.3728
Example 9.

Similar with Example 8, Γ0 and a(x,u) are replaced by Γ0={(μ0,φ)|μ0=0.5,0φ2π}, Γ1={(μ1,φ)μ1=1.0,0φ2π}, and a(x,u)=1/1-u2, respectively.

The exact solution of Example 9 is u=sin(2coshμcosφ/f0(cosh2μ+cos2φ)). The numerical results are given in Table 3 and Figure 2(a).

The relationship between θk and convergence rate (N=10, m=4, and M=16).

θ k    k 0 1 2 3 4 5 6 9
0.18 e 4.4218 E - 02 3.8978 E - 02 3.6443 E - 02 3.4711 E - 02 3.3392 E - 02 3.2341 E - 02 3.1481 E - 02 2.9668 E - 02
e h 5.2395 E - 03 2.5350 E - 03 1.7316 E - 03 1.3185 E - 03 1.0519 E - 03 8.5974 E - 04 5.0325 E - 04
q h 2.0668 1.4640 1.3133 1.2535 1.2235 1.1860

0.38 e 4.4218 E - 02 3.6741 E - 02 3.3193 E - 02 3.1073 E - 02 2.9709 E - 02 2.8795 E - 02 2.8167 E - 02 2.7174 E - 02
e h 7.4769 E - 03 3.5473 E - 03 2.1199 E - 03 1.3646 E - 03 9.1335 E - 04 6.2830 E - 04 2.3314 E - 04
q h 2.1078 1.6733 1.5535 1.4941 1.4537 1.3635

0.50 e 4.4218 E - 02 3.5398 E - 02 3.1599 E - 02 2.9601 E - 02 2.8465 E - 02 2.7784 E - 02 2.7357 E - 02 2.6751 E - 02
e h 8.8194 E - 03 3.7989 E - 03 1.9979 E - 03 1.1359 E - 04 6.8102 E - 04 4.2741 E - 04 1.3535 E - 05
q h 2.3216 1.9014 1.7588 1.6680 1.5934 1.4114

0.58 e 4.4218 E - 02 3.4503 E - 02 3.0685 E - 02 2.8865 E - 02 2.7913 E - 02 2.7377 E - 02 2.7055 E - 02 2.6609 E - 02
e h 9.7143 E - 03 3.8180 E - 03 1.8204 E - 03 9.5190 E - 04 5.3571 E - 04 3.2246 E - 04 9.9192 E - 05
q h 2.5443 2.0974 1.9124 1.7769 1.6613 1.4090

0.65 e 4.4218 E - 02 3.3720 E - 02 2.9983 E - 02 2.8358 E - 02 2.7564 E - 02 2.7135 E - 02 2.6882 E - 02 2.6530 E - 02
e h 1.0497 E - 02 3.7371 E - 03 1.6248 E - 03 7.9382 E - 04 4.2892 E - 04 2.5372 E - 04 7.9261 E - 05
q h 2.8089 2.3000 2.04687 1.8507 1.6905 1.3951

(a)   The relationship between θk and convergence rate (N=10, m=4 and M=16). (b) The relationship between ε and convergence rate (N=10, m=4, M=16 and θk=0.58). (c) The relationship between meshes and convergence rate (N=10, θk=0.65).

Example 10.

We assume the exterior domain Ωc with elliptical boundary Γ0={(μ0,φ)|μ0=0.8,0φ2π}, Γμ1={(μ1,φ)|μ1=2.0,0φ2π}. Now, we consider the problem (89)-(x(εa(x,u)ux)+y(a(x,u)uy))=f(x),-(x(εa(x,u)ux)+y(a(x,u)uy))=in  Ωi,u=0,on  Γ0,εnxa0(u)ux+nya0(u)uy=𝒦1(u(μ1,φ)),on  Γμ1, when a(x,u)=1/(1+u2), f0=1.25 and (90)f=2(1-ε)sinhμsinφ(3cosh2μcos2φ-sinh2μsin2φ)(cosh2μ+cos2φ)3.

The exact solution of Example 10 is u=tan(2sinhμsinφ/f0(cosh2μ+cos2φ)). The numerical results are given in Table 4 and Figure 2(b).

The relationship between ε and convergence rate (N=10, m=4, M=16, and θk=0.58).

ε    k 0 1 2 3 4 5 6 9
0.75 e 2.3433 E - 01 1.2450 E - 01 7.4989 E - 02 5.2822 E - 02 4.7623 E - 02 4.6278 E - 02 4.5525 E - 02 4.4564 E - 02
e h 1.0983 E - 01 4.9507 E - 02 2.2168 E - 02 1.0239 E - 02 4.9334 E - 03 2.5048 E - 03 4.9131 E - 04
q h 2.2185 2.2333 2.1651 2.0754 1.9696 1.5930

0.5 e 2.3120 E - 01 1.3855 E - 01 9.5116 E - 02 7.4367 E - 02 6.3994 E - 02 5.8518 E - 02 5.6670 E - 02 5.5578 E - 02
e h 9.2654 E - 02 4.3431 E - 02 2.0749 E - 02 1.0373 E - 02 5.4758 E - 03 3.0715 E - 03 7.8777 E - 04
q h 2.1334 2.0931 2.0003 1.8943 1.7828 1.4837

0.05 e 3.1906 E - 01 2.6051 E - 01 2.3278 E - 01 2.1796 E - 01 2.0941 E - 01 2.0414 E - 01 2.0068 E - 01 1.9529 E - 01
e h 7.1017 E - 02 2.7730 E - 02 1.4816 E - 02 8.5529 E - 03 5.2765 E - 03 3.4545 E - 04 1.2856 E - 04
q h 2.5610 1.8716 1.7323 1.6209 1.5274 1.3363

0.005 e 3.6584 E - 01 3.2417 E - 01 3.0784 E - 01 2.9940 E - 01 2.9434 E - 01 2.9100 E - 01 2.8867 E - 01 2.8468 E - 01
e h 1.9591 E - 01 1.6329 E - 02 1.8640 E - 02 1.2301 E - 02 4.9778 E - 03 2.3344 E - 03 9.9644 E - 04
q h 11.9976 0.8760 1.5153 2.4712 2.1324 1.2896

0.0005 e 3.7328 E - 01 3.4165 E - 01 3.3061 E - 01 3.2510 E - 01 3.2187 E - 01 3.1978 E - 01 3.1833 E - 01 3.1583 E - 01
e h 1.1714 E - 01 3.6606 E - 02 1.1542 E - 02 3.2283 E - 03 2.0883 E - 03 1.4491 E - 03 6.3191 E - 04
q h 3.2001 3.1715 3.5754 1.5459 1.4411 1.2749
Example 11.

Similar with Example 10, Γ0, Γ1, and a(x,u) are replaced by Γ0={(μ0,φ)|μ0=0.5,0φ2π}, Γ1={(μ1,φ)μ1=1.5,0φ2π}, and a(x,u)=1/1-u2, respectively. And we take (91)f=2(1-ε)coshμcosφ(cosh2μcos2φ-3sinh2μsin2φ)(cosh2μ+cos2φ)3. The exact solution of Example 11 is u=sin(2coshμcosφ/f0(cosh2μ+cos2φ)). The numerical results are given in Tables 5 and 6 and Figure 2(c).

From the numerical results, one obtains that the numerical errors can be affected by the choice of relaxation factor θk, the coefficient ε, and the location of the artificial boundary. In Tables 1, 2, 5, and 6 and Figure 2(c), the relationship between the meshes and the convergence rate is presented. We obtain that the convergence rate is independent of the finite element mesh size. In Table 3 and Figure 2(c), the convergence rates for different relaxation factor θk are compared. The results indicate that the choice of the relaxation factor is very important for the performance of the nonoverlapping alternating method. On the other hand, the convergence rate is not sensitive to the relaxation factor θk between the interval (0.5,0.67). What’s more, as it is shown in Table 4 and Figure 2(c), for the anisotropic problems, the choice of ε can also affect the convergence rate.

The relationship between meshes and convergence rate (N=10, θk=0.65, and ε=0.75).

( m , M) Error Iteration number
0 1 2 3 4 5 6 9
(2,8) e 0 6.6354 E - 01 3.8411 E - 01 2.8271 E - 01 2.3876 E - 01 2.1771 E - 01 2.0659 E - 01 2.0013 E - 01 1.9132 E - 01
e 3.2474 E - 01 2.0835 E - 01 1.6359 E - 01 1.4308 E - 01 1.3278 E - 01 1.2713 E - 01 1.2375 E - 01 1.1900 E - 01
e h 1.5452 E - 01 5.6859 E - 02 2.5207 E - 02 1.2187 E - 02 6.4127 E - 03 3.6744 E - 03 1.0834 E - 03
q h 2.7176 2.2557 2.0684 1.9004 1.7452 1.3973

(4,16) e 0 4.4653 E - 01 1.7098 E - 01 8.8097 E - 02 5.8527 E - 02 4.7483 E - 02 4.2925 E - 02 4.0759 E - 02 3.8380 E - 02
e 2.5970 E - 01 9.5295 E - 02 5.1105 E - 02 3.9408 E - 02 3.4740 E - 02 3.2516 E - 02 3.1348 E - 02 2.9955 E - 02
e h 1.7293 E - 01 5.2160 E - 02 2.0094 E - 02 8.4650 E - 03 3.8880 E - 03 1.9662 E - 03 4.5470 E - 04
q h 3.3154 2.5958 2.3738 2.1772 1.9774 1.5007

The relationship between meshes and convergence rate (N=10, θk=0.65, and ε=0.50).

(m, M) Error Iteration number
0 1 2 3 4 5 6 9
(2,8) e 0 8.2535 E - 01 4.8503 E - 01 3.7829 E - 01 3.3445 E - 01 3.1417 E - 01 3.0370 E - 01 2.9771 E - 01 2.8963 E - 01
e 4.0734 E - 01 2.6879 E - 01 2.2125 E - 01 2.0008 E - 01 1.8968 E - 01 1.8407 E - 01 1.8075 E - 01 1.7609 E - 01
e h 2.0397 E - 01 6.6268 E - 02 2.7907 E - 02 1.2785 E - 02 6.3537 E - 03 3.4410 E - 03 1.0625 E - 03
q h 3.0780 2.3746 2.1829 2.0122 1.8465 1.3787

(4,16) e 0 5.3524 E - 01 1.8866 E - 01 1.1017 E - 01 8.6488 E - 02 7.8123 E - 02 7.4538 E - 02 7.2733 E - 02 7.0595 E - 02
e 3.0655 E - 01 1.1062 E - 01 7.6720 E - 02 6.5692 E - 02 6.0842 E - 02 5.8486 E - 02 5.7219 E - 02 5.5639 E - 02
e h 2.3531 E - 01 5.4156 E - 02 1.7431 E - 02 6.7638 E - 03 2.9679 E - 03 1.4447 E - 03 3.7294 E - 04
q h 4.3449 3.1069 2.5771 2.2790 2.0543 1.4481
Acknowlegments

This work is supported by the National Natural Science Foundation of China, Contact/Grant no. 11071109 and the Foundation for Innovative Program of Jiangsu Province, Contact/Grant no. CXZZ12_0383 and CXZZ11_0870.

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