A Nonoverlapping Domain Decomposition Method for an Exterior Anisotropic Quasilinear Elliptic Problem in Elongated Domains

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 anduniqueness result for the original problem. Finally, somenumerical examples are presented to illustrate the feasibility of the method.


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 [1][2][3][4] 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 [6][7][8], 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 [11][12][13][14][15] by virtue of the nonoverlapping domain decomposition method.
Let Ω be an elongated, bounded, and simple connected domain in R 2 with sufficiently smooth boundary Ω = Γ 0 , Ω  = R 2 /Ω.We consider the numerical solution to the exterior anisotropic quasilinear problem with  >  > 0 or  =  = 1, x = (, ), (⋅, ⋅), and (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  is the thermal conductivity of the medium and  is the temperature field; the field of compressible flow, where  is the density and  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  is the heat conductivity.One can also refer to [11][12][13][14]16] for more details.Following [11,13], suppose that the given function (⋅, ⋅) satisfies 0 <  0 ≤  (x, ) ≤  1 , ∀ ∈ R, and for almost all x ∈ Ω  , Where the two constants  0 ,  1 ∈ R. In the following, we suppose that the function (x) ∈  2 (Ω  ) has compact support, that is, there exists a constant  0 > 0, such that We also assume that  (x, ) =  0 () , when |x| ≥  0 .
Step 2. Solve a Dirichlet boundary value problem in the exterior domain Ω 2 : Step 3. Solve a mixed boundary value problem in the interior domain Ω 1 : Step 4. Update the boundary value 0 <   < 1, Step 5. Put  =  + 1, turn to Step 2. The relaxation factor   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 (x, ) =  which is independent of x and , [6,8,19] have discussed the corresponding problems by this technique.Now, we introduce the so-called Kirchhoff transformation [20] Then, we have From ( 7), we have that  satisfies the following problem 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.

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 Ω 2 .For this purpose, we need to discuss some properties between elliptic coordinates (, ) and Cartesian coordinates (, ) first.The relationship between the two coordinates can be expressed as below with  0 = √  2 −  2 ,  =  0 cosh  1 , and  =  0 sinh  1 .
Following from [18], we have the following.
Theorem 1.The transformation between elliptic coordinates and Cartesian coordinates in (14) possesses the following property.
(2) For  ∈  2 (R 2 ), the following holds (3) For the exterior domain Ω 2 , we have where  refers to the unit exterior normal vector on Γ  1 (regarded as the inner boundary of Ω 2 ).
Proof.The conclusions 1 and 2 can be obtained by direct computation.And 3 follows from the property

Natural Integral Equation for
is the solution to problem (13) and the value | ||= 1 is given, that is, Then, based on the natural boundary reduction, there are the Poisson integral formulas or and the natural integral equation or the definition of  0 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 with with

Variational Problem and Convergence
Analysis of the Algorithm and the corresponding norms The boundary value problem ( 8) is equivalent to the following variational problem.
Then, problem (44) is equivalent to Since it is difficult to estimate the convergence rate for a general unbounded domain Ω  , we here let Ω  be an exterior domain of an ellipse Γ 0 , with  =  0 , and Γ 1 is taken as stated in Section 1.We introduce the following conclusion first.

Lemma 2. If 𝑢 is the solution of
where Ω 1 is the elliptical ring domain between Γ 0 and Γ 1 , then there exist a unique  ∈  1 (Ω 1 ) and × e i +  0 +  0 ( −  0 ) . (48) Proof.The result can be obtained directly from (46) by the separation of variables.

Discrete Nonoverlapping Alternating Algorithm and Convergence Analysis.
Divide the arc Γ  1 into  parts, and take a finite element subdivision in Ω 1 such that their nodes on Γ  1 are coincident.That is, we make a regular and quasiuniform triangulation T ℎ on Ω 1 , such that where  is a (curved) triangle and ℎ is the maximal diameter of triangles.Let Then, the approximate problem of (39) can be written as follows with In practice, the sum of ( 63) is truncated to a finite number of terms .By the hypothesis of (⋅, ⋅), it is not difficult to know the following result.
For this purpose, for ,  ∈ , we assume that with Next, we show that D(; , ) ≥ 0, for any  ∈ .For any given  ∈ , let us consider the following auxiliary problem in Ω 1 : From the analysis in Section 2.1, we know that the solution  to problem (70) satisfies Multiplying (70) by  and integrating over Ω 1 , we have Letting  2 =   2 +   2 , 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
with  3 a positive constant.
Proof.Taking  equal to  in (39), by Lemma 4, one has So, the right-hand side is nonnegative on the circle of radius  which is defined by Applying Brouwer's fixed point theorem [21] yields the existence of   in   , with norm less than , such that Since the sequence {  }  is bounded by  in , there exists a subsequence which converges weakly to  in   .Using the compactness of the imbedding of  1/2 (Γ 1 ) into  3 (Γ 1 ), we obtain that  is a solution to problem (39).Now, we show the uniqueness of the solution.Let  and  be two solutions to this problem.Then, taking  equal to  and  in (39), respectively, and combining with Lemma 4, one obtains Since  2 >  0 , 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: where U is a vector whose components are function values at nodes on Γ 1 , and V is a vector whose components are function values at interior nodes of Ω 1 .The matrix  = () = (  11  12  21  22 ) is the stiffness matrix obtained from finite element in Ω 1 , while  ℎ =  ℎ (| Γ 1 ) follows from the natural boundary element method on Γ 1 .
Problem (82) can also be rewritten as follows: Then, we have the iterative algorithm with By condition (3), one obtains that  is a positive definite matrix, so we know that  −1 22 exists.Now, we let  ℎ =  (1)  ℎ +  ℎ be the discrete analogue of the Steklov-Poincaré operator on Γ 1 , with  (1)  ℎ =  11 −  12  −1 22  21 and B = − 12  −1 22 b.Then, similar to the proof of Theorems 7.6 and 7.7 in [6], we have that the alternating algorithm (84)-( 85) is equivalent to the preconditioned Richardson iteration: And we also have the following convergence result.Theorem 7. If 0 < min   ≤ max   < 1, then the discrete nonoverlapping alternating method (84)-( 85) is convergent, and both the convergence rate and the condition number of [ (1)  ℎ ] −1  ℎ are independent of the mesh size ℎ.
We give the following four examples.Examples 8 and 11 show the relationship between meshes and convergence rate for the cases  =  = 1 and  >  > 0, respectively.Example 9 focuses on the effect of the relaxation factor   to the convergence rate.And Example 10 wants to show the relationship between the coefficients ,  and the convergence rate.
The exact solution of Example 8 is  = tan(2 sinh  sin / 0 (cosh 2 + cos 2)).For different meshes and  1 , we have the following results ranked in Tables 1 and 2. The exact solution of Example 9 is  = sin(2 cosh  cos / 0 (cosh 2 + cos 2)).The numerical results are given in Table 3 and Figure   (90) The exact solution of Example 10 is  = tan(2 sinh  sin / 0 (cosh 2 + cos 2)).The numerical results are given in Table 4 and Figure   (91) The exact solution of Example 11 is  = sin(2 cosh  cos / 0 (cosh 2 + cos 2)).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   , the coefficient , 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   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   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  can also affect the convergence rate.