Research Article | Open Access
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 and uniqueness result for the original problem. Finally, some numerical examples are presented to illustrate the feasibility of the method.
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 [1–4] 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 [6–8], 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 [11–15] by virtue of the nonoverlapping domain decomposition method.
Let be an elongated, bounded, and simple connected domain in with sufficiently smooth boundary , . We consider the numerical solution to the exterior anisotropic quasilinear problem with or , , , and 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 [11–14, 16] for more details.
Following [11, 13], suppose that the given function satisfies Where the two constants . In the following, we suppose that the function has compact support, that is, there exists a constant , such that
We also assume that Now, we introduce an elliptical artificial boundary enclosing such that
Then, is divided into two nonoverlapping subdomains and (see Figure 1), where denotes the bounded domain between and , and refers to the unbounded domain outside . The original problem (1) is decomposed into two subproblems in domains and with . We have the nonoverlapping domain decomposition algorithm.
Step 1. Choose an initial value , and put .
Step 2. Solve a Dirichlet boundary value problem in the exterior domain :
Step 3. Solve a mixed boundary value problem in the interior domain :
Step 4. Update the boundary value ,
Step 5. Put , 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 which is independent of and , [6, 8, 19] have discussed the corresponding problems by this technique. Now, we introduce the so-called Kirchhoff transformation  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.
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 . 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 , , and . Following from , we have the following.
Theorem 1. The transformation between elliptic coordinates and Cartesian coordinates in (14) possesses the following property.(1)The Jacobian determinant of (14) is if and only if .(2)For , the following holds (3)For the exterior domain , we have where refers to the unit exterior normal vector on (regarded as the inner boundary of ).
Proof. The conclusions 1 and 2 can be obtained by direct computation. And 3 follows from the property
2.1. Natural Integral Equation for
Assume that is the solution to problem (13) and the value 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 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 , , .
2.2. Natural Integral Equation for
Now, we assume that can be expressed in the form: , with . We also assume that is the solution to problem (13) and the value is given, namely, Let and . Then, the boundary is changed by the elliptic boundary ; the unit exterior normal vector on is By the above transformation, problem (13) changes into
This is the right problem we talked in Section 2.1. Similar with (14), we let with Then, just the same as the problem discussed in Section 2.1, we have the natural integral equation on where is the unit exterior normal vector on . From (13), we obtain Combining (11), (34), and (35), we obtain the exact artificial boundary condition of on
3. Variational Problem and Convergence Analysis of the Algorithm
3.1. The Equivalent Variational Problem
Now, we consider problem (8). We will use denoting the standard Sobolev spaces. and refer to the corresponding norms and seminorms. Especially, we define , , and . Let us introduce the space and the corresponding norms
3.2. Convergence Analysis of the Method in Continuous Case
From (10)-(11), the original problem (1) can be changed to Then, we let and . Problem (43) becomes where and are the corresponding changes from and , respectively. Let be extended to , , . Then, problem (44) is equivalent to
Since it is difficult to estimate the convergence rate for a general unbounded domain , we here let be an exterior domain of an ellipse , with , and is taken as stated in Section 1. We introduce the following conclusion first.
Lemma 2. If is the solution of where is the elliptical ring domain between and , then there exist a unique and
Proof. The result can be obtained directly from (46) by the separation of variables.
Proof. We assume that the exact solution to problem (1) is , and we let , , and . Then, following (7)–(8), we let and . From (24), we assume that . By the natural integral equation (27), we have
By Lemma 2, one obtains
with . From (51), confines on can be expressed as
Then, we have
Let , then and
Assuming that , then .
If , , then or equally
By the trace theorem, we have From (54), one also has with . Assuming that , then .
For , , the convergence result can be obtained similarly with (55)–(57). Therefore, for , the nonoverlapping domain decomposition method is convergent.
3.3. Discrete Nonoverlapping Alternating Algorithm and Convergence Analysis
Divide the arc into parts, and take a finite element subdivision in such that their nodes on are coincident. That is, we make a regular and quasiuniform triangulation on , such that where is a (curved) triangle and 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.
Lemma 4. There exists a constant which has a different meaning in a different place and is related to and , such that
Proof. From (3) and (40), one can obtain that
For the natural integral equation, one can obtain the following result
For this purpose, for , we assume that with , . Then, we have
Combining property (3), Cauchy’s inequality, and the trace theorem, we have
Next, we show that , for any . For any given , let us consider the following auxiliary problem in :
From the analysis in Section 2.1, we know that the solution to problem (70) satisfies Multiplying (70) by and integrating over , we have
Letting , 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 to problem (39) satisfies with a positive constant.
Theorem 6. Problem (39) is uniquely solvable.
Proof. Since the space is separable (indeed, it is a closed subspace of the space which is separable), there exist increasing sequences of finite-dimensional Hilbert subspaces of such that
We define a mapping by where stands for the scalar product on . Since the function is bounded and the trace of the function in on belongs to , hence at least to from the Sobolev embedding, each mapping is well defined and continuous on . What’s more, by Lemmas 4 and 5, we obtain So, the right-hand side is nonnegative on the circle of radius which is defined by Applying Brouwer’s fixed point theorem  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 into , 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 , 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 is a vector whose components are function values at nodes on , and is a vector whose components are function values at interior nodes of . The matrix is the stiffness matrix obtained from finite element in , while follows from the natural boundary element method on .
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 exists. Now, we let be the discrete analogue of the Steklov-Poincaré operator on , with and . 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:
And we also have the following convergence result.
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 And we let denote the maximal error between the iteration and , that is, and 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 and , 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.
Example 8. We assume the exterior domain with elliptical boundary , . Now, we consider the problem
when , , and .
The exact solution of Example 8 is . For different meshes and , we have the following results ranked in Tables 1 and 2.