Solution of Exterior Quasilinear Problems Using Elliptical Arc Artificial Boundary

In this paper, the method of artificial boundary conditions for exterior quasilinear problems in concave angle domains is investigated. Based on the Kirchhoff transformation, the exact quasiliner elliptical arc artificial boundary condition is derived. Using the approximate elliptical arc artificial boundary condition, the finite element method is formulated in a bounded region. The error estimates are obtained. The effectiveness of our method is showed by some numerical experiments.


Introduction
In many fields of scientific and engineering computing, such as heat transfer, magnetostatics or compressible flow, it is necessary to deal with problems in unbounded domains. The method of artificial boundary conditions [1,2], which is also named coupled of finite element method and natural boundary element method [3][4][5] or DtN finite element method [6,7], is a normal method used to solve this kind of problem numerically.
The method can be summarized in the following four steps: (i) By introducing an artificial boundary Γ , truncate the original infinite domain Ω into two subdomains: a finite computational subdomain Ω and an unbounded residual subdomain Ω . (ii) By dissecting the problem in Ω , obtain a relation on Γ involving the solution and its derivatives. (iii) Use the relation as an approximate boundary condition on Γ , to earn a well posed problem confined in Ω . (iv) Use finite element method or other numerical methods to solve the problem in Ω .
The relation derived in (ii) and used in (iii) is called an artificial boundary condition, natural integral equation or DtN map. Natural boundary reduction reduces the boundary value problem into a hypersingular integral equation on the artificial boundary. It has many advantages, such as the positive-definite symmetry of stiffness matrices, the stability of approximate solutions, and can be coupled with the finite element method naturally and directly. The method has been used to solve linear problems in last century. It has also been successfully extended to nonlinear problems in recent years.
Suppose Ω is an infinite domain with a concave angle , and 0 2 . The boundaries of Ω are disintegrated into three disjoint parts: Γ , Γ and Γ (see Figure 1), i.e., ∂Ω Γ ∪ Γ ∪ Γ, Γ ∩ Γ ∅, Γ ∩ Γ ∅ and Γ ∩ Γ ∅. The boundary Γ is a simple smooth curve part, Γ and Γ are two half lines. We consider the following quasilinear problem where , and are given functions with some properties which will be decribed later.
Problem (1) has numerous physical applications, e.g., in the field of magnetostatics, where is the magnetic scalar potential and is the magnetic permeability; in the field of compressible flow, where is the velocity potential and is the density. There have been many numerical results about problems of this kind in bounded domains, for example, the existence and uniqueness of weak solution [8,9], the finite element method [10][11][12], the mixed finite element method [13][14][15], the discontinuous Galerkin finite element method [16][17][18], the weak Galerkin finite element method [19], and the adaptive finite element method [20,21].
The circular artificial boundary was used for exterior quasilinear problems in early years [22,23]. The elliptical artificial boundary was generalised later for elongated domains problems [24]. The circular arc boundary was often selected for problems in unbounded domains with concave angles [25], but for the problems in elongated concave angle domains, an elliptical arc boundary which leads to a smaller computational domain is much better than the circular arc case (see Figure 2).  In this paper, we propose a new method of elliptical arc artificial boundary conditions for the numerical solution of quasilinear problems in exterior elongated domains with concave angles.
Suppose that the given function •,• satisfies [8] 0 , 1 , ∀ ∈ ℝ, and for almost all ∈ Ω, where , and are three positive constants. We also assume that , are continuous. Additionally, we suppose that ∈ Ω has compact support, i.e., there exists a constant 0, such that Moreover, we assume that The rest of this paper is organized as follows. We derive the exact quasilinear elliptical arc artificial boundary condition in Section 2. In Section 3, we formulate the finite element approximation and give an new error estimate. In Section 4, we give some numerical experiments to show the efficiency and feasibility of our method. Some conclusions are given in Section 5.
In the following sections, we denote as a general positive constant independent of , and ℎ, where and ℎ will be defined in Sections 2 and 3, respectively. The constant has different meaning in different place.

Exact Quasilinear Elliptical Arc Artificial Boundary Condition
We first introduce the elliptical arc artificial boundary Γ , | 1, , ∈ Ω, 0 to enclose supp , which divides Ω into a bounded computational domain Ω and an unbounded domain Ω (see Figure 2b). Let 2 denote the distance between two foci of the previous ellipse, we introduce the elliptic co-ordinates , such that artificial boundary Γ coincides with elliptical arc , | , 0 , where √ , ln . Thus, the Cartesian co-ordinates , are related to the elliptic co-ordinates , , that is cosh cos , sinh sin . Then, problem (1) can be rewritten in the following coupled form: and are continuous on the artificial boundary Γ , We introduce the so-called Kirchhoff transformation [26]: , for ∈ Ω .
The transformation is invertible because is a positive function. Notice that we can transform the quasilinear problem (7) into the following linear problem Suppose is the solution of problem (11). By Fourier series expansion, we have where , cos , 0,1,2, ⋯.
It is easy to obtain Since , where cosh cos , and , we obtain the exact artificial boundary condition of on Γ : The difference between this artificial boundary condition and that in circular arc case is only a factor [25], so it does not increase the computational complexity of the stiff matrix from artificial boundary. In the meantime, an elliptical arc artificial boundary is advantageous in that it may be used to enclose some narrow region with concave angle efficiently, so it is much better than the circular arc one. By the exact quasilinear artificial boundary condition (17), we have Let ∈ Ω | | 0 , then problem (18) is equivalent to the variational problem as follows: Find ∈ , such that ; , where ; , .
In practice, we must truncate the infinite series in (17) Consider the following approximation problem: It is equivalent to the variational problem as follows: where For ∈ ℝ, we introduce the equivalent definition of Sobolev spaces Γ as follows [27]: The norm of Γ can be defined as follows: Then, we have the following results.
In the same way, we obtain Next, we show that ; , and ; , are semi-definite. For any given ∈ , we consider the following auxiliary problem: the solution of the above problem satisfies 0 1 , .
In the same way, we obtain ; , 0.
This completes the proof. □

Finite Element Approximation
Suppose ℎ is a regular and quasi-uniform triangulation on Ω , s.t.
Proof. From (2), it is easy to shown that the bilinear form ; , is -elliptic and bounded in , i.e., there exist constants , 0, such that Combining with Lemma 1, we can deduce that ; , ; , is also -elliptic and bounded in . By (3), , is uniformly Lipschitz continuous with respect to . Since these conditions hold, it is known [8] that variational problem (19) has a unique solution ∈ for all ∈ Ω . In the same way, we obtain that the variational problems (25) and (30) are uniquely solvable. □ We let , ∈ Ω and ∈ be the solution of problems (19), (25) and (30), respectively. We also assume that ℎ ⊂ ∩ 1,2 Ω for some ∈ 0,1 .
Moreover, we require that → is a family of finite dimensional subspaces of ∩ Ω , s.t.
Following the convergence theory in [5,8], we have the following result: Moreover, we can obtain the following lemma.

Lemma 3.
Suppose be the solution of (19) and be the solution of (25). Then, we have Proof. From (2) Next, we give the error estimates. We assume that the solution of problem (1) satisfies For simplicity, we define the following notation ; , ≜ ; , ; , , Then, problems (19), (25) and (30) where 0 is a sufficient large constant.
We assume that Let : → ′ denote the canonical injection. We obtain that operator : → ′ defined by ,0 is also compact since is compactly embedded in Ω . Thus the Fredholm alternative applies for . We have that : → ′ is an isomorphism.

Lemma 6.
There exists a constant 0 independent of ℎ, such that We define a nonlinear mapping : → as follows.
is the unique solution of for any given ∈ . Let By the fact that is quasi-uniform, according to [30], we have the inverse inequality, as follows: Combining the definition of with (41) and (49), we obtain ‖ ‖ , , 1.
This means that ∈ . By the definition of , (43) can be rewritten as Then, by (40), Lemmas 5 and 6, we have This implies that : Ω be a solution of problem (1), with 0, 2. And we also assume that | ∈ Γ and satisfies (39). With sufficiently small ℎ, the finite element equation (30) has an approximate solution ∈ , such that Proof. From Lemma 7 and Brouwer's fixed point theorem, there exists ∈ , such that . By Lemma 5, we deduce that is a solution of (30). Furthermore, by (41) and ∈ , we obtain For any ∈ , from Lemma 3, we have | ; , ; , | It follows from (25) Combining (51) with (52), we obtain results are given in Table 1 Figure 6 shows Mesh ℎ of subdomain Ω 2 . The numerical results are given in Table 2    The exact solution of original problem is sin . Figure 9 shows Mesh ℎ of subdomain Ω . The numerical results are given in Table 3, Figures 10 and 11.    the location of the artificial boundary. Numerical experiments are identical with the theoretical analysis and show that the proposed method is very effective.

Conclusions
In this paper, we propose an artificial boundary method using elliptical arc artificial boundary for exterior quasilinear problems in concave angle domains. Based on the Kirchhoff transformation, we obtain the exact and a series of approximate boundary conditions. We formulate the finite element approximation in a bounded region using the approximate elliptical arc artificial boundary condition. We also provide error estimates depend on the finite element mesh, the order of the artificial boundary condition and the location of artificial boundary. Our numerical examples show the efficiency of our method.
An elliptical arc artificial boundary is advantageous in that it may be used to enclose slender obstacles with a concave angle efficiently, so that only a small computational domain in the immediate vicinity of the obstacle is need. It is much better than the circular arc one since it does not increase the computational complexity of the stiff matrix from artificial boundary. Using the elliptical arc boundary condition we proposed in this paper, one can design other numerical methods, for example, the non-overlapping and overlapping domain decomposition methods to solve the exterior quasilinear problems in concave angle domains. The results in this paper extend many related results about the numerical methods for quasilinear problems in unbounded domains.