Abstract
In this paper, the exact artificial boundary conditions for quasi-linear problems in semi-infinite strips are investigated. Based on the Kirchhoff transformation, the exact and approximate boundary conditions on a segment artificial boundary are derived. The error estimate for the finite element approximation with the artificial boundary condition is obtained. Some numerical examples show the efficiency of this method.
1. Introduction
The quasi-linear problems in semi-infinite strips have many physical applications in the field of magnetostatics or compressible flow around an obstacle in a channel. There have been many relevant works about quasi-linear problems in bounded domains, for example, the Galerkin approximations [1, 2], the finite element method [3, 4], and the mixed finite element method [5–7] for quasi-linear problems. One can refer to [8–10], for more related works.
The artificial boundary method [11, 12], which is also called coupling of the finite element method with natural boundary reduction [13–15] or the DtN method [16, 17], is a common method to deal with quasi-linear problems in unbounded domains. In the last decade, artificial boundaries of various shapes have been derived for quasi-linear problems in unbounded domains. Circular [18, 19] and elliptical [20] artificial boundaries are for two-dimensional problems, spheroidal artificial boundaries [18] are for three-dimensional problems, and circular arc artificial boundaries [21] are for problems in concave angle domains.
The purpose of this paper is to propose an artificial boundary method of using a segment artificial boundary for quasi-linear problems in semi-infinite strips. The segment artificial boundary we proposed in this paper is different with the circular artificial boundary in [18]. We also obtain an error estimate in Section 3, which was not discussed in [18].
Let be a strip, and is the width of the channel . The boundary of domain is decomposed into three disjoint parts: ,, and (see Figure 1). We introduce a Cartesian coordinate system , such that the ray coincides with the -axis.
We consider the following quasi-linear problem:where is the unit exterior normal vector on or and and are two given functions.
Suppose that and are continuous, and satisfies [1]where are two positive constants, andwhere is a positive constant. We also assume that has compact support, i.e., there exists a constant , such that
Additionally, we suppose that
The rest of the paper is organized as follows. In Section 2, we derive the exact artificial boundary condition on a segment. In Section 3, we discuss the finite element approximation and a new error estimate. In Section 4, we give some numerical examples to show the efficiency of the method. The conclusions are given in Section 5.
2. Exact Quasi-Linear Artificial Boundary Condition
We introduce a segment artificial boundary to enclose , which divides into a bounded domain and an unbounded domain (see Figure 2).
Then, the original problem (1) can be described in the coupled form:where , , , and .
We introduce the Kirchhoff transformation [22]:
Since is a positive function, transformation (9) is invertible. Notice that
Then, quasi-linear problem (1) can be transformed into a linear problem as follows:
By the natural boundary reduction [13–15], we know that the solution of problem (11) has the Fourier series expansion:where
We differentiate (12) with respect to and set to obtain
Sincewe have the exact artificial boundary condition of on :
By the exact artificial boundary condition (16), we obtain
Suppose ; then, problem (17) is equivalent to the following variational problem:where
For any , we have the following equivalent definition of Sobolev spaces [23]:
The norm of can be defined as follows:
Then, we obtain the following lemma.
Lemma 1. The bilinear form is symmetric, continuous, and semidefinite on .
Proof. For , we suppose thatTaking the derivative with respect to and , we obtainThen, we haveNext, we show that is semidefinite. For any given , we consider the auxiliary problem as follows:The solution of the above problem satisfiesWe multiply (27) by and integrate over ; then, we haveThis completes the proof.
In practice, we have to truncate the infinite series in (16) by finite terms; letConsider the approximation problemProblem (31) is equivalent to the following variational problem:whereSimilar with Lemma 1, we have
Lemma 2. The bilinear form is symmetric, continuous, and semidefinite on .
3. Finite Element Approximation
Suppose that is a quasi-uniform and regular triangulation of such thatwhere is a (curved) triangle and is the maximal diameter of the triangles. Let
We consider the approximation problem of (32):
Theorem 1. The variational problems (18), (32), and (36) are uniquely solvable.
Proof. By (2), we haveThis means that is coercive and bounded in . From Lemma 1, we obtain that is also coercive and bounded in . By (3), we get that is uniformly Lipschitz continuous with respect to . Under these conditions, referring to [1], we obtain that variational problem (18) has a unique solution , for all . It is easy to deduce that problems (32) and (36) are uniquely solvable in the same way.
We assume and are the solutions of problems (18), (32), and (36), respectively. We also suppose thatAdditionally, we require that is a family of finite-dimensional subspaces of , such that, for anywhere is independent of .
Then, we obtain that the continuous piecewise polynomial spaces (35) satisfy condition (38). Moreover, if we assume , where is the interpolation operator, then, by (40), we obtainFollowing the convergence theory in [4, 15], we have the result as follows:Furthermore, we have the following lemma.
Lemma 3. Suppose is the solution of (18) and is the solution of (32); we have
Proof. From (2) and Theorem 2, we haveFor , we supposeThen,From , we have that is bounded in . So, there exists a subsequence , s.t. . Then, similar with Lemma 3,4 in [18], we obtain (43).
Then, we have the following convergence theorem.
Theorem 2. Let , and assumptions (38)–(40) be satisfied. Then,
Next, we deduce the error estimates. We suppose that the solution of problem (1) satisfies
For simplicity, we also define some notations as follows:
Then, problems (18), (32), and (36) can be replaced by some simple forms, respectively. Moreover, we introduce the following bilinear form and
Suppose is the dual space of . By (2) and continuity of , we obtain that is bounded on . Then, there exists an operator such that
Similar to Lemma 2.2 in [19], we have the lemma as follows:
Lemma 4. The following inequality,holds, where is a sufficient large constant.
We suppose that
Assume is the canonical injection. Since is compactly embedded in , we obtain that the operator defined by is also compact. Then, we deduce that is an isomorphism.
By conditions (19), (52), and (53) and Theorem 10.1.2 in [24], there exists, s.t. the following inf-sup condition is satisfied:where is a constant independent of .
We define the Galerkin projection with respect to and :
Then, we obtainwhere
Lemma 5. is a solution of (36) if and only if the following equation,holds, wherewith .
Proof. Suppose . Then, by (32) and (36) andwe can obtain the desired result.
SupposeThen, from [19, 25], we have the following lemma.
Lemma 6. There exists a constant independent of , such that
We denote a nonlinear mapping , which satisfies that is the unique solution offor any given . Suppose
Then, we obtain the lemma as follows.
Lemma 7. The nonlinear mapping is a continuous mapping from to .
Proof. By (62), we haveCombining (64) with (54), we deduce that the mapping is continuous, i.e.,For any ,Since is regular and quasi-uniform, according to [26], we have the following inverse inequality:By the definition of , (56), and (69), we obtainThis implies that . Under the definition of , (62) can be rewritten asThen, by (54) and Lemmas 5 and 6, we obtainThis means that .
Theorem 3. Suppose is a solution of problem (1), where . We also assume that and satisfies (53). With sufficiently small , problem (36) has an approximate solution , such that
Proof. By Brouwer’s fixed-point theorem and Lemma 7, there exists , such that . From Lemma 5, we deduce that is a solution of (36). Moreover, by (56) and , we haveFor any , according to Lemma 3, we obtainFrom (32), we haveLet ; then,By (18), (52), and (53) and [24], we haveCombining (74) with (78), we obtainThis completes the proof.
4. Numerical Examples
In this section, we computed some numerical examples by the method developed in Sections 2 and 3 to test the efficiency of the method.
Example 1. We take , , , , and . The exact solution of original problem is . Let be the artificial boundaries. Figure 3 shows the mesh of subdomain . The numerical results are given in Table 1 and Figures 4 and 5.
From the numerical results, we can deduce that the finite element mesh, the location of artificial boundary, and the truncation terms of series can affect the numerical errors. It is obvious that our method is very effective.
5. Conclusions
In this paper, we propose a method of artificial boundary conditions for quasi-linear problems in semi-infinite strips by using a segment artificial boundary. The exact and approximate artificial boundary conditions are given based on the Kirchhoff transformation. A new error estimate for the finite element approximation with the approximate artificial boundary condition is obtained. Finally, some numerical examples show the efficiency of this method. The quasi-linear problem, we considered in this paper, is a two-dimensional problem. Based on the proposed method, one can design some artificial boundary conditions for three-dimensional problem; we shall report on progress in some of these directions in a future publication.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was funded by the National Natural Science Foundation of China (no. 11371198).