On the Coupled of NBEM and CFEM for an Anisotropic Quasilinear Problem in an Unbounded Domain with a Concave Angle ()
1. Introduction
In this paper, we propose the method of coupling curved edge finite element (CEFE) and natural boundary element (NBE) to solve an anisotropic quasilinear problem in an unbounded domain. The CEFE-NBE method is based on the artificial boundary method [1] [2] , namely the coupling of edge finite element (EFE) and natural boundary element (NBE) method [3] [4] . The EFE-NBE method has been used to solve many linear problems [5] [6] and is generalized to solve quasilinear problems [7] [8] [9] [10] [11] . We find the CEFE-NBE method also has been used to solve some linear problems [12] [13] [14] [15] , so we try to generalize it to quasilinear boundary value problem.
The standard procedure of the method of coupling curved edge finite element and natural boundary element can be described as follows. We introduce an artificial boundary to divide the original domain into two subregions, a bounded inner region and an unbounded one with special boundary, which are solved by the curved edge finite element and the boundary element method respectively.
Suppose Ω is an infinite domain with a concave angle
, and
; the boundaries of Ω are disintegrated into three disjoint parts:
, the boundary Γ is a simple smooth curve part, Γ1 and Γ2 are two half lines. We have
We consider the following quasilinear problem
Suppose that the given function
satisfies:
, and for almost all
, (1)
with two constants
;
, and for almost all
, (2)
with a constant
.
This problem has many physical applications in the field of continuum mechanics, we also assume that
are continuous and suppose that the given function
has compact support, i.e., there exists a constant
, such that
(3)
Furthermore, we assume that
, when
. (4)
The rest of the paper is organized as follows. In section 2, we obtain the exact quasilinear elliptical arc artificial boundary condition. In section 3, we give the equivalent variational problems and the finite element approximations and prove the well-posedness and the convergence results of the reduced problems. In section 4, we give some numerical examples to show the efficiency and feasibility of this method.
2. Natural Boundary Reduction
Now, we introduce an artificial boundary
(5)
with
.
Let us introduce the artificial boundary Γ0 which divides Ω into two regions (see Figure 1), a bounded domain Ω1 and an unbounded domain Ω2. Then the problem (1) can be rewritten in the coupled form:
(6)
(7)
and
are continuous on
, where
,
,
and
, and
is the unit exterior normal vector on
.
Particularly,
is independent of x and u when
, the problem (7) is simplified to the linear exterior elliptic problem [8] .
We introduce the so-called Kirchhoff transformation: [8]
(8)
which gives
(9)
and
Figure 1. Artificial boundary of area Ω.
(10)
From (7), that w satisfies the following problem:
(11)
Assume that
is the solution of the problem (13), and the value
is given
we introduce
, the boundary
is changed to the elliptic arc boundary
, the unit exterior normal vector on
is
The next, we need to discuss the relationship between elliptic coordinates
and cartesian coordinates
, the relationship can be expressed as below:
(12)
where
The problem (11) is transformed into
(13)
The based on the natural boundary reduction [1] [2] , there are the Poisson integral formulas [9] .
(14)
and the natural integral equation
(15)
From (9), we obtain
(16)
Combining (8), (9) and (15), we obtain the exact artificial boundary condition of u on
,
(17)
By the exact quasilinear artificial boundary condition (17), the original problem confines in
can be defined as follows
(18)
3. Variational Problem and Finite Element Approximation
3.1. The Equivalent Variational Problems
Now, we consider the problem (18). First, we will use
denoting the standard Sobolev spaces,
and
denoting the corresponding norms and semi-norms. In particular, we denote
,
and
.
Let us introduce the space
(19)
and the corresponding norms
The boundary value problem (18) is equivalent to the following variational problem
(20)
with
(21)
(22)
(23)
Lemma 1 [4] There exists
, such that
In practice, we need to truncate the series in (17) for some nonnegative integer N, that is
(24)
with
(25)
Then, we consider the following approximate problem
(26)
The problem (26) is equivalent to the following variational problem
(27)
where
(28)
Similar with lemma 1, we have:
Lemma 2 There exists a constant
, such that
3.2. Finite Element Approximation
Firstly, we divide the arc
into M parts and take an edge finite element subdivision
in
, so that their nodes on
are the same. The curved edge element subdivision
. means that we make a regular
on
, and then we replace the standard triangles T whose two vertices lie on the boundary
with a curved triangles
whose one edge coincides with the boundary.
Secondly, we let
, and use
to denote the vertex of a curved triangles
, and
to denote the curved edge. We use area coordinates
to represent a standard triangle T corresponding to the curved triangle
,
(29)
where
, then, based on the linear transformation of the T, we make the following nonlinear transformation, so that the curved triangle
can be mapped to the reference element
one-by-one,
(30)
where
(31)
(32)
and
,
,
,
,
,
,
.
In the end, we get a curved edge element subdivision by the above method in
, and it maps a standard triangle in the plane
to a curved triangle one-by-one.
Next, we construct the coordination element, i.e. approaching space
. Triples
are used to define the finite element, and use
to denote the space of first degree polynomials and the space of zero degree polynomials on
. The following mappings exist
(33)
and use
to denote a composite mapping, and
is a set of linear functions, we denote
[12] (supplement: see reference [12] for meaning of
).
According to the reference [12] , we propose:
Lemma 3. If
and the edge finite element are regular, the difference error of each curved triangular element is estimated as follows
(34)
where C is a constant,
is interpolation operator,
,
is well-determined.
Proof: we assume
is regular, from reference [12] , when
, we have
In the affine condition, according to various properties of
, we obtain the difference error of each curved triangular element,
Let
, combined with (29), we obtain
(35)
The approximate problem of (27) can be written as
(36)
Similar with existence and uniqueness in [4] , we have:
Lemma 4. The variational problems (20), (27) and (36) are uniquely solvable.
From now on, let
, and
be the solution of problems (20), (27) and (36) respectively. We also assume that
(37)
and we require that
is a family of finite-dimensional subspaces of
, such that
(38)
(39)
where
is independent of h.
The continuous piecewise polynomial spaces, such as
, satisfy the condition (37), and if let
, where
is the interpolation operator, we have
and we can obtain the following lemma.
Theorem 1 [5] Suppose
, and
be the solution of problems (20), (27) and (36) respectively,
. According to the references [4] , we know
(40)
Proof: For
, we assume that
Then, we have the following result
From references [5] [8] [9] , we know
then, we obtain
Like [8] , we introduce operators:
, we have
for
, we have
Therefore,
4. Numerical Example
Example 1. We assume
with
. According to references [8] [9] [10] , we make
for easy analysis. We introduce an elliptical boundary
We take our numerical results for problem (1), with
Furthermore, we let
for some integer
. The exact solution “u” is solved with
.
The numerical results are given in Table 1 and Table 2 (Figures 2-4).
Table 1. Error value when straight triangular element is used for finite element.
Table 2. Error value of curved edge element in finite element.
Figure 2. Error value of straight triangular element in finite element with N = 10,
.
Figure 3. Error value of curved edge element in finite element with N = 10,
.
Figure 4. Comparison of error value in Figure 2 and Figure 3 with mesh = h/4.
These are the three error analysis diagrams of Example 1.
Example 2. We assume
with
. Similar with example 1,
. We introduce an elliptical boundary
We take our numerical results for problem (1), with
Furthermore, we let
for some integer
. The exact solution “u” is solved with
.
The numerical results are given in Table 3 and Table 4 (Figures 5-7).
Figure 5. Error value of straight triangular element in finite element with N = 10,
.
Figure 6. Error value of curved edge element in finite element with N = 10,
.
Table 3. Error value when straight triangular element is used for finite element.
Table 4. Error value of curved edge element in finite element.
Figure 7. Comparison of error value in Figure 5 and Figure 6 with mesh = h/2.
These are the three error analysis diagrams of Example 2.
The numerical examples show that the errors can be affected by the location of the artificial boundary and the order of the artificial boundary condition, and it can be reduced by refining the mesh and optimizing finite element subdivision methods. The numerical results are coincident with the theoretical analysis and show the efficiency of the coupling method.
Acknowledgements
This research was funded by the National Natural Science Foundation of China (No. 11401296). The authors are grateful for them.