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 [
Let
Problem (
Following [
We also assume that
Then,
The illustration of domains
Choose an initial value
Solve a Dirichlet boundary value problem in the exterior domain
Solve a mixed boundary value problem in the interior domain
Update the boundary value
Put
The relaxation factor
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
The transformation between elliptic coordinates and Cartesian coordinates in (
The Jacobian determinant of (
For
For the exterior domain
The conclusions 1 and 2 can be obtained by direct computation. And 3 follows from the property
Assume that
From (
Now, we assume that
This is the right problem we talked in Section
Now, we consider problem (
The boundary value problem (
And,
From (
Since it is difficult to estimate the convergence rate for a general unbounded domain
If
The result can be obtained directly from (
If
We assume that the exact solution to problem (
Then, we have
Let
Assuming that
If
By the trace theorem, we have
For
Divide the arc
There exists a constant
From (
For this purpose, for
Combining property (
Next, we show that
From the analysis in Section
Letting
We are now in the position to show the existence and uniqueness result for this type of problem (
Any solution
Taking
Problem (
Since the space
We define a mapping
Now, we show the uniqueness of the solution. Let
From the discrete problem (
Problem (
By condition (
And we also have the following convergence result.
If
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 (
We give the following four examples. Examples
We assume the exterior domain
The exact solution of Example
The relationship between meshes and convergence rate (

Error  Iteration number  

0  1  2  3  4  5  6  9  
(2,8) 










— 









—  —  1.7980  1.8620  1.8077  1.7220  1.6296  1.3992  
 
(4,16) 










— 









—  —  2.2192  2.3452  2.2467  2.1233  1.9925  1.5865 
The relationship between meshes and convergence rate (

Error  Iteration number  

0  1  2  3  4  5  6  9  
(2,8) 










— 









—  —  1.6309  1.6194  1.5564  1.4880  1.4248  1.2857  
 
(4,16) 










— 









—  —  1.8516  1.8787  1.7868  1.6828  1.5860  1.3728 
Similar with Example
The exact solution of Example
The relationship between


0  1  2  3  4  5  6  9 

0.18 










— 






 

—  — 







 
0.38 










— 






 

—  — 







 
0.50 










— 






 

—  — 







 
0.58 

















 

— 








 
0.65 










— 






 

—  — 






(a) The relationship between
We assume the exterior domain
The exact solution of Example
The relationship between


0  1  2  3  4  5  6  9 

0.75 










— 






 

—  — 







 
0.5 










— 






 

—  — 







 
0.05 










— 






 

—  — 







 
0.005 










— 






 

—  — 







 
0.0005 










— 






 

—  — 






Similar with Example
From the numerical results, one obtains that the numerical errors can be affected by the choice of relaxation factor
The relationship between meshes and convergence rate (

Error  Iteration number  

0  1  2  3  4  5  6  9  
(2,8) 

















 







 

—  —  2.7176  2.2557  2.0684  1.9004  1.7452  1.3973  
 
(4,16) 

















 

— 






 

—  —  3.3154  2.5958  2.3738  2.1772  1.9774  1.5007 
The relationship between meshes and convergence rate (
( 
Error  Iteration number  

0  1  2  3  4  5  6  9  
(2,8) 

















 







 

—  —  3.0780  2.3746  2.1829  2.0122  1.8465  1.3787  
 
(4,16) 

















 







 

—  —  4.3449  3.1069  2.5771  2.2790  2.0543  1.4481 
This work is supported by the National Natural Science Foundation of China, Contact/Grant no. 11071109 and the Foundation for Innovative Program of Jiangsu Province, Contact/Grant no. CXZZ12_0383 and CXZZ11_0870.