Embedding the irregular doubly connected domain into an annular regular region, the unknown functions can be approximated by the barycentric Lagrange interpolation in the regular region. A highly accurate regular domain collocation method is proposed for solving potential problems on the irregular doubly connected domain in polar coordinate system. The formulations of regular domain collocation method are constructed by using barycentric Lagrange interpolation collocation method on the regular domain in polar coordinate system. The boundary conditions are discretized by barycentric Lagrange interpolation within the regular domain. An additional method is used to impose the boundary conditions. The least square method can be used to solve the overconstrained equations. The function values of points in the irregular doubly connected domain can be calculated by barycentric Lagrange interpolation within the regular domain. Some numerical examples demonstrate the effectiveness and accuracy of the presented method.
In physics, mechanics, and other disciplines, Poisson equation or Laplace equation is used as the governing equation to describe electric potential, temperature, and many other physical quantities. The functions satisfying Laplace equation are called potential functions and the problems of electric potential, temperature, and so forth are also known as potential problems. In engineering problems, we need to inevitably solve potential problems in complex regions and the doubly connected domain composed of two closed curves is a typical complex region. Therefore, how to precisely solve potential problems in complex regions is an important issue in the field of numerical calculation.
The finite element method (FEM) is an effective numerical method for solving potential problems in complex domains [
Collocation method without element division and numerical integration is a truly meshless method. The collocation method has been widely applied to the field of engineering numerical calculation [
Embedding an irregular domain into a regular region, such as rectangular and disk, it is an effective method for solving boundary value problems of partial differential equation in complex regions [
The paper is organized as follows. In Section
Consider the potential problem on irregular domain as shown in Figure
The doubly connected domain and its regular domain.
In system of polar coordinates, the governing equation and boundary conditions of the potential problem are the following:
Given a function
The barycentric Lagrange interpolation basis function is defined as follows:
As a result, (
Embedding the irregular computational region
The computational nodes and function values can be formed into three
The barycentric Lagrange interpolation of the function
From formula (
By formula (
Here,
The boundary conditions (
For numerical analysis in doubly connected domain, we need some additional conditions to ensure the singlevaluedness and smoothness of function
The first and second conditions in (
Owing to the additional conditions discretized on the computational nodes, we can use replacement method to apply the additional conditions [
Combining (
Using least square method to solve (
In this section, we present some numerical experiments to verify the methods developed in the earlier sections. The method is validated by employing exact solutions with known boundary conditions and evaluating the computational errors. The computational programs compile using Matlab. The overconstrained equation (
In numerical analysis, the second kind Chebyshev points on the interval
Consider the doubly connected domain composed of two concentric circles. The radii of internal and external circles are
The absolute error and relative error with the different number of computational nodes in radial and ring direction are listed in Table
Computational error of regular domain collocation method under different number of nodes in Example


Absolute error  Relative error 

9  15 


9  21 


11  15 


11  21 


15  15 


19  15 


23  15 


31  15 


The relations of nodes number and computational errors with different types of nodes. (a) Chebyshev point; (b) equidistant nodes.
Consider the doubly connected domain composed of the kite external curve
Boundary conditions are determined by analytical solution
The absolute error and relative error with the different number of computational nodes in radial and ring direction are listed in Table
In another highly accurate numerical method, the collocation Trefftz method, the absolute error of
Computational error of regular domain collocation method under different number of nodes in Example


Absolute error  Relative error 

9  15 


9  21 


9  31 


9  41 


11  15 


11  21 


11  31 


11  41 


The doubly connected domain and its regular domain in Example
Distribution of computational nodes on regular and doubly connected domains in Example
The error distribution of nodes using regular domain collocation method in Example
The image of numerical solutions on irregular domain in Example
Consider the doubly connected domain composed of the epitrochoid external curve
Boundary conditions are determined by analytical solution
The doubly connected domain and its regular domain in Example
The error distribution of nodes using regular domain collocation method in Example
The image of numerical solutions on irregular domain in Example
Consider the eccentric annular region composed of the external eccentric circle
Boundary conditions are determined by analytical solution
The doubly connected domain and its regular domain in Example
The error distribution of nodes using regular domain collocation method in Example
The image of numerical solutions on irregular domain in Example
Consider the doubly connected domain composed of the external ellipse
Boundary conditions are determined by analytical solution
The doubly connected domain and its regular domain in Example
The error distribution of nodes using regular domain collocation method in Example
The image of numerical solutions on irregular domain in Example
Regular domain collocation method is an effective method to solve the potential problem on doubly connected domain of complex boundary and has the very high calculation precision. Barycentric interpolation collocation method can be applied to solve the boundary value problem of differential equations on irregular region by regular domain collocation method. So the application scope of barycentric interpolation collocation method is expanded.
The key problem is how to discrete and impose boundary conditions in the regular domain collocation method. Using barycentric Lagrange interpolation method, a stabilized, highprecision interpolation method, we can accurately and conveniently discretize boundary conditions on irregular boundary. Numerical calculation indicates that if the boundary point is less than the maximum of the radial nodes and circular nodes, the resulting coefficient matrix of algebraic equation is not column full rank. As a result, we cannot get numerical solution.
Regular domain collocation method proposed in this paper can be directly applied to solve the differential equation boundary value problem on the irregular simply connected region.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the support of the National Basic Research Program of China (Grant no. 2010CB732002), the National Natural Science Foundation of China (Grants nos. 51179098 and 51379113), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20120131110031), and the Program for New Century Excellent Talents in University of Ministry of Education of China (Grant no. NCET122009).