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# Schwarz alternating methods for anisotropic problems with prolate spheroid boundaries

- Zhenlong Dai
^{1}, - Qikui Du
^{1}Email author and - Baoqing Liu
^{2}

**Received:**3 May 2016**Accepted:**12 August 2016**Published:**26 August 2016

## Abstract

The Schwarz alternating algorithm, which is based on natural boundary element method, is constructed for solving the exterior anisotropic problem in the three-dimension domain. The anisotropic problem is transformed into harmonic problem by using the coordinate transformation. Correspondingly, the algorithm is also changed. Continually, we analysis the convergence and the error estimate of the algorithm. Meanwhile, we give the contraction factor for the convergence. Finally, some numerical examples are computed to show the efficiency of this algorithm.

## Keywords

- Schwarz alternating algorithm
- Exterior anisotropic problem
- Prolate ellipsoidal
- Artificial boundary
- Iteration method

## Background

How to deal with boundary value problems has always been a essential part of partial differential equation. Finite difference method (FDM) (Evans 1977) and finite element method (FEM) (Brenner and Scott 1996) are the most widely used method to solve PDE numerically. These two methods become in vain when it comes to the problem over unbounded domain. To overcome this, boundary element method (BEM), which can reduce the dimension of the computational domain and is suitable for solving problems in the unbounded domains, is proposed in Feng (1980). Although, it is better to handle BEM with infinite regions, it doesn’t work so well as FEM in bounded ones. Hence, the coupling of BEM and FEM becomes the interest of researchers. Papers MacCamy and Marin (1980), Hsiao and Porter (1986), Wendland (1986), Costabel (1987), Han (1990) had focused on this method. In 1983, Feng firstly proposed a direct and natural coupling method. Later in the same year, Feng and Yu (1983) formally named the method as natural boundary element method (NBEM). Meanwhile, the DtN method, which has the similar principle with NBEM, is proposed in Keller and Givoli (1989), Grote and Keller (1995). Du and Yu (2001), Hu and Yu (2001), Gatica et al. (2003), Koyama (2007), Koyama (2009), Das and Mehrmann (2016), Das and Natesan (2014), Das (2015) and references therein present the applications of this methods.

Among the reasons that effects the NBEM, the shape of artificial boundary is the essential one. Classically, circle (Givoli and Keller 1989) and spherical (Grote and Keller 1995; Wu and Yu 1998, 2000a) are chosen as the artificial boundaries. Few papers Grote and Keller (1995), Wu and Yu (2000b), Huang and Yu (2006) focus on the special artificial boundaries. These papers also proved the classic artificial boundaries were not suitable for the problem with irregular shape. On the other hand, the coupling of FEM and BEM are not enough as the performance of computer developed. The domain decomposition method (DDM) (Brenner and Scott 1996), which separates the infinite region as sum of bounded one and unbounded one with an artificial boundary on which an iteration method is constructed in, is applied on the NBEM (Yu 1994). Wu and Yu (2000b) applied this method over an infinite region. Continually, Huang et al. (2009) and Luo et al. (2013) applied this method in different problems.

In this paper, we consider the anisotropic harmonic problem over an exterior three-dimensional domain. A Schwartz alternating method is designed for the numerical solution with prolate artificial boundaries.

The outline of the paper is as follows. In “Schwarz alternating algorithm based on NBR” section, we divide the original domain \(\Omega \) into two overlapping subdomains \(\Omega _1\) and \(\Omega _2\) by choosing two artificial boundaries \(\Gamma _1\) and \(\Gamma _2\), then we construct the Schwarz alternating algorithm. We prove the convergence of the algorithm in “Convergence of the algorithm” section. The convergence rate of the algorithm is analysed in the “Analysis of the convergence rate” section. In “The error estimates of the algorithm” section, we deduce the error estimates of the discrete algorithm. In “Numerical results” section, numerical examples are computed to express the advantages of this method. Finally, we give some conclusions in “Conclusions” section.

## Schwarz alternating algorithm based on NBR

*g*is a given function that satisfies \(g\in H^{1/2}(\Gamma _0)\), and \(r=\sqrt{x^2+y^2+z^2}\). The third item of Eq. (1) keeps the existence and uniqueness of the solution.

Setting the initial value of function \(u_2^{(0)}\) on boundary \(\Gamma _1\) as \(u_2^{(0)}|_{\Gamma _1}=0\). Hence, we can solve the problem (2). Furthermore, with the limitation of \(u_1^{(1)}\) on \(\Gamma _2\), one solves the problem (3). Sequentially, we solve the problem in \(\Omega _1\) again with substituting the value of solution \(u_2^{(2)} \) on \(\Gamma _1\). Then , we repeat the steps for \(k=1,2,\ldots \) and so on.

*u*of (7) restricted on \(\Gamma'_1\) can be expressed as

## Convergence of the algorithm

*V*. Both functions of \(V_1\) and \(V_2\) can be extended into

*V*. For example, we can extend \(u \in V_1\) by zero in \(\Omega' {\setminus} \Omega'_1\) to a function in

*V*.

*V*. Thus (14) is equivalent to

###
**Theorem 1**

*There exists a constant*\(\alpha \), \(0\le \alpha <1\),

*such that*

It is obvious to conclude \(\alpha \) keeps the convergence of Schwarz alternating method. In the next section, we will prove the contraction factor \(\alpha \).

## Analysis of the convergence rate

###
**Lemma 1**

*Let*

*where*

*n*,

*m*

*are both nonnegative integers. If*\(0\le m<n\),

*then*\(P_{nm}(x)\)

*has*\(n-m\)

*different zeros*\(-1 = \alpha _1\le \alpha _2\le \cdots \le \alpha _{n-m} = 1\)

*with*\(\alpha _i=-\alpha _{n-m-(i-1)}, \quad i=1,\ldots ,n-m-1.\)

###
**Lemma 2**

*If*\(\mu >\mu _0\),

*then we conclude*

*and*

###
*Proof*

On the other hand, (21) can be easily proved by the proposition of Huang and Yu (2006),

###
**Theorem 2**

*Suppose*\(g_0\)

*is continuous on*\(\Gamma _0\)

*and*(16)

*holds. If we apply the Schwarz alternating algorithm given in*“

*Schwarz alternating algorithm based on NBR*”

*section, then*

*and*

*hold true, the constant*\(C_i\) \((i=1,2)\)

*depend only on*\(g_0\)

*and*\(\displaystyle \frac{Q^m_n(\cosh \mu _i)}{Q^m_n(\cosh \mu _0)}\)

*while*

###
*Proof*

Obviously, (23) can be proved with similar process. Finally, the theorem is proved.

###
*Remark*

The convergence is related on the overlapping part of \(\Omega'_1\) and \(\Omega'_2\). From Theorem 2, we conclude the larger the overlapping part is, the smaller the contraction factor \(\alpha \) will be, which identically means the faster the Schwarz alternating algorithm converging.

## The error estimates of the algorithm

*V*by zero extension. Therefore, we have the discrete Schwarz alternating algorithm as

*V*. Hence, \(A_h(\Omega'_2)\subset V_2\subset V.\) We have the following variational problem on the discrete space

###
**Theorem 3**

*For the discrete Schwarz alternating algorithm and the constant*\(\alpha \)

*in*

*Theorem*1,

*the following error estimates hold*

## Numerical results

Some numerical examples are computed to show the efficiency of our algorithm in this section. Using the method developed in “Schwarz alternating algorithm based on NBR” section. The linear elements is used in the computation of FEM. Computationally, we consider on three meshes: Mesh I, Mesh II and Mesh III. Each mesh is a refinement of its former one, especially as Mesh I is the primary. The refinement is defined as each of elements of the former mesh is divided into eight similar shape equally.

*e*and \(e_h\) denote the maximal error of all node functions on \(\Gamma _{1h}\), respectively, i.e.,

###
*Example 1*

The relation between convergence rate and mesh: \(\mu _1=1.5\), \(\mu _2=1.25\)

Mesh | k | Number of iteration and corresponding values | |||||
---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | ||

I |
| 2.4726E−1 | 9.0403E−2 | 5.4826E−2 | 8.0814E−3 | 8.0782E−3 | 8.0774E−3 |

\(e_h\) | – | 2.8013E−2 | 3.6179E−3 | 7.2392E−4 | 1.5669E−4 | 3.6362E−4 | |

\(q_h\) | – | – | 77.4294 | 4.9977 | 4.6200 | 4.3092 | |

II |
| 8.6794E−2 | 4.0215E−3 | 3.1259E−5 | 2.9243E−5 | 2.9104E−5 | 2.9100E−5 |

\(e_h\) | – | 1.0366E−4 | 3.4624E−6 | 3.1645E−7 | 2.8591E−7 | 2.8503E−7 | |

\(q_h\) | – | – | 29.9437 | 10.9409 | 1.1068 | 1.0031 | |

III |
| 1.6827E−3 | 9.2546E−4 | 7.4972E−5 | 7.4802E−5 | 7.4792E−5 | 7.4753E−5 |

\(e_h\) | – | 9.2858E−4 | 7.6389E−5 | 6.6424E−6 | 5.9675E−6 | 5.5203E−6 | |

\(q_h\) | – | – | 12.1564 | 11.5004 | 1.1131 | 1.0817 |

The relation between convergence rate and overlapping degree (Mesh II)

\(\mu _1\) | \(\mu _2\) |
| Number of iteration and corresponding values | |||||
---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | |||

1.5 | 1.2 |
| 6.4728E−2 | 4.6532E−3 | 3.4571E−5 | 2.6119E−5 | 2.6084E−5 | 2.6002E−5 |

\(e_h\) | – | 2.0222E−3 | 1.2045E−4 | 4.5076E−5 | 9.0874E−6 | 9.0244E−6 | ||

\(q_h\) | – | – | 16.7890 | 3.8033 | 4.9290 | 1.0660 | ||

1.5 | 1.0 |
| 4.5186E−2 | 1.0521E−3 | 9.0705E−5 | 5.4413E−5 | 1.2218E−5 | 1.2103E−5 |

\(e_h\) | – | 1.3736E−3 | 4.8967E−5 | 2.6640E−7 | 1.4184E−7 | 7.5349E−7 | ||

\(q_h\) | – | – | 28.0516 | 18.3810 | 2.7813 | 2.8248 | ||

1.5 | 0.8 |
| 1.4825E−3 | 6.7734E−4 | 9.2125E−5 | 1.8249E−5 | 5.6719E−6 | 5.5017E−6 |

\(e_h\) | – | 6.4936E−4 | 2.1429E−5 | 1.2093E−6 | 8.2674E−8 | 1.0827E−8 | ||

\(q_h\) | – | – | 30.3022 | 17.7197 | 14.62807 | 7.6359 |

From Table 1, we can see the convergence is really fast. Both *e* and \(e_h\) are smaller than them on former mesh. And the Fig. 2 shows us the errors converge rapidly. Both of them reveal that the fine the mesh, the faster the convergence. The numbers of Table 2 testify the remark in “The error estimates of the algorithm” section. By taking different \(\mu _1\) and \(\mu _2\), we chose 3 couples of artificial boundaries. Geometrically, the bigger the \(|\mu _1-\mu _2|\), the bigger the overlapping domain. Within the same triangular partition (Mesh II), we conclude that the bigger the overlapping domain, the faster the convergence.

###
*Example 2*

The relation between convergence rate and mesh: \(\mu _1=2.5\), \(\mu _2=2.0\)

Mesh | k | Number of iteration and corresponding values | |||||
---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | ||

I |
| 2.1078E−2 | 8.4562E−3 | 5.9623E−3 | 4.6782E−3 | 4.6511E−3 | 4.6407E−3 |

\(e_h\) | 9.0022E−4 | 3.0713E−5 | 2.1630E−6 | 1.5593E−6 | 1.1858E−6 | ||

\(q_h\) | 29.3106 | 14.1992 | 1.3871 | 1.3150 | |||

II |
| 8.3741E−3 | 7.6501E−3 | 4.6829E−3 | 9.4296E−4 | 8.6241E−4 | 8.5788E−4 |

\(e_h\) | – | 7.7637E−4 | 1.4383E−6 | 3.7605E−8 | 9.6070E−9 | 2.4529E−9 | |

\(q_h\) | – | – | 53.9787 | 38.2471 | 3.9143 | 3.9166 | |

III |
| 1.8257E−3 | 5.4865E−4 | 4.2731E−5 | 3.5722E−5 | 3.5605E−5 | 3.5592E−5 |

\(e_h\) | – | 1.0350E−6 | 5.2502E−9 | 1.2387E−10 | 3.6938E−11 | 5.0933E−11 | |

\(q_h\) | – | – | 197.1280 | 51.8669 | 11.4751 | 6.2403 |

The relation between convergence rate and overlapping degree (Mesh II)

\(\mu _1\) | \(\mu _2\) |
| Number of iteration and corresponding values | |||||
---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | |||

2.5 | 1.8 |
| 7.4537E−3 | 8.6547E−4 | 4.6829E−4 | 9.5781E−5 | 8.7710E−5 | 8.7058E−5 |

\(e_h\) | – | 6.0775E−7 | 4.7353E−8 | 5.3837E−9 | 6.2859E−10 | 5.6858E−10 | ||

\(q_h\) | – | – | 12.8344 | 8.7955 | 8.5647 | 1.1055 | ||

2.5 | 1.6 |
| 2.4832E−3 | 7.6489E−4 | 5.4952E−5 | 3.6848E−5 | 2.6981E−5 | 2.6773E−5 |

\(e_h\) | – | 2.9321E−7 | 1.1713E−8 | 5.8642E−10 | 2.8518E−10 | 2.1763E−10 | ||

\(q_h\) | – | – | 25.0324 | 19.9742 | 2.0563 | 1.3104 | ||

2.5 | 1.4 |
| 5.4377E−4 | 7.6811E−5 | 6.8129E−6 | 8.1056E−7 | 8.0859E−7 | 8.05378E−7 |

\(e_h\) | – | 4.2367E−7 | 6.0310E−9 | 1.0814E−10 | 1.9075E−11 | 9.2494E−12 | ||

\(q_h\) | – | – | 70.2475 | 55.76912 | 5.6694 | 2.06226 |

The data of Tables 3 and 4 show us a good convergence. And the analysis of the numbers can be similar to Example 1.

## Conclusions

In this paper, we construct a Schwarz alternating algorithm for the anisotropic problem on the unbounded domain. The algorithm uses the DDM based on FEM and natural boundary element method. The theoretical analysis shows its convergence is first-order. Further, the rate of convergence is dependent on the overlapping domain. Some numerical examples testify the theoretical conclusions. We can investigate the Schwarz alternating algorithm for anisotropic problem with three different parameters over unbounded domain. Full details and results will be given in a future publication.

## Declarations

### Authors’ contributions

All authors completed this paper together. All authors read and approved the final manuscript.

### Acknowledgements

All authors are greatly indebted to the referees as the valuable suggestions and comments.This work was subsidized by the National Natural Science Foundation of China (11371198, 11401296), Jiangsu Provincial Natural Science Foundation of China (BK20141008), Natural science fund for colleges and universities in Jiangsu Province (14KJB110007).

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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