Robin邊界的使用情況如在表面間的傳輸非交疊處理領域分解的情況反覆的程序被 P.L .Lions介紹了，而且稍後藉著許多的作家討論。在所有的這些討論，流出的權衡和解決方案的痕跡中與反覆的步驟數字無關。最近Douglas和Huang已經考慮延長在那一個權衡已經仰賴反覆的索引而且證明類似那個為交互－方向重複發生使用假的－時間的一個周期的集中的一個加速率行走結果。組合的方法已經以奇異領域受到在橢圓形的相等的邊界價值問題的研究方面的許多注意。
這一張紙的物體是把這個加速的程序延伸到在一個扇形領域上的一個奇異問題。我們表示集中率類似那個發生因為交互－方向重複產生。所以一個組合的方法和權衡參數的方法學可能用來解決一個奇異問題。

The use of Robin boundary conditions as interfacial transmission conditions in nonoverlapping domain decomposition iterative procedures was introduced by P.~L.~Lions and later discussed by a number of authors. In all of these discussions, the weighting of the flux and the trace of the solution were independent of the iterative step number. Recently, Douglas and Huang have considered an extension in which the weighting has depended on the iterative index and proved that an acceleration in the convergence rate similar to that occurring for alternating-direction iteration using a cycle of pseudo-time steps results. Combined methods have received a lot of attention on the study of the boundary value problems of elliptic equation with singularities domain. The objects of this paper are to extend this accelerated procedure to a singularity problem on a sectorial domain. We show that the convergence rate similar to that occurring for alternating-direction iteration results. Also, a combined method and the methodology of weighting parameter can be used to solve a singularity problem.

1. Introdution 3
2. The mixed Schwarz method 4
3. The model differential problem 6
4. A model finite difference problem 8
5. Boundary Approximation Method 12
5.1Analytic solutions 12
5.2Formulate the solution of N-D model 15
6. A practical problem 19
7. Numerical Experiments 25
Reference 33

1. B. Despres. de decomposition de domains pour les problems de propagation dondes en regime harmonique. Ph.D. thesis, Universities Paris IX Dauphine, UER Mathematiques de la D'ecision 1991.
2. J. Douglas, Jr., and C.-S. Huang. An accelerated domain decomposition procedure based on Robin transmission conditions. BIT, 37:3 (1997), 678--686.
3. J. Douglas, Jr., and C.-S. Huang. Accelerated domain decomposition procedures for mixed methods based on Robin transmission conditions. Calcolo, 35, (1998)131--147.
4. J. Douglas, Jr., P. J. Paes~Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial differential
equations by mixed finite element methods, Numer. Math., 65 (1993) 95--108.
5. Z. C. Li, Numerical methods for ellepticboundary values problems with singularities, Ph.D. thsis, Dept. Math. Appl. Math., Univ. Toronto, May 1986.
6. Z. C. Li, and G. P. Liang. On the simplified hybrid-combined method, Mathematics of Computation, 41:163 (1983) 13--25.
7. P. L. Lions. On the Schwarz alternating method III: a variant for nonoverlapping subdomains, In ``Domain Decomposition Methods for Partial Differential Equations", T. F. Chan, R. Glowinski, J. Periaux, and O. B. Widlund, eds., pages 202--223. SIAM, Philadelphia, PA, 1990.
8. Z. C. Li, R. Meathon and P. Sermer, Boundary methods for solving elliptic problems with singularities and interfaces, SIAM J. Numer. Anal. 24 (1987) 487-498.
9. Z. C. Li and R. Meathon, Error and stability analysis of boundary methods for elliptic problems with interfaces, Math. of Comput. 54 (1990) 41-61.
10. Z. C. Li and T. T. Lu, Singularities and treatments of elliptic boundary value problems, Math. of Comput. 31 (2000) 97-145.