曾闽丽^1,2, 张国凤³

1. 兰州大学数学与统计学院, 兰州 730000;
2. 莆田学院数学学院, 福建莆田 351100;
3. 兰州大学数学与统计学院, 兰州 730000

收稿日期:2015-07-26出版日期:2016-12-15发布日期:2016-10-13


基金资助:国家自然科学基金（11271174，11471175，11511130015）；福建省自然科学基金（2016J05016）；福建省教育厅基金（JAT160450）和莆田学院校内科研基金（2015061，2016021，2016075）资助项目.

A NEW SPLITTING ITERATIVE TECHNIQUE FOR THE SADDLE POINT SYSTEM FROM A VELOCITY TRACKING PROBLEM

Zeng Minli^1,2, Zhang Guofeng³

1. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China;
2. School of Mathematics, Putian University, Putian 351100, Fujian, China;
3. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received:2015-07-26Online:2016-12-15Published:2016-10-13







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有限元离散一类速度追踪问题后得到具有鞍点结构的线性系统，针对该鞍点系统，本文提出了一种新的分裂迭代技术.证明了新的分裂迭代方法的无条件收敛性，详细分析了新的分裂预条件子对应的预处理矩阵的谱性质.数值结果验证了对于大范围的网格参数和正则参数，新的分裂预条件子在求解有限元离散速度追踪问题得到的鞍点系统时的可行性和有效性.
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[1] Bai Z Z, Li G Q. Restrictively preconditioned conjugate gradient methods for systems of linear equations[J]. IMA J. Numer. Anal., 2003, 23(4):561-580. [2] Bai Z Z, Golub G H, Ng M K. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems[J]. SIAM J. Matrix Anal. Appl., 2003, 24(3):603-626. [3] Bai Z Z, Parlett B N, Wang Z Q. On generalized successive overrelaxation methods for augmented linear systems[J]. Numer. Math. 2005, 102(1):1-38. [4] Bai Z Z, Wang Z Q. On parameterized inexact Uzawa methods for generalized saddle point problems[J]. Linear Algebra Appl., 2008, 428(11):2900-2932. [5] Benzi M, Guo X P. A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations[J]. Appl. Numer. Math., 2011, 61(1):66-76. [6] Benzi M, Ng M K, Niu Q. A relaxed dimensional factorization preconditioner for the incompressible Navier-Stokes equations[J]. J. Comput. Phys., 2011, 230(16):6185-6202. [7] Benzi M, Golub G H, Liesen J. Numerical solution of saddle point problems[J]. Acta Numer., 2005, 14(1):1-137. [8] Bramble J H, Pascisk J E, Vassilev A T. Analysis of the inexact Uzawa algorithm for saddle point problems[J]. SIAM J. Numer. Anal., 1997, 34(3):1072-1092. [9] Cao Y, Yao L Q and Jiang M Q. A relaxed HSS preconditioner for saddle point problems from meshfree discretization[J]. J. Comput. Math., 2013, 31:398-421. [10] Cao Y, Miao S X, Cui Y S. A relaxed splitting preconditioner for generalized saddle point problems[J]. Comput. Appl. Math., 2014:1-15. [11] Cao Y, Du J, Niu Q. Shift-splitting preconditioners for saddle point problems[J]. J. Comput. Appl. Math., 2014, 272:239-250. [12] Cao Y, Yao L Q, Jiang M Q. A modified dimensional split preconditioner for generalized saddle point problems[J]. J. Comput. Appl. Math., 2013, 250:70-82. [13] Chen F, Jiang Y L. A generalization of the inexact parameterized Uzawa methods for saddle point problems[J]. Appl. Math. Comput., 2008, 206:765-771. [14] Chen C R, Ma C F. A generalized shift-splitting preconditioner for saddle point problems[J]. Appl. Math. Letters, 2015, 43:49-55. [15] Elman H C, Ramage A, Silvester D J. Algorithm 866:IFISS, a Matlab toolbox for modelling incompressible flow[J]. ACM Tran. Math. Soft., 2007, 33(2):14. [16] Hinze M, Pinnau R, Ulbrich M, Ulbrich S. Optimization with PDE Constraints[M]. Springer, New York, 2009. [17] Horn R A, Johnson C R. Matrix Analysis, Second edition[M]. Cambridge University Press, Cambridge, 2012. [18] Kellogg R B. Another alternating-direction-implicit method[J]. J. Korean Soc. Ind. Appl. Math., 1963, 11:976-979. [19] Kollmann M, Zulehner W. A robust preconditioner for distributed optimal control for Stokes flow with control constraints[J]. Springer Berlin Heidelberg, 2013:771-779. [20] Pearson J W. On the role of commutator arguments in the development of parameter-robust preconditioners for Stokes control problems[J]. Technical Report., 2013. [21] Rees T, Wathen A J. Preconditioning iterative methods for the optimal control of the Stokes equations[J]. SIAM J. Sci. Comput., 2011, 33(5):2903-2926. [22] Nocedal J, Wright S. Numerical Optimization[M]. Springer, New York, 1999 [23] Pan J Y, Ng M K, Bai Z Z. New preconditioners for saddle point problems[J]. Appl. Math. Comput., 2006, 172(2):762-771. [24] Saad Y, Schultz M H. GMRES:a generalized minimal residual algorithm for solving nonsymmetric linear systems[J]. SIAM J. Statist. Sci. Comput. 1986, 7(3):856-869. [25] Saad Y. Iterative Methods for Sparse Linear Systems (2nd edn)[M]. SIAM, Philadelphia, PA, 2003. [26] Simoncini V, Benzi M. Spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for saddle point problems[J]. SIAM J. Matrix Anal. Appl., 2004, 26(2):377-389. [27] Stoll M, Wathen A. All-at-once solution of time-dependent Stokes control[J]. J. Comput. Phys., 2013, 232(1):498-515. [28] Tan N B, Huang T Z, Hu Z J. A Relaxed Splitting Preconditioner for the Incompressible Navier-Stokes Equations[J]. J. Appl. Math., 2012, 2012:1-12. [29] Zhang J L, Gu C Q, Zhang K. A relaxed positive-definite and skew-Hermitian splitting preconditioner for saddle point problems[J]. Appl. Math. Comput., 2014, 249:468-479.

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