JCM
Journal of Computational Mathematics ›› 2021, Vol. 39 ›› Issue (5): 733-754.DOI: 10.4208/jcm.2006-m2020-0037

Previous Articles     Next Articles

MODIFIED ALTERNATING POSITIVE SEMIDEFINITE SPLITTING PRECONDITIONER FOR TIME-HARMONIC EDDY CURRENT MODELS

Yifen Ke, Changfeng Ma   

  1. College of Mathematics and Informatics&FJKLMAA, Fujian Normal University, Fuzhou 350117, China
  • Received:2020-02-15 Revised:2020-05-29 Online:2021-09-15 Published:2021-10-15
  • Supported by:
    The authors would like to thank the referees for the comments and constructive suggestions which are valuable in improving the quality of the paper. This research is supported by the National Key Research and Development Program of China (Nos. 2019YFC0312003 and 2018YFC1504200) and the National Natural Science Foundation of China (Nos. 11901098 and U1839207).
PDF ( 30 ) Cited

Yifen Ke, Changfeng Ma. MODIFIED ALTERNATING POSITIVE SEMIDEFINITE SPLITTING PRECONDITIONER FOR TIME-HARMONIC EDDY CURRENT MODELS[J]. Journal of Computational Mathematics, 2021, 39(5): 733-754.

In this paper, we consider a modified alternating positive semidefinite splitting preconditioner for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current model. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied for both simple and general topology. Numerical results demonstrate the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.
Key words: Time-harmonic eddy current model, Saddle point problem, Eigenvalue distribution, Preconditioner

CLC Number: 

[1] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci., 21(1998), 823-864.
[2] M. Arioli and G. Manzini. A null space algorithm for mixed finite-element approximations of Darcy's equation. Commun. Numer. Meth. Engng., 18(2002), 645-657.
[3] K. Arrow, L. Hurwicz, and H. Uzawa. Studies in Linear and Nonlinear Programming. Stanford University Press, Stanford, 1958.
[4] Z.Z. Bai. Optimal parameters in the HSS-like methods for saddle-point problems. Numer. Linear Algebra Appl., 16(2009), 447-479.
[5] Z.Z. Bai. Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models. Numer. Linear Algebra Appl., 19(2012), 914-936.
[6] Z.Z. Bai and G.H. Golub. Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal., 27(2007), 1-23.
[7] Z.Z. Bai, G.H. Golub, and M.K. Ng. Hermitian and skew-Hermitian splitting methods for nonHermitian positive definite linear systems. SIAM J. Matrix Anal. Appl., 24(2003), 603-626.
[8] Z.Z. Bai, G.H. Golub, and J.Y. Pan. Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math., 98(2004), 1-32.
[9] Z.Z. Bai and M.K. Ng. On inexact preconditioners for nonsymmetric matrices. SIAM J. Sci. Comput., 26(2005), 1710-1724.
[10] Z.Z. Bai and Z.Q. Wang. On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl., 428(2008), 2900-2932.
[11] M. Benzi. Solution of equality-constrained quadratic programming problems by a projection iterative method. Rend. Mat. Appl., 13(1993), 275-296.
[12] M. Benzi, G.H. Golub, and J. Liesen. Numerical solution of saddle point problems. Acta Numer., 14(2005), 1-137.
[13] J.H. Bramble, J.E. Pasciak, and A.T. Vassilev. Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal., 34(1997), 1072-1092.
[14] J.H. Bramble, J.E. Pasciak, and A.T. Vassilev. Uzawa type algorithms for nonsymmetric saddle point problems. Math. Comp., 69(2000), 667-689.
[15] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15, Springer-Verlag, New York, 1991.
[16] A. Bossavit. On the condition ‘h normal to the wall’ in magnetic field problems. Internat. J. Numer. Methods Engrg., 24(1987), 1541-1550.
[17] Z.H. Cao. Fast Uzawa algorithms for solving non-symmetric stabilized saddle point problems. Numer. Linear Algebra Appl., 11(2004), 1-24.
[18] Y. Cao, S. Li, and L.Q. Yao. A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems. Appl. Math. Lett., 49(2015), 20-27.
[19] Y. Cao, L.Q. Yao, M.Q. Jiang, and Q. Niu. A relaxed HSS preconditioner for saddle point problems from meshfree discretization. J. Comput. Math., 31(2013), 398-421.
[20] Y. Cao and S.C. Yi. A class of Uzawa-PSS iteration methods for nonsingular and singular nonHermitian saddle point problems. Appl. Math. Comput., 275(2016), 41-49.
[21] F. Chen and Y.L. Jiang. A generalization of the inexact parameterized Uzawa methods for saddle point problems. Appl. Math. Comput., 206(2008), 765-771.
[22] C.R. Chen and C.F. Ma. A generalized shift-splitting preconditioner for singular saddle point problems. Appl. Math. Comput., 269(2015), 947-955.
[23] X.L. Cheng. The inexact Uzawa algorithm for saddle point problem. Appl. Math. Lett., 13(2000), 1-3.
[24] X.L. Cheng. On the nonlinear inexact Uzawa algorithm for saddle-point problems. SIAM J. Numer. Anal., 37(2000), 1930-1934.
[25] X.L. Cheng and J. Zou. An inexact Uzawa-type iterative method for solving saddle point problems. Int. J. Comput. Math., 80(2003), 55-64.
[26] M.R. Cui. A sufficient condition for the convergence of the inexact Uzawa algorithm for saddle point problems. J. Comput. Appl. Math., 139(2002), 189-196.
[27] M.R. Cui. Analysis of iterative algorithms of Uzawa type for saddle point problems. Appl. Numer. Math., 50(2004), 133-146.
[28] H.C. Elman and G.H. Golub. Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal., 31(1994), 1645-1661.
[29] N. Gould, D. Orban, and T. Rees. Projected Krylov methods for saddle-point systems. SIAM J. Matrix Anal. Appl., 35(2014), 1329-1343.
[30] Q.Y. Hu and J. Zou. Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems. Numer. Math., 93(2002), 333-359.
[31] N. Huang, C.F. Chang, and Y.J. Xie. An inexact relaxed DPSS preconditioner for saddle point problem. Appl. Math. Comput., 265(2015), 431-447.
[32] Y.F. Ke and C.F. Ma. Spectrum analysis of a more general augmentation block preconditioner for generalized saddle point matrices. BIT, 56(2016), 489-500.
[33] Y.F. Ke and C.F. Ma. An inexact modified relaxed splitting preconditioner for the generalized saddle point problems from the incompressible Navier-Stokes equations. Numer. Algor., 75(2017), 1103-1121.
[34] Y.F. Ke and C.F. Ma. The dimensional splitting iteration methods for solving saddle point problems arising from time-harmonic eddy current models. Appl. Math. Comput., 303(2017), 146-164.
[35] Y.F. Ke and C.F Ma. A low-order block preconditioner for saddle point linear systems. Comput. Appl. Math., 37(2018), 1959-1970.
[36] Y.F. Ke amd C.F. Ma, A new relaxed splitting preconditioner for the generalized saddle point problems from the incompressible Navier-Stokes equations. Comput. Appl. Math., 37(2018), 515-524.
[37] I.N. Konshin, M.A. Olshanskii, and Y.V. Vassilevski. ILU preconditioners for nonsymmetric saddle-point matrices with application to the incompressible Navier-Stokes equations. SIAM J. Sci. Comput., 37(2015), A2171-A2197.
[38] P.R. Kotiuga. Topological considerations in coupling magnetic scalar potentials to stream functions describing surface currents. IEEE T. Magn., 25(1989), 2925-2927.
[39] C.X. Li, S.L. Wu, and A.Q. Huang. A relaxed splitting preconditioner for saddle point problems. J. Numer. Math., 23(2015), 361-368.
[40] Q.B. Liu, G.L. Chen, and C.Q. Song. Preconditioned HSS-like iterative method for saddle point problems. J. Comput. Math., 32(2014), 442-455.
[41] Z.R. Ren and Y. Cao. An alternating positive-semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models. IMA J. Numer. Anal., 36(2016), 922-946.
[42] A.A. Rodríguez and R. Hernández. Iterative methods for the saddle-point problem arising from the HC/EI formulation of the eddy current problem. SIAM J. Sci. Comput., 31(2009), 3155-3178.
[43] A.A. Rodríguez, R. Hiptmair, and A. Valli. A hybrid formulation of eddy current problems. Numer. Meth. Part. D. E., 21(2005), 742-763.
[44] A.A. Rodríguez and A. Valli. Eddy Current Approximation of Maxwell Equations:Theory, Algorithms and Applications. Springer, Milan, 2010.
[45] H.A. Van der Vorst. Iterative Krylov Methods for Large Linear Systems. Cambridge University Press, Cambridge, 2003.
[46] Z.Q. Wang. Optimization of the parameterized Uzawa preconditioners for saddle point matrices. J. Comput. Appl. Math., 226(2009), 136-154.
[47] A.L. Yang and Y.J. Wu. The Uzawa-HSS method for saddle-point problems. Appl. Math. Lett., 38(2014), 38-42.
[48] J.L. Zhang, C.Q. Gu, and K. Zhang. A relaxed positive-definite and skew-Hermitian splitting preconditioner for saddle point problems. Appl. Math. Comput., 249(2014), 468-479.
[49] G.F. Zhang, Z.R. Ren, and Y.Y. Zhou. On HSS-based constraint preconditioners for generalized saddle-point problems. Numer. Algor., 57(2011), 273-287.
[50] M.Z. Zhu, G.F. Zhang, Z. Zheng, and Z.Z. Liang. On HSS-based sequential two-stage method for non-Hermitian saddle point problems. Appl. Math. Comput., 242(2014), 907-916.
[1] Mei Yang, Ren-Cang Li. HEAVY BALL FLEXIBLE GMRES METHOD FOR NONSYMMETRIC LINEAR SYSTEMS [J]. Journal of Computational Mathematics, 2022, 40(5): 711-727.
[2] Lu Zhang, Qifeng Zhang, Hai-wei Sun. A FAST COMPACT DIFFERENCE METHOD FOR TWO-DIMENSIONAL NONLINEAR SPACE-FRACTIONAL COMPLEX GINZBURG-LANDAU EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(5): 708-732.
[3] Daniele Boffi, Zhongjie Lu, Luca F. Pavarino. ITERATIVE ILU PRECONDITIONERS FOR LINEAR SYSTEMS AND EIGENPROBLEMS [J]. Journal of Computational Mathematics, 2021, 39(4): 633-654.
[4] Kaibo Hu, Ragnar Winther. WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS [J]. Journal of Computational Mathematics, 2021, 39(3): 333-357.
[5] Maohua Ran, Chengjian Zhang. A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS [J]. Journal of Computational Mathematics, 2020, 38(2): 239-253.
[6] Davod Hezari, Vahid Edalatpour, Hadi Feyzollahzadeh, Davod Khojasteh Salkuyeh. ON THE GENERALIZED DETERIORATED POSITIVE SEMI-DEFINITE AND SKEW-HERMITIAN SPLITTING PRECONDITIONER [J]. Journal of Computational Mathematics, 2019, 37(1): 18-32.
[7] Yang Cao, Zhiru Ren, Linquan Yao. IMPROVED RELAXED POSITIVE-DEFINITE AND SKEW-HERMITIAN SPLITTING PRECONDITIONERS FOR SADDLE POINT PROBLEMS [J]. Journal of Computational Mathematics, 2019, 37(1): 95-111.
[8] Xinhui Shao, Chen Li, Tie Zhang, Changjun Li. A MODIFIED PRECONDITIONER FOR PARAMETERIZED INEXACT UZAWA METHOD FOR INDEFINITE SADDLE POINT PROBLEMS [J]. Journal of Computational Mathematics, 2018, 36(4): 579-590.
[9] Davide Forti, Alfio Quarteroni, Simone Deparis. A PARALLEL ALGORITHM FOR THE SOLUTION OF LARGE-SCALE NONCONFORMING FLUID-STRUCTURE INTERACTION PROBLEMS IN HEMODYNAMICS [J]. Journal of Computational Mathematics, 2017, 35(3): 363-380.
[10] Xin He, Maya Neytcheva, Cornelis Vuik. ON PRECONDITIONING OF INCOMPRESSIBLE NON-NEWTONIAN FLOW PROBLEMS [J]. Journal of Computational Mathematics, 2015, 33(1): 33-58.
[11] Qingbing Liu, Guoliang Chen, Caiqin Song. PRECONDITIONED HSS-LIKE ITERATIVE METHOD FOR SADDLE POINT PROBLEMS [J]. Journal of Computational Mathematics, 2014, 32(4): 442-455.
[12] Xiaoying Zhang, Yumei Huang. ON BLOCK PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS [J]. Journal of Computational Mathematics, 2014, 32(3): 272-283.
[13] Minli Zeng, Guofeng Zhang. A NEW PRECONDITIONING STRATEGY FOR SOLVING A CLASS OF TIME-DEPENDENT PDE-CONSTRAINED OPTIMIZATION PROBLEMS [J]. Journal of Computational Mathematics, 2014, 32(3): 215-232.
[14] Hongtao Fan, Bing Zheng. THE GENERALIZED LOCAL HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR THE NON-HERMITIAN GENERALIZED SADDLE POINT PROBLEMS [J]. Journal of Computational Mathematics, 2014, 32(3): 312-331.
[15] Qiang Niu, Michael Ng. NUMERICAL STUDIES OF A CLASS OF COMPOSITE PRECONDITIONERS [J]. Journal of Computational Mathematics, 2014, 32(2): 136-151.
Viewed
Full text


Abstract

Journal of Computational Mathematics
Contact Us
Editor-in-Chief: Zhiming Chen
Electronic: 1991-7139
ISSN Print: 0254-9409
Issues: 6 / year
Institute of Computational Mathematics and Scientific/ Engineering Computing
Chinese Academy of Sciences
Zhongguancun East Rd. No.55
Beijing 100190, China
Phone: 86-10-82541418
Email: jcm@lsec.cc.ac.cn
×

扫码分享

京公网安备11010802040203号      京ICP备05002806号-10　　The journal is sponsored by ICMSEC　　Copyrights © ICMSEC