A dimension expanded preconditioning technique for block two-by-two linear equations

• Autorzy: Luo Wei-Hua , Carpentieri Bruno , Guo Jun

Identyfikatory

DOI

10.1515/dema-2023-0260

• Języki publikacji: EN

Abstrakty

EN

In this article, we introduce a novel block preconditioner for block two-by-two linear equations by expanding the dimension of the coefficient matrix. Theoretical results on the eigenvalues distribution of the preconditioned matrix are obtained, and a feasible implementation is discussed. Some numerical examples, including the solution of the Navier-Stokes equations, are presented to support the theoretical findings and demonstrate the preconditioner’s efficiency.

Słowa kluczowe

EN

block two-by-two; linear equations; preconditioner; Navier-Stokes equations

PL

równania liniowe; kondycjoner; równania Naviera-Stokesa

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2023
• Tom: Vol. 56, nr 1
• Strony: art. no. 20230260
• Opis fizyczny: Bibliogr. 33 poz., tab., wykr.

Twórcy

autor

Luo Wei-Hua

• School of Mathematics and Physics, Hunan University of Arts and Science, Changde, Hunan 415000, P. R. China

autor

Carpentieri Bruno

• Faculty of Engineering, Free University of Bozen-Bolzano, Bolzano, Italy

autor

Guo Jun

• College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 611130, P. R. China

Bibliografia

• [1] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer Science & Business Media, NewYork, 2012.
• [2] Z. Chen, Finite Element Methods and Their Applications, Springer, Berlin, 2005.
• [3] G. Bao and W. Sun, A fast algorithm for the electromagnetic scattering from a large cavity, SIAM J. Sci. Comput. 27 (2005), 553–574.
• [4] W. Van Dijk and F. M. Toyama, Accurate numerical solutions of the time-dependent Schrödinger equation, Phys. Rev. E. 75 (2007), 036707, 1–10.
• [5] J.-H. Zhang and H. Dai, A new block preconditioner for complex symmetric indefinite linear systems, Numer. Algorithms 74 (2017), 889–903.
• [6] Y. Cao and Z.-R. Ren, Two variants of the PMHSS iteration method for a class of complex symmetric indefinite linear systems, Appl. Math. Comput. 264 (2015), 61–71.
• [7] K. Chen, Matrix Preconditioning Techniques and Applications, Cambridge University Press, Cambridge, 2005.
• [8] T. A. Davis and Y. Hu, The University of Florida sparse matrix collection, ACM Trans. Math. Software 38 (2011), no. 1, 1–25.
• [9] M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix. Anal. Appl. 26 (2004), no. 1, 20–41.
• [10] Z.-Z. Bai, G. H. Golub, and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix. Anal. Appl. 24 (2003), no. 3, 603–626.
• [11] Y. Cao, Z.-R. Ren, and Q. Shi, A simplified HSS preconditioner for generalized saddle point problems, BIT Numer. Math. 56 (2016), 423–439.
• [12] Y. Cao, L.-Q. Yao, and M.-Q. Jiang, A modified dimensional split preconditioner for generalized saddle point problems, J. Comput. Appl. Math. 250 (2013), 70–82.
• [13] Y. Cao, S.-X. Miao, and Y.-S. Cui, A relaxed splitting preconditioner for generalized saddle point problems, Comput. Appl. Math. 34 (2015), 865–879.
• [14] M. Benzi and X.-P. Guo, A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations, Appl. Numer. Math. 61 (2011), no. 1, 66–76.
• [15] M. Benzi, M. Ng, N. Qiang, and W. Zhen, A relaxed dimensional factorization preconditioner for the incompressible Navier-Stokes equations, J. Comput. Phys. 230 (2011), 6185–6202.
• [16] J.-L. Zhang, An efficient variant of HSS preconditioner for generalized saddle point problems, Numer. Linear Algebra Appl. 25 (2018), no. 4, 1–14.
• [17] Z.-Z. Bai, M. K. Ng, and Z.-Q. Wang, Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix. Anal. Appl. 31 (2009), no. 2, 410–433.
• [18] G.-F. Zhang, Z.-R. Ren, and Y.-Y. Zhou, On HSS-based constraint preconditioners for generalized saddle point problems, Numer. Algorithms, 57 (2011), 273–287.
• [19] Y. Cao, M.-Q. Jiang, and Y.-L. Zheng, A splitting preconditioner for saddle point problems, Numer. Linear Algebra Appl. 18 (2011), no. 5, 875–895.
• [20] 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), no. 5, A2171–A2197.
• [21] J.-H. Zhang and H. Dai, A new splitting preconditioner for the iterative solution of complex symmetric indefinite linear systems, Appl. Math. Lett. 49 (2015), 100–106.
• [22] Q.-Q. Shen and Q. Shi, A variant of the HSS preconditioner for complex symmetric indefinite linear systems, Comput. Math. Appl. 75 (2018), no. 3, 850–863.
• [23] M. Frigo, N. Castelletto, and M. Ferronato, A relaxed physical factorization preconditioner for mixed finite element coupled poromechanics, SIAM J. Sci. Comput. 41 (2019), no. 4, B694–B720.
• [24] M. J. Gander, Q. Niu, and Y. Xu, Analysis of a new dimension-wise splitting iteration with selective relaxation for saddle point problems, BIT Numer. Math. 56 (2016), no. 2, 441–465.
• [25] Y. Cao, J.-L. Dong, and Y.-M. Wang, A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier-Stokes equation, J. Comput. Appl. Math. 273 (2015), 41–60.
• [26] Z.-Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput. 75 (2006), 791–815.
• [27] W. Chao, T.-Z. Huang, and C. Wen, A new preconditioner for indefinite and asymmetric matrices, Appl. Math. Comput. 219 (2013), 11036–11043.
• [28] H. Chen, X. Li, and Y. Wang, A splitting preconditioner for a block two-by-two linear system with applications to the bidomain equations, J. Comput. Appl. Math. 321 (2017), 487–498.
• [29] Q. Zheng and L. Lu, A shift-splitting preconditioner for a class of block two-by-two linear systems, Appl. Math. Lett. 66 (2016), no. 3, 54–60.
• [30] M. Masoudi and D. K. Salkuyeh, An extension of the positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems, Comput. Math. Appl. 79 (2020), no. 8, 2304–2321.
• [31] Y. Cao, A block positive-semidefinite splitting preconditioner for generalized saddle point linear systems, J. Comput. Appl. Math. 374 (2020), 112787.
• [32] W.-H. Luo, X.-M. Gu, and B. Carpentieri, A dimension expanded preconditioning technique for saddle point problems, BIT Numer. Math. 62 (2022), no. 4, 1983–2004.
• [33] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, PA, 2003.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).