Plasma and Fusion Research

Volume 17, 2403014 (2022)

Regular Articles

Linear-System Solver for EFG-Type Saddle-Point Problem without Using QR Decomposition
Atsushi KAMITANI, Teruou TAKAYAMA, Ayumu SAITOH and Hiroaki NAKAMURA1,2)
Yamagata University, Yamagata 992-8510, Japan
National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan
Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
(Received 19 December 2021 / Accepted 16 February 2022 / Published 18 March 2022)


A novel method is proposed for solving an EFG-type Saddle-Point (EFG-SP) problem. Although the null-space method and the variable-reduction method (VRM) were developed as a solver of a saddle-point problem, both methods are extremely time-consuming in solving an EFG-SP problem. This is attributable to the QR decomposition that is indispensable for both methods. For the purpose of resolving this problem, the improved Variable-Reduction Method (iVRM) is formulated without using the QR decomposition. A numerical code has been developed for solving an EFG-SP problem with the iVRM, the VRM and the ICCG method. By means of the code, the performance of the three methods is investigated numerically. The results of computations show that, from the standpoint of convergence property and computational speed, the iVRM is even superior to either of the VRM and the ICCG method.


computer simulation, element-free Galerkin method, Krylov space method, meshless approach, QR decomposition, saddle-point problem

DOI: 10.1585/pfr.17.2403014


  • [1] T. Belytschko, Y.Y. Lu and L. Gu, Int. J. Numer. Meth. Eng. 37, 229 (1994).
  • [2] M. Rozložník, Saddle-Point Problems and Their Iterative Solution (Birkhäuser, Switzerland, 2018) Chaps. 1 and 4.
  • [3] M. Benzi, G.H. Golub and J. Liesen, Acta Numer. 14, 1 (2005).
  • [4] A. Kamitani, T. Takayama, A. Saitoh and S. Ikuno, J. Adv. Simulat. Sci. Eng. 4, 117 (2018).
  • [5] A. Kamitani, T. Takayama and A. Saitoh, Int. J. Appl. Electromagn. Mech. 59, 157 (2019).
  • [6] G.H. Golub and C.F. Van Loan, Matrix Computations, 4th ed. (Johns Hopkins University Press, Maryland, 2013) p. 246.
  • [7] A. Kamitani, T. Takayama, T. Itoh and H. Nakamura, Plasma Fusion Res. 6, 2401074 (2011).