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Spectral analysis of the preconditioned system for the 3 × 3 block saddle point problem

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Abstract

In this work, we consider some preconditioning techniques for a class of 3 × 3 block saddle point problems, which arise from finite element methods for solving time-dependent Maxwell equations and some other applications. We propose an exact block diagonal preconditioner for solving the symmetric saddle point problem and its nonsymmetric form. We show that the corresponding preconditioned systems have six different eigenvalues. For the needs of practical application, we also present a class of inexact block diagonal preconditioners for solving the saddle point problems. For the symmetric system, we estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrix, respectively. For the nonsymmetric system, we derive some explicit and sharp bounds on the real and complex eigenvalues. Numerical experiments are presented to demonstrate the effectiveness and robustness of all these new preconditioners.

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Acknowledgments

The authors thank the anonymous reviewers for their valuable comments and suggestions that helped improve the quality of this paper.

Funding

The work was supported by the National Postdoctoral Program for Innovative Talents (Grant No. BX201600182), China Postdoctoral Science Foundation (Grant No. 2016M600141), National Natural Science Foundation of China (Grant No. 11071041), National Science Foundation of China (41725017), and National Basic Research Program of China under grant number 2014CB845906. It is also partially supported by the CAS/CAFEA international partnership Program for creative research teams (No. KZZD-EW-TZ-19 and KZZD-EW-TZ-15), Strategic Priority Research Program of the Chinese Academy of Sciences (XDB18010202).

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Authors and Affiliations

  1. College of Mathematics and Informatics, Fujian Key Laborotary of Mathematical Analysis and Applications, Fujian Normal University, Fuzhou, 350117, People’s Republic of China

    Na Huang & Chang-Feng Ma

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  1. Na Huang
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  2. Chang-Feng Ma
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Correspondence to Chang-Feng Ma.

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Huang, N., Ma, CF. Spectral analysis of the preconditioned system for the 3 × 3 block saddle point problem. Numer Algor 81, 421–444 (2019). https://doi.org/10.1007/s11075-018-0555-6

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