To improve the performance of block triangular (BT) preconditioner, we develop a two-parameter BT (TPBT) preconditioner for a double saddle point problem arising from liquid crystal directors modeling. Theoretical analysis shows that all the eigenvalues of the TPBT preconditioned coefficient matrix are real and located in an interval (0, 2) no matter which value the spectral radius of matrix D− 1CA− 1CT is chosen. Moreover, an upper bound of the degree of the minimal polynomial of the TPBT preconditioned coefficient matrix is also obtained. Inasmuch as the efficiency of the TPBT preconditioner depends on the values of its parameters, we further derive a class of fast and effective formulas to compute the quasi-optimal values of the parameters involved in the TPBT preconditioner. Finally, numerical results are reported to illustrate the feasibility and the efficiency of the TPBT preconditioner.
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The authors are very much indebted to the referees for their constructive and valuable comments and suggestions which greatly improved the original manuscript of this paper.
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11471150, 11401281) and the National Key Research and Development Program of China (Grant No. 2018YFC0406600).
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Zhu, JL., Wu, YJ. & Yang, AL. A two-parameter block triangular preconditioner for double saddle point problem arising from liquid crystal directors modeling. Numer Algor (2021). https://doi.org/10.1007/s11075-021-01142-5
- Double saddle point problem
- TPBT preconditioner
- Eigenvalue analysis
- Preconditioned matrix
- Optimal parameter
Mathematics Subject Classification (2010)