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^− 1C^T 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.

为了提高块三角形 (BT) 预处理器的性能，我们开发了一种双参数 BT (TPBT) 预处理器，用于解决液晶导向器建模引起的双鞍点问题。理论分析表明，无论矩阵D ^{− 1} C A ^{− 1} C ^T的谱半径是哪个值，TPBT 预处理系数矩阵的所有特征值都是实数且位于区间 (0, 2)被选中。此外，还得到了TPBT预处理系数矩阵的最小多项式的次数的上界。由于TPBT预处理器的效率取决于其参数的值，我们进一步推导出一类快速有效的公式来计算TPBT预处理器中涉及的参数的准最优值。最后，报告了数值结果以说明 TPBT 预处理器的可行性和效率。