在工程应用的诸多研究中,如涉及流体力学问题、电磁学问题、最优化问题等相关计算时,常常需要求解一类具有特殊结构的鞍点问题[1-5]:
(1) |
其中:A∈Rm×m为非对称的正定矩阵;B∈Rm×n(m≥n)为列满秩矩阵; x, f∈ Rm; y, g∈Rn;BΤ为B的转置矩阵;f, g为已知向量.鞍点系统呈现一定病态,使得一些经典求解方法难以实现,因此如何行之有效地解决这类问题是近年来国内外学者们研究的热点.通常做法是采用定常迭代与非定常迭代双线并行,结合相应的预条件处理技术, 其中包括Uzawa类方法[6-9],SOR类方法[10-18],HSS类方法[19-21],Krylov子空间类方法[21],以及相应的预处理方法.
1 MSOR-Like方法的迭代格式和收敛性分析基于HS分裂,令A=H+S, 其中
其中:
(2) |
整理得如下迭代算法:
(3) |
将式(3)称之为求解鞍点问题的MSOR-Like方法.
若令λ为MSOR-Like方法迭代矩阵Mω的特征值,(x y)T∈Rm+n为相应特征向量,则有
(4) |
引理1 设A∈Rm×m为非对称正定矩阵,
证明 (反证法)若λ=1,由式(4)可得
引理2 在引理1条件下,设λ为迭代矩阵Mω的任一特征值,(x y)T∈Rm+n为Mω的相应特征向量,必有x≠0.若y=0,则|λ|<1.
证明 ① (反证法)若x=0,由式(4)可得By=0.由于B为列满秩矩阵,因而有y=0, 与(x y)T为迭代矩阵Mω的特征向量矛盾,故x≠0.② 若x≠0,y=0,由式(4)可得(1-ω-λ)Hx-λωSx=0.用
引理3[1] 实一元二次方程λ2-bλ+c=0两根之模均小于1,当且仅当方程系数满足|c|<1,且 |b|<1+c.
定理1 设A∈Rm×m为非对称的正定矩阵,
证明 设λ为迭代矩阵Mω的任一特征值,
整理并同时左乘
推论1 在定理1条件下,若MSOR-Like方法迭代格式收敛,则松弛参数范围可进一步化简为
证明 由于H和BQ-1BΤ是对称正定矩阵,则
考虑模型Stokes问题,有对流扩散方程:
其中:Ω为(0, 1)×(0, 1)
将上述问题用迎风差分格式离散化处理,可得形式如下的鞍点问题:
其中:
令A=H+S, 其中
取
数值实验结果表 1~表 3表明,对于不同的预优矩阵Q,MSOR-Like方法只有收敛速度的区别,没有收敛性的影响.因而可以得出,在解决非对称鞍点问题上,该方法是切实有效的.
本文以SOR-Like方法为基础,融合HS分裂思想,将经典鞍点问题的求解方法推广到特殊鞍点问题的求解,给出了一种具有新型分裂迭代格式的MSOR-Like方法,理论分析和数值算例验证了MSOR-Like方法在解决非对称鞍点问题时的有效性.
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