Modified restrictive preconditioners for double saddle point problems arising from liquid crystal director modeling

We further investigate the double saddle point problem arising from liquid crystal directors modeling. First, we transform the original linear system into a symmetric positive definite linear system by decomposing the double saddle point matrix. Then, we use the conjugate gradient method to solve this equivalent linear system. By utilizing the characteristics of the submatrices, we construct two modified restrictive (MR) preconditioners, and propose an economic restrictively preconditioned conjugate gradient (RPCG) method. Furthermore, we discuss the spectral properties of the corresponding preconditioned matrices and draw a conclusion that their spectral distributions are within a subset of the interval. Finally, we demonstrate the validity of the MR preconditioners with numerical examples from the liquid crystal director models. The results show that the MR preconditioners are more effective than some preconditioners, such as restrictive preconditioners and other classical preconditioners for liquid crystal directors modeling.