A block upper triangular preconditioner for double saddle point problems arising from liquid crystal modeling

To address the double saddle point problems that emerge in liquid crystal modeling, this article proposes the utilization of the preconditioned generalized minimal residual method for their resolution. Initially, a block upper triangular (BUT) splitting of the coefficient matrix is introduced to facilitate this objective. Subsequently, the BUT preconditioner, incorporating relevant parameters, is logically derived. Furthermore, the convergence of the iteration method when employing the preconditioner is delved into in great detail. To better demonstrate the preconditioner’s efficacy, the formula for calculating the parameters has also been established. Ultimately, the numerical example confirms that the robustness of the proposed block preconditioner exceeds that of the four existing block preconditioners, especially in the case of large dimensions.