A block preconditioner with two parameters for double saddle point systems in liquid crystal director modeling

Technical Summary: A Block Preconditioner with Two Parameters for Double Saddle Point Systems in Liquid Crystal Director Modeling

Problem Statement
The paper addresses the numerical solution of double saddle point systems arising from the finite element discretization of liquid crystal director modeling. The physical problem involves minimizing a free energy functional subject to constraints, which, when solved via Lagrange multipliers and a Newton-type scheme, yields a linear system of the form $\tilde{A}u = d$. The coefficient matrix $\tilde{A}$ is a $3 \times 3$ block matrix with specific structural properties: the diagonal blocks $A$ and $D$ are symmetric positive definite (SPD), $B$ has full row rank, and $C$ is rank-deficient. These systems are notoriously difficult to solve due to their indefinite nature and large scale, necessitating efficient iterative solvers and robust preconditioning strategies.

Methodology
The authors propose a novel block preconditioner, termed BPT (Block Preconditioner with Two parameters), designed specifically for this class of double saddle point systems. The methodology is structured as follows:

1. Preconditioner Formulation: The method is based on a specific splitting of the coefficient matrix $\tilde{A} = Q_{BPT} - L_{BPT}$, where $Q_{BPT}$ serves as the preconditioner. This matrix incorporates two tunable positive parameters, $\alpha$ and $\beta$, and is structured to allow for efficient inversion.
2. Convergence Analysis: The paper establishes sufficient conditions for the convergence of the underlying stationary iterative scheme defined by the splitting. Using spectral analysis of the iteration matrix, the authors derive a condition involving the parameters $\alpha, \beta$ and the spectral properties of the sub-matrices ($A, B, C, D$). They demonstrate that under the assumption $A \succ C^T D^{-1} C$ (which is verified numerically for the problem class), the method converges unconditionally for any $\alpha, \beta > 0$.
3. Spectral Analysis: A detailed spectral analysis of the preconditioned matrix $Q_{BPT}^{-1}\tilde{A}$ is performed. The authors prove that the preconditioned matrix possesses an eigenvalue of 1 with algebraic multiplicity at least $p$ (where $p$ is the dimension of the $D$ block). The remaining eigenvalues are shown to be real and positive if the parameters satisfy $\frac{\alpha}{\beta} \leq \frac{1}{4}\lambda_{\min}^2(A)$. Crucially, the spectrum is shown to be clustered near unity, a property that accelerates the convergence of Krylov subspace methods.
4. Parameter Selection Strategy: To avoid manual tuning, the paper derives algebraic estimates for $\alpha$ and $\beta$ by minimizing the trace of the difference between the preconditioned matrix and the identity. These estimates, denoted $\alpha_{est}$ and $\beta_{est}$, rely on the traces of $C^T \hat{D}^{-1} C$ and $B D_A^{-1} B^T$, where $\hat{D}$ and $D_A$ are diagonal approximations of $D$ and $A$.
5. Implementation: The preconditioner is integrated into the Flexible Generalized Minimal Residual (FGMRES) method. This choice is necessary because the linear subsystems within the preconditioner application (involving $A + (\alpha + 1/\beta)B^T B$ and $D$) are solved inexactly using the Conjugate Gradient (CG) method.

Key Contributions

• Novel Preconditioner: Introduction of a two-parameter block preconditioner tailored for liquid crystal director models, extending existing block triangular approaches.
• Theoretical Guarantees: Rigorous proof of convergence for the iterative scheme and characterization of the eigenvalue distribution of the preconditioned matrix, showing clustering near 1 under specific parameter constraints.
• Parameter Estimation: Development of a practical, data-driven strategy for selecting the parameters $\alpha$ and $\beta$ without requiring prior knowledge of the exact eigenvalues of the system matrices.
• Robustness: Demonstration that the method maintains high performance and solution accuracy even as the problem size increases and mesh resolution is refined.

Experimental Results
The authors conducted numerical experiments on "off-state" and "on-state" liquid crystal problems using mesh sizes ranging from approximately 20,000 to over 2.6 million degrees of freedom. The proposed BPT preconditioner was compared against two state-of-the-art alternatives: the Two-Parameter Block Triangular (TPBT) preconditioner and the Block Upper Triangular (BUT) preconditioner.

• Iteration Counts: The BPT preconditioner consistently required fewer iterations (often just 2) compared to TPBT and BUT (which required 3–5 iterations) across all tested sizes.
• Scalability: Unlike the competing methods, where iteration counts and solution errors (ERR) tended to degrade as the mesh was refined, the BPT method maintained constant iteration counts and high solution fidelity.
• Computational Efficiency: Due to the reduced iteration counts, the BPT method achieved lower CPU times across the board.
• Eigenvalue Distribution: Spectral plots confirmed that the BPT preconditioner clustered eigenvalues tightly around 1, whereas other methods showed broader distributions.

Significance and Claims
The paper claims that the proposed BPT preconditioner offers a superior balance of computational efficiency and robustness for double saddle point systems in liquid crystal modeling. The authors emphasize that their method not only accelerates convergence but also preserves solution accuracy at large scales where other methods fail. By providing a theoretically grounded strategy for parameter selection and demonstrating unconditional convergence under realistic assumptions, the work offers a reliable tool for solving the complex linear systems inherent in liquid crystal director simulations. The results suggest that the method is particularly effective for large-scale problems where maintaining precision and minimizing iteration counts are critical.