The fourth-order PDE that models the density variation of smectic A liquid crystals presents unique challenges in its (numerical) analysis beyond more common fourth-order operators, such as the classical biharmonic. While the operator is positive definite, the equation has a "wrong-sign" shift, making it somewhat more akin to an indefinite Helmholtz operator, with lowest-energy modes consisting of plane waves. As a result, for large shifts, the natural continuity, coercivity, and inf-sup constants degrade considerably, impacting standard error estimates. In this paper, we analyze and compare three finite-element formulations for such PDEs, based on $H^2$-conforming elements, the $C^0$ interior penalty method, and a mixed finite-element formulation that explicitly introduces approximations to the gradient of the solution and a Lagrange multiplier. The conforming method is simple but is impractical to apply in three dimensions; the interior-penalty method works well in two and three dimensions but has lower-order convergence and (in preliminary experiments) seems difficult to precondition; the mixed method uses more degrees of freedom, but works well in both two and three dimensions, and is amenable to monolithic multigrid preconditioning. Our analysis reveals different behaviours of the error bounds with the shift parameter and mesh size for the different schemes. Numerical results verify the finite-element convergence for all discretizations, and illustrate the trade-offs between the three schemes.

翻译：描述近晶A型液晶密度变化的四阶偏微分方程，在数值分析中带来了不同于经典双调和算子等常见四阶算子的独特挑战。尽管该算子是正定的，但方程中存在"符号相反"的偏移项，使其更类似于不定亥姆霍兹算子，最低能量模式由平面波构成。因此，当偏移较大时，自然的连续性、强制性和inf-sup常数显著退化，影响了标准误差估计。本文分析并比较了三种针对此类偏微分方程的有限元公式：基于$H^2$协调元、$C^0$内罚方法和一种显式引入解的梯度近似和拉格朗日乘子的混合有限元公式。协调方法简单但难以在三维中应用；内罚方法在二维和三维中表现良好，但收敛阶较低且（初步实验中）难以预处理；混合方法使用了更多自由度，但在二维和三维中均表现良好，且适用于整体多重网格预处理。我们的分析揭示了不同格式下误差界随偏移参数和网格尺寸变化的差异性。数值结果验证了所有离散化方法的有限元收敛性，并展示了三种方案之间的权衡。