Finite-element discretization of the smectic density equation

TL;DR

This paper compares three finite-element methods for solving a challenging fourth-order PDE modeling smectic A liquid crystal density variations, analyzing their stability, convergence, and computational trade-offs in different dimensions.

Contribution

It provides a detailed analysis and comparison of three finite-element formulations for a complex fourth-order PDE with a shift, highlighting their advantages and limitations.

Findings

All schemes achieve finite-element convergence.

Mixed method is effective in 2D and 3D with good preconditioning.

Interior-penalty method has lower-order convergence and preconditioning challenges.

Abstract

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.