Finite-element discretization of the smectic density equation

The density variation of smectic A liquid crystals is modelled by a fourth-order PDE, which exhibits two complications over the biharmonic or other typical $H^2$-elliptic fourth-order problems. First, the equation involves a ``Hessian-squared'' (div-div-grad-grad) operator, rather than a biharmonic (div-grad-div-grad) operator. Secondly, while positive-definite, the equation has a ``wrong-sign'' shift, making it somewhat more akin to a Helmholtz operator, with lowest-energy modes arising from certain plane waves, than an elliptic one. 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. Numerical results verify the finite-element convergence for all discretizations, and illustrate the trade-offs between the three schemes.