  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.
