No abstract

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finiteelement discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed.

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 positivedefinite, 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 Finite-element discretization of the smectic density equation 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.