  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
finite-element 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.
