  The dual formulation for linear elasticity, in contrast to the primal
formulation, is not affected by locking, as it is based on the stresses as main
unknowns. Thus it is quite attractive for nearly incompressible and
incompressible materials. Discretization with mixed finite elements will lead
to -- possibly large -- linear saddle point systems with a particular
structure. Whereas efficient multigrid methods exist for solving problems in
mixed plane elasticity, to the knowledge of the authors, no multigrid methods
are readily available for the general dual formulation. Two are the main
challenges in constructing a multigrid method for the dual formulation for
linear elasticity. First, in the incompressible limit, the matrix block related
to the stress is semi-positive definite. Second, the stress belongs to
$\textbf{H}_{\text{div}}$ and standard smoothers, working for $\textbf{H}^1$
regular problems, cannot be applied. We present a novel patch-based smoother
for the dual formulation for linear elasticity. We discuss different types of
local boundary conditions for the patch subproblems. Based on our
patch-smoother, we build a multigrid method for the solution of the resulting
saddle point problem and investigate its efficiency and robustness. Numerical
experiments show that Robin conditions best fit the multigrid framework,
leading eventually to multigrid performance.
