Patch-smoother and multigrid for the dual formulation for linear elasticity

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. Whereas efficient multigrid methods exist for solving problems in mixed plane elasticity for nearly incompressible materials, we propose a multigrid method that is also stable in the incompressible limit. There are two 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 positive semidefinite. Second, the stress belongs to (Formula presented.) and standard smoothers, working for (Formula presented.) 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 Dirichlet and Robin conditions work best and eventually lead to textbook multigrid performance.