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 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.