We consider multigrid methods for stabilized Q1/P0, P2/P0 and Q2/P1 elements for elasticity, where the hydrostatic pressure is modeled by discontinuous lower order approximations. In the implementation the pressure can be eliminated by static condensation. Then, the resulting system is symmetric and positive definite, and standard multigrid methods can be applied. In addition, we introduce a special construction for the smoother to obtain robustness of the multigrid convergence in the incompressible limit.
The stabilized elements are applied to Prandtl-Reuß-plasticity. The basis for the plasticity computations builds a flexible finite element library which supports adaptive multigrid methods for various discretizations. This is coupled by an abstract interface for the material evaluation at every Gauß-point. The equation in time is discretized by the implicit Euler method, and every time step is solved with a modified Newton method where the defect and the tangent operator is evaluated by a radial return algorithm. We show that the approximation and the multigrid convergence are improved by the stabilization.
The algorithm is realized with the software package UG, which is fully supported in parallel. The performance is demonstrated by several examples in 2D with assumed plain strain and in 3D. We investigate the resulting displacements, stresses and hardening parameters after a complete loading cycle, comparing different material laws (perfect plasticity, isotropic linear and exponential hardening) and comparing a full 3D geometry with a reduced 2D geometry.