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Patch‐smoother and multigrid for the dual formulation for linear elasticity
Author(s) -
Rovi Gabriele,
Krause Rolf
Publication year - 2021
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6845
Subject(s) - multigrid method , linear elasticity , mathematics , elasticity (physics) , discretization , linear system , compressibility , mathematical analysis , mathematical optimization , saddle point , finite element method , geometry , partial differential equation , physics , mechanics , thermodynamics
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 toH divand standard smoothers, working forH 1regular 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.