Two-level preconditioning of pure displacement non-conforming FEM systems (Q2760370)

The authors study numerically almost incompressible materials by using a plane linear elasticity problem with Dirichlet boundary conditions. To avoid the locking effect appearing if low-order conforming finite elements (FE) are used and Poisson ratio \(\nu \) tends to 0.5, the authors approximate the problem by means of Crouzeix-Raviart linear non-conforming FE. As the condition number of the resulting stiffness matrix goes to infinity if \(\nu \to 0.5\), the goal is to construct a locking-free preconditioner. NEWLINENEWLINENEWLINEThe proposed construction is based on a two-level basis of FE space. As finite element spaces on successive levels of mesh refinements are not nested for the Crouzeix-Raviart non-conforming FE, each coarse triangle is refined into four similar triangles, and forms a macroelement the interior nodal unknowns of which can be eliminated. It gives rise to a Schur complement written in block form. By taking the diagonal part of a block and introducing a scaling parameter \(\omega \), the authors define the preconditioner. Local spectral analysis is applied to determine \(\omega \) and, consequently, some other parameters, as e.g. the constant in the strengthened CSB inequality. The derived estimates for the condition number of the preconditioned stiffness matrix are uniform with respect to \(\nu \in (0,0.5)\) and to the mesh-size parameter. Numerical tests illustrate the theory and convergence rate of the method. Uniform meshes and simple convex polygonal domains are assumed in the theory and calculations, and the effects of non-uniform meshes and more complex geometries are not commented.