Two-level Preconditioning of Pure Displacement Non-conforming FEM Systems

摘要

This paper is concerned with the pure displacement problem of planar linear elasticity. Our interest is focused on the locking-free FEM approximation of the problem in the case of almost incompressible material. Crouzeix-Raviart linear finite elements are implemented. An optimal order pure algebraic multiplicative two-level preconditioner for the related stiffness matrix is developed. The proposed construction is based on a proper hierarchical basis of the FEM space. It is important to note that the nodal FEM spaces corresponding to successive levels of mesh refinements are not nested for the non-conforming elements under consideration. Local spectral analysis is applied to determine the scaling parameter of the preconditioner as well as to estimate the related constant in the strengthened CBS inequality. The derived estimates are uniform with respect to the Poisson ratio v implied by (0, 0.5). A set of numerical tests is presented to illustrate the accuracy of the FEM aproximation. and the convergence rate of the two-level PCG method.