Aggregation-based multilevel preconditioning of non-conforming FEM elasticity problems

Abstract

Preconditioning techniques based on various multilevel extensions of two-level splittings of finite element (FE) spaces lead to iterative methods which have an optimal rate of convergence and computational complexity with respect to the number of degrees of freedom. This article deals with the construction of algebraic two-level and multilevel preconditioning algorithms for the Lamé equations of elasticity, which are discretized by Crouzeix-Raviart non-conforming linear finite elements on triangles. An important point to note is that in the non-conforming case the FE spaces corresponding to two successive levels of mesh refinements are not nested. To handle this, a proper aggregation-based two-level basis is considered, which enables us to fit the general framework of the two-level preconditioners and to generalize the method to the multilevel case. The derived estimate of the constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality is uniform with respect to both, mesh anisotropy and Poisson ratio, including the almost incompressible case. © Springer-Verlag Berlin Heidelberg 2006.