Algebraic multigrid for finite element elasticity equations: Determination of nodal dependence via edge matrices and two-level convergence

We study an algebraic multigrid (AMG) method for solving elliptic finite element equations of linear elasticity problems. In this method, which has been proposed in [J.K. Kraus, SIAM J. Sci. Comput., 30 (2008), pp. 505–524], the coarsening is based on so-called edge matrices, which allows to generalize the concept of strong and weak connections, as used in classical AMG, to “algebraic vertices” that accumulate the nodal degrees of freedom in case of vector-field problems. The major contribution of this work is related to the investigation of the measure for the nodal dependence and on the generation of the edge matrices, which are the basic building blocks of this method. A natural measure is the cosine of the abstract angle between the two subspaces spanned by the basis functions corresponding to the respective algebraic vertices. Another original contribution of this work is a two-level convergence analysis of the method. The presented numerical results cover also problems with jumps in the Young’s modulus of elasticity and orthotropic materials.