Improving algebraic multigrid interpolation operators for linear elasticity problems

Linear systems arising from discretizations of systems of partial differential equations can be challenging for algebraic multigrid (AMG), as the design of AMG relies on assumptions based on the near-nullspace properties of scalar diffusion problems. For elasticity applications, the near-nullspace of the operator includes the so-called rigid body modes (RBMs), which are not adequately represented by the classical AMG interpolation operators. In this paper we investigate several approaches for improving AMG conver-gence on linear elasticity problems by explicitly incorporating the near-nullspace modes in the range of the interpolation. In particular, we propose two new methods for extending any initial AMG interpolation operator to exactly fit the RBMs based on the introduction of additional coarse degrees of freedom at each node. Though the methodology is general and can be used to fit any set of near-nullspace vectors, we focus on the RBMs of linear elasticity in this paper. The new methods can be incorporated easily into existing AMG codes, do not require matrix inversions, and do not assume an aggregation approach or a finite element framework. We demonstrate the effectiveness of the new interpolation operators on several 2D and 3D elasticity problems.