Subspace Correction Methods for Linear Elasticity : Algebraic Multigrid & Subspace Corrections for almost incompressible Materials

The monograph at hand deals with the numerical solution of the equations of linear elasticity. We focus on first order discretizations of those equations by means of linear finite elements. In order to solve the arising discrete problems efficiently we exploit the framework of subspace correction methods. First we present an overview of existing subspace correction methods. Later we introduce in detail a variant, namely "algebraic multigrid based on computational molecules" (AMGm), being specifically suitable for standard discretizations of the equations of linear elasticity. Since the standard linear discretization suffers from locking if the material becomes incompressible, we consider a varational formulation based on "reduced integration" using piecewise linear finite elements in a second part. This discretization can be shown to be stable in the incompressible limit. In order to solve the arising discrete problem efficiently we discuss a preconditioner based on subspace corrections. This preconditioner is based on a suitable splitting of the vector space. With proper numerical methods for the subproblems we are able to set up an efficient and robust preconditioner.