On parallel solution of linear elasticity problems: Part I: theory

The discretised linear elasticity problem is solved by the preconditioned conjugate gradient (pcg) method. Mainly we consider the linear isotropic case but we also comment on the more general linear orthotropic problem. The preconditioner is based on the separate displacement component (sdc) part of the equations of elasticity. The preconditioning system consists of two or three subsystems (in two or three dimensions) also called inner systems, each of which is solved by the incomplete factorisation pcg-method i.e. we perform inner iterations. A finite element discretisation and node-numbering giving high degree of partial parallelism with equal processor load for the solution of these systems by the MIC(0) pcg method is presented. In general, the incomplete factorisation requires an M-matrix. This property is studied for the elasticity problem. The rate of convergence of the pcg-method is analysed for different preconditionings based on the sdc-part of the elasticity equations. In the following two parts of this trilogy we will focus more on parallelism and implementation aspects.