An Optimal Domain Decomposition Preconditioner for the Finite Element Solution of Linear Elasticity Problems

For linear elasticity problems the finite element method is an extremely successful method to model complicated structures. The successful implementation requires the solution of very large, sparse, positive definite linear systems of algebraic equations. A new technique for solving these systems using the preconditioned conjugate gradient method is proposed. Using ideas from both additive Schwarz methods and iterative substructuring methods, it is proven that the condition number of the resulting system does not grow as the substructures are made smaller and the mesh is refined. This result holds for two and three dimensions. Numerical experiments have been performed to demonstrate the power of this method. For linear elasticity problems in two dimensions the condition numbers are observed numerically to be less than four when using a regular mesh.