Iterative Substructuring Methods for Spectral Element Discretizations of Elliptic Systems I: Compressible Linear Elasticity

An iterative substructuring method for the system of linear elasticity in three dimensions is introduced and analyzed. The pure displacement formulation for compressible materials is discretized with the spectral element method. The resulting stiiness matrix is symmetric and positive deenite. The method proposed provides a domain decomposition preconditioner constructed from local solvers for the interior of each element, and for each face of the elements and a coarse, global solver related to the wire basket of the elements. As in the scalar case, the condition number of the preconditioned operator is independent of the number of spectral elements and grows as the square of the logarithm of the spectral degree. AMS(MOS) subject classiications. 65N30, 65N35, 65N55 1. Introduction. Finite element discretizations of problems in structural mechanics require the solution of large and sparse linear systems of equations. In the past, such systems have often been solved by direct methods. These methods are limited by their high arithmetical costs, memory requirements, and poor scalability on parallel computers. In order to overcome these limitations, a great deal of research has in recent years focused on the design and analysis of eecient iterative methods. Solvers which combine a Krylov space accelerator with a robust preconditioner have been shown to outperform direct solvers in large three-dimensional elasticity computations ; see Dickinson and Forsyth 11], Farhat and Roux 14] and the references therein. Domain decomposition provides some of the best preconditioners for elliptic problems; see Smith, Bjjrstad and Gropp 35] for a general introduction and Le Tallec 20] and Farhat and Roux 14] for a discussion of domain decomposition in structural mechanics. In this paper, we will focus on the system of linear elasticity in three dimensions. For more general problems and methods in nonlinear elasticity, we refer to Ciarlet 10] and Le Tallec 21]. Iterative substructuring methods form an important family of domain decomposition algorithms, with origins in the direct substructuring techniques developed in the structural analysis community over several decades. When an iterative substructuring method is used, the domain of the elliptic problem is decomposed into nonoverlapping subdomains. After the elimination of the interior variables, the discrete problem for the interface variables, known as the Schur complement system, is solved iteratively by