Dual-primal FETI methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations, as in primal iterative substructuring methods, while most of the constraints are enforced by Lagrange multipliers, as in one-level FETI methods. These methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems. The purpose of this article is to develop strategies for selecting these constraints, which are enforced throughout the iterations, such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them. The algorithms are described in terms of a change of basis that has proven to be quite robust in practice. A theoretical analysis is provided for the case of linear elasticity, and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the Young's modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain. The strategies have already proven quite successful in large-scale implementations of these iterative methods.
Block-triangular preconditioners for a class of saddle point problems with a penalty term are considered. An important example is the mixed formulation of the pure displacement problem in linear elasticity. It is shown that the spectrum of the preconditioned system is contained in a real, positive interval and that the interval bounds can be made independent of the discretization and penalty parameters. This fact is used to construct bounds of the convergence rate of the GMRES method with respect to an energy norm. Numerical results are given for GMRES and BI-CGSTAB. Key words. saddle point problems, penalty term, block-triangular preconditioners, linear elasticity, almost incompressible materials AMS subject classifications. 65F10, 65N22, 65N30 PII. S1064827596303624 1. Introduction. Many problems in the engineering sciences lead to saddle point problems. Important examples are the Stokes equations of fluid dynamics, modeling the flow of an incompressible viscous fluid, and mixed formulations of problems from linear elasticity, e.g., for almost incompressible materials; cf. Braess [4] or Brezzi and Fortin [10]. These problems can be analyzed in the framework of saddle point problems with a penalty term.It is well known that (lower-order) conforming finite element discretizations of the displacement formulation of linear elasticity problems can suffer from the locking phenomenon; i.e., the convergence rate of the finite element method deteriorates when the Poisson ratio ν of the material approaches 1/2; see Babuška and Suri [1]. Although the resulting linear system is symmetric and positive definite, it has a condition number that degenerates when ν approaches 1/2. When this linear problem is solved iteratively, the dependence upon the Poisson ratio can lead to a deterioration of the convergence rate of the iterative method. One approach to avoiding these problems is to use a reformulation in terms of a saddle point problem with a penalty term; cf. Brezzi and Fortin [10]. In this case, the penalty term depends on the Poisson ratio, or, alternatively, on the Lamé parameter λ; cf. Brezzi and Fortin [10] and section 4. The resulting stiffness matrix is symmetric and indefinite. Since saddle point problems generally lead to indefinite systems, we encounter an additional difficulty when solving the resulting linear problems iteratively.In this article, we focus on the construction of an iterative method for certain saddle point problems with a penalty term. We analyze a block-triangular preconditioner for
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