Dual-primal FETI methods for linear elasticity problems in three dimensions are considered. These are nonoverlapping domain decomposition methods where some primal continuity constraints across subdomain boundaries are required to hold throughout the iterations, whereas most of the constraints are enforced by Lagrange multipliers. An algorithmic framework for dualprimal FETI methods is described together with a transformation of basis to implement the primal constraints. Numerical results obtained from a parallel implementation of these algorithms applied to a model benchmark problem with structured meshes and to problems with more complicated geometries from industrial and biological applications using unstructured meshes are provided. These results show that the presented dual-primal FETI algorithms are numerical and parallel scalable. Introduction.Dual-primal FETI (FETI-DP) methods are the most recent members of the family of finite element tearing and interconnecting (FETI) domain decomposition methods. The FETI methods, introduced by Farhat and Roux [18], are all dual iterative substructuring methods for partial differential equations. In these methods the original domain, on which the given partial differential equation has to be solved, is decomposed into nonoverlapping subdomains. The intersubdomain continuity is then enforced by Lagrange multipliers across the interface defined by the subdomain boundaries. For further results and references, see, e.g., [17,13,5,34,24,41]. In FETI-DP methods, some continuity constraints on the primal displacement variables are forced to hold throughout iterations, as in primal substructuring algorithms, while the other constraints are enforced by the use of Lagrange multipliers, as in standard one-level FETI methods. The primal constraints have to be chosen such that the local subproblems become invertible and such that a parallel scalable method is obtained; the primal constraints provide a coarse problem for these domain decomposition methods. Different choices of primal constraints are possible; see the discussion below. Each of these choices defines a different coarse space and thus a different member of the family of FETI-DP algorithms.FETI-DP algorithms were introduced by Farhat et al. in [14] for linear elasticity problems in the plane and then extended by Farhat, Lesoinne, and Pierson [15] to three-dimensional elasticity problems; see also Pierson [37]. The first theoretical analysis for two-dimensional, scalar elliptic partial differential equations of second and fourth order with only small coefficient jumps across the subdomain boundaries was given by Mandel and Tezaur [35]; it was shown that the condition number is bounded polylogarithmically as a function of the dimension of the individual subregion
Abstract. In the theory for domain decomposition algorithms of the iterative substructuring family, each subdomain is typically assumed to be the union of a few coarse triangles or tetrahedra. This is an unrealistic assumption, in particular if the subdomains result from the use of a mesh partitioner, in which case they might not even have uniformly Lipschitz continuous boundaries. The purpose of this study is to derive bounds for the condition number of these preconditioned conjugate gradient methods which depend only on a parameter in an isoperimetric inequality, two geometric parameters characterizing John and uniform domains, and the maximum number of edges of any subdomain. A related purpose is to explore to what extent well-known technical tools previously developed for quite regular subdomains can be extended to much more irregular subdomains. Some of these results are valid for any John domain, while an extension theorem, which is needed in this study, requires that the subdomains have complements which are uniform. The results, so far, are complete only for problems in two dimensions. Details are worked out for a FETI-DP algorithm and numerical results support the findings. Some of the numerical experiments illustrate that care must be taken when selecting the scaling of the preconditioners in the case of irregular subdomains.Key words. domain decomposition, preconditioners, iterative substructuring, dual-primal FETI, John and uniform domains, fractal subdomains AMS subject classifications. 65F10, 65N30, 65N55DOI. 10.1137/070688675 1. Introduction. In the theory of domain decomposition methods of iterative substructuring type, we typically assume that each subdomain is quite regular, e.g., the union of a small set of coarse triangles or tetrahedra; see, e.g., [36, Assumption 4.3]. However, such an assumption is unlikely to hold especially if the subdomains result from using a mesh partitioner, in which case the subdomain boundaries might not even be uniformly Lipschitz continuous, i.e., the number of patches which cover ∂Ω, and in each of which the boundary is the graph of a Lipschitz continuous function, might not be uniformly bounded independently of the finite element mesh size. We also note that the shape of the subdomains is likely to change if the mesh size is altered and a mesh partitioner is used. The purpose of this paper is to develop a theory for domain decomposition methods under much weaker assumptions on the partitioning and to categorize the rate of convergence in terms of a few geometric parameters. We will denote the nonoverlapping subdomains by Ω i and the interface between them by Γ.So far, complete results have been obtained only for problems in the plane. To simplify our presentation, we will also focus on scalar elliptic problems of the following
Highly scalable parallel domain decomposition methods for elliptic partial differential equations are considered with a special emphasis on problems arising in elasticity. The focus of this survey article is on Finite Element Tearing and Interconnecting (FETI) methods, a family of nonoverlapping domain decomposition methods where the continuity between the subdomains, in principle, is enforced by the use of Lagrange multipliers. Exact onelevel and dual‐primal FETI methods as well as related inexact dual‐primal variants are described and theoretical convergence estimates are presented together with numerical results confirming the parallel scalability properties of these methods. New aspects such as a hybrid onelevel FETI/FETI‐DP approach and the behavior of FETI‐DP for anisotropic elasticity problems are presented. Parallel and numerical scalability of the methods for more than 65 000 processor cores of the JUGENE supercomputer is shown. An application of a dual‐primal FETI method to a nontrivial biomechanical problem from nonlinear elasticity, modeling arterial wall stress, is given, showing the robustness of our domain decomposition methods for such problems.
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