Journal ArticleDOI
Dual-Primal FETI Methods for Three-dimensional Elliptic Problems with Heterogeneous Coefficients
TLDR
It is shown that the condition numbers of several of the dual-primal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains, the mesh size, and jumps in the coefficients.Abstract:
In this paper, certain iterative substructuring methods with Lagrange multipliers are considered for elliptic problems in three dimensions. The algorithms belong to the family of dual-primal finite element tearing and interconnecting (FETI) methods which recently have been introduced and analyzed successfully for elliptic problems in the plane. The family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms. Particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual-primal preconditioner. It is shown that the condition numbers of several of the dual-primal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains, the mesh size, and jumps in the coefficients. These results closely parallel those of other successful iterative substructuring methods of primal as well as dual type.read more
Citations
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Journal ArticleDOI
An algebraic theory for primal and dual substructuring methods by constraints
TL;DR: It is shown that commonly used properties of the transfer operators in fact determine the operators uniquely, and it is proved that the eigenvalues of the preconditioned problems are the same.
Journal ArticleDOI
Convergence of a balancing domain decomposition by constraints and energy minimization
TL;DR: A convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts and the main result is a bound on the condition number based on inequalities involving the matrices of the preconditionser.
Journal ArticleDOI
Dual‐primal FETI methods for linear elasticity
Axel Klawonn,Olof B. Widlund +1 more
TL;DR: The purpose of this article is to develop strategies for selecting 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.
Convergence of a Balancing Domain Decomposition by constraints and energy minimization Dedicated to Professor Ivo Marek on the occasion of his 70th birthday.
Jan Mandel,Clark R. Dohrmann +1 more
TL;DR: In this article, a convergence theory for a substructuring preconditioner based on constrained energy minimization concepts is presented, where the constraints include values at corners and optionally averages on edges and faces.
Journal ArticleDOI
FETI‐DP, BDDC, and block Cholesky methods
Jing Li,Olof B. Widlund +1 more
TL;DR: With the new formulation of these algorithms, a simplified proof is provided that the spectra of a pair of FETI‐DP and BDDC algorithms, with the same set of primal constraints, are essentially the same.
References
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Journal ArticleDOI
The construction of preconditioners for elliptic problems by substructuring. I
TL;DR: This paper develops a technique which utilizes earlier methods to derive even more efficient preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems.
Journal ArticleDOI
FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method
TL;DR: This paper presents a dual–primal formulation of the FETI‐2 concept that eliminates the need for that second set of Lagrange multipliers, and unifies all previously developed one‐level and two‐level FETi algorithms into a single dual‐primal FetI‐DP method.
Journal ArticleDOI
Optimal convergence properties of the FETI domain decomposition method
TL;DR: This paper shows that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the FETI method are equivalent to the construction and solution of a coarse problem that propagates the error globally, accelerates convergence, and ensures a performance that is independent of the number ofSubdomains.
Book
Iterative Methods for the Solution of Elliptic Problems on Regions Partitioned Into Substructures
TL;DR: A class of iterative methods is discussed in which these subproblems are solved by direct methods, while the interaction across the curves or surfaces which divide the region is handled by a conjugate gradient method.
Journal ArticleDOI
A scalable dual‐primal domain decomposition method
TL;DR: This work blends dual and primal domain decomposition approaches to construct a fast iterative method for the solution of large-scale systems of equations arising from the finite element discretization of second- and fourth-order partial differential equations.