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88
Dualprimal FETI methods for threedimensional elliptic problems with heterogeneous coefficients
 SIAM J. Numer. Anal
, 2002
"... 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 FETI methods which have recently been introduced and analyzed successfully for elliptic problems i ..."
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Cited by 80 (13 self)
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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 FETI methods which have recently 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 nding 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 coeÆcients. These results closely parallel those for other successful iterative substructuring methods of primal as well as dual type.
An Algebraic Theory for Primal and Dual Substructuring Methods by Constraints
, 2004
"... FETI and BDD are two widely used substructuring methods for the solution of large sparse systems of linear algebraic equations arizing from discretization of elliptic boundary value problems. The two most advanced variants of these methods are the FETIDP and the BDDC methods, whose formulation does ..."
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Cited by 63 (10 self)
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FETI and BDD are two widely used substructuring methods for the solution of large sparse systems of linear algebraic equations arizing from discretization of elliptic boundary value problems. The two most advanced variants of these methods are the FETIDP and the BDDC methods, whose formulation does not require any information beyond the algebraic system of equations in a substructure form. We formulate the FETIDP and the BDDC methods in common framework as methods based on general constraints between the substructures, and provide a simplified algebraic convergence theory. The basic implementation blocks including transfer operators are common to both methods. It is shown that commonly used properties of the transfer operators in fact determine the operators uniquely. Identical algebraic condition number bounds for both methods are given in terms of a single inequality, and, under natural additional assumptions, it is proved that the eigenvalues of the preconditioned problems are the same. The algebraic bounds imply the usual polylogarithmic bounds for finite elements, independent of coefficient jumps between substructures. Computational experiments confirm the theory.
Balancing NeumannNeumann methods for incompressible Stokes equations
, 2001
"... Balancing NeumannNeumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the ..."
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Cited by 43 (9 self)
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Balancing NeumannNeumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing NeumannNeumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the infsup constants of the discrete problem and of the coarse subproblem. Numerical results show that the method is quite fast; they are also fully consistent with the theory.
Robust FETIDP methods for heterogeneous three dimensional elasticity problems
 COMPUT. METHODS APPL. MECH. ENGRG
, 2005
"... Iterative substructuring methods with Lagrange multipliers are considered for heterogeneous linear elasticity problems with large discontinuities in the material stiffnesses. In particular, results for algorithms belonging to the family of dualprimal FETI methods are presented. The core issue of ..."
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Cited by 27 (2 self)
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Iterative substructuring methods with Lagrange multipliers are considered for heterogeneous linear elasticity problems with large discontinuities in the material stiffnesses. In particular, results for algorithms belonging to the family of dualprimal FETI methods are presented. The core issue of these algorithms is the construction of an appropriate global problem, in order to obtain a robust method which converges independently of the material discontinuities. In this article, several necessary and sufficient conditions arising from the theory are numerically tested and confirmed. Furthermore, results of numerical experiments are presented for situation which are not covered by the theory, such as curved edges and material discontinuities not aligned with the interface, and an attempt is made to develop rules for these cases.
Analysis of FETI methods for multiscale PDEs
 Numer. Math
"... www.oeaw.ac.at www.ricam.oeaw.ac.at Analysis of FETI methods for multiscale elliptic PDEs ..."
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Cited by 23 (11 self)
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www.oeaw.ac.at www.ricam.oeaw.ac.at Analysis of FETI methods for multiscale elliptic PDEs
Adaptive Selection of Face Coarse Degrees of Freedom in the BDDC and the FETIDP Iterative Substructuring Methods
, 2006
"... We propose a class of method for the adaptive selection of the coarse space of the BDDC and FETIDP iterative substructuring methods. The methods work by adding coarse degrees of freedom constructed from eigenvectors associated with intersections of selected pairs of adjacent substructures. It is as ..."
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Cited by 20 (4 self)
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We propose a class of method for the adaptive selection of the coarse space of the BDDC and FETIDP iterative substructuring methods. The methods work by adding coarse degrees of freedom constructed from eigenvectors associated with intersections of selected pairs of adjacent substructures. It is assumed that the starting coarse degrees of freedom are already sufficient to prevent relative rigid body motions in any selected pair of adjacent substructures. A heuristic indicator of the the condition number is developed and a minimal number of coarse degrees of freedom is added to decrease the indicator under a given threshold. It is shown numerically on 2D elasticity problems that the indicator based on pairs of substructures with common edges predicts the actual condition number reasonably well, and that the method can select adaptively the hard part of the problem and concentrate computational work there to achieve good convergence of the iterations at a modest cost.
Domain decomposition preconditioners for linear–quadratic elliptic optimal control problems
, 2004
"... ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquad ..."
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Cited by 19 (4 self)
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ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs. We prove that for a class of problems the performance of the twolevel versions of our preconditioners is independent of the mesh size and of the subdomain size. 1.
The mosaic of high performance Domain Decomposition Methods for Structural Mechanics: Formulation, interrelation and numerical efficiency of primal and dual methods
 Comput. Methods Appl. Mech. Engrg
, 2003
"... numerical efficiency of primal and dual methods ..."
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