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62
Convergence of a Balancing Domain Decomposition by Constraints and Energy Minimization
, 2002
"... A convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts. The preconditioner is formulated as an Additive Schwarz method and analyzed by building on existing results for Balancing Domain Decomposition. The main result is a bound on the c ..."
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Cited by 46 (10 self)
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A convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts. The preconditioner is formulated as an Additive Schwarz method and analyzed by building on existing results for Balancing Domain Decomposition. The main result is a bound on the condition number based on inequalities involving the matrices of the preconditioner. Estimates of the usual form C(1 + log²(H/h)) are obtained under the standard assumptions of substructuring theory. Computational results demonstrating the performance of method are included.
A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions
 SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
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Cited by 36 (10 self)
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Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
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 36 (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.
Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. Arxiv preprint arXiv:1105.1131
, 2011
"... Abstract. An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local ” subspaces and a global “coarse ” space is developed. Particular applications of this abstract framework include practically important probl ..."
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Cited by 15 (7 self)
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Abstract. An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local ” subspaces and a global “coarse ” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Efendiev and Galvis [Multiscale Model. Simul., 8 (2010), pp. 1461–1483] developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman’s problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided. 1.
Analysis of FETI Methods for Multiscale PDEs
 Numer. Math
, 2008
"... Abstract. In this paper we study a variant of the finite element tearing and interconnecting (FETI) method which is suitable for elliptic PDEs with highly heterogeneous (multiscale) coefficients α(x); in particular, coefficients with strong variation within subdomains and/or jumps that are not align ..."
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Cited by 14 (8 self)
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Abstract. In this paper we study a variant of the finite element tearing and interconnecting (FETI) method which is suitable for elliptic PDEs with highly heterogeneous (multiscale) coefficients α(x); in particular, coefficients with strong variation within subdomains and/or jumps that are not aligned with the subdomain interfaces. Using energy minimisation and cutoff arguments we can show rigorously that for an arbitrary (positive) coefficient function α ∈ L ∞ (Ω) the condition number of the preconditioned FETI system can be bounded by C(α) (1 + log(H/h)) 2 where H is the subdomain diameter and h is the mesh size, and where the function C(α) depends only on the coefficient variation in the vicinity of subdomain interfaces. In particular, if αΩi varies only mildly in a layer Ωi,η of width η near the boundary of each of the subdomains Ωi, then C(α) = O((H/η) 2), independent of the variation of α in the remainder Ωi\Ωi,η of each subdomain and independent of any jumps of α across subdomain interfaces. The quadratic dependency of C(α) on H/η can be relaxed to a linear dependency under stronger assumptions on the behaviour of α in the interior of the subdomains. Our theoretical findings are confirmed in numerical tests.
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 11 (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.
BDDC and FETIDP under Minimalist Assumptions
, 2008
"... The FETIDP, BDDC and PFETIDP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method co ..."
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Cited by 8 (5 self)
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The FETIDP, BDDC and PFETIDP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that PFETIDP and BDDC are in fact the same. The FETIDP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETIDP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example.
ANALYSIS OF FETI METHODS FOR MULTISCALE PDES  PART II: INTERFACE VARIATION
, 2009
"... In this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting (FETI) method and a variant thereof, allfloating FETI. We consider the scalar elliptic equation in a two or threedimensional domain with a highly heterogeneous (multiscale) diffusi ..."
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Cited by 7 (6 self)
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In this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting (FETI) method and a variant thereof, allfloating FETI. We consider the scalar elliptic equation in a two or threedimensional domain with a highly heterogeneous (multiscale) diffusion coefficient. This coefficient is allowed to have large jumps not only across but also along subdomain interfaces and in the interior of the subdomains. In other words, the subdomain partitioning does not need to resolve any jumps in the coefficient. Under suitable assumptions, we can show that the condition numbers of the onelevel and the allfloating FETI system are robust with respect to strong variations in the contrast in the coefficient. We get only a dependence on some geometric parameters associated with the coefficient variation. In particular, we can show robustness for socalled face, edge, and vertex islands in highcontrast media. As a central tool we prove and use new weighted Poincaré and discrete Sobolev type inequalities that are explicit in the weight. Our theoretical findings are confirmed in a series of numerical experiments.