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.

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.

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 FETI-DP and the BDDC methods, whose formulation does not require any information beyond the algebraic system of equations in a substructure form. We formulate the FETI-DP 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.

Abstract. The purpose of this paper is to extend the BDDC (balancing domain decomposition by constraints) algorithm to saddle-point problems that arise when mixed finite element methods are used to approximate the system of incompressible Stokes equations. The BDDC algorithms are iterative substructuring methods, which form a class of domain decomposition methods based on the decomposition of the domain of the differential equations into nonoverlapping subdomains. They are defined in terms of a set of primal continuity constraints, which are enforced across the interface between the subdomains and which provide a coarse space component of the preconditioner. Sets of such constraints are identified for which bounds on the rate of convergence can be established that are just as strong as previously known bounds for the elliptic case. In fact, the preconditioned operator is effectively positive definite, which makes the use of a conjugate gradient method possible. A close connection is also established between the BDDC and FETI-DP algorithms for the Stokes case.

. In this paper, a dual-primal FETI method is developed for incompressible Stokes equations approximated by mixed nite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, solving the indenite Stokes problem is reduced to solving a symmetric positive denite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by a Krylov space method with a Dirichlet preconditioner. At each step of the iteration, both subdomain problems and a coarse problem on the coarse subdomain mesh are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the product of the inverse of the inf-sup constant of the discrete problem and the square of the logarithm of the number of unknowns in the individual subdomain problems. Illustrative numerical results are presented by solving a lid driven cavity problem. Key words. domain decomposition, Stokes, FETI, dual-primal methods AMS subject classications. 65N30, 65N55, 76D07 1.

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 cut-off 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 α|Ωi 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.

We propose a class of method for the adaptive selection of the coarse space of the BDDC and FETI-DP 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.

In this paper we discuss domain decomposition parallel iterative solvers for highly heterogeneous problems of flow and transport in porous media. We are particularly interested in highly unstructured coefficient variation where standard periodic or stochastic homogenisation theory is not applicable. When the smallest scale at which the coefficient varies is very small it is often necessary to scale up the equation to a coarser grid to make the problem computationally feasible. Standard upscaling or multiscale techniques, require the solution of local problems in each coarse element, leading to a computational complexity that is at least linear in the global number N of unknowns on the subgrid. Moreover, except for the periodic and the isotropic random case, a theoretical analysis of the accuracy of the upscaled solution is not yet available. Multilevel iterative methods for the original problem on the subgrid, such as multigrid or domain decomposition, lead to similar computational complexity (i.e. O(N)) and are therefore a viable alternative. However, previously no theory was available guaranteeing the robustness of these methods to large coefficient variation. We review a sequence of recent papers where simple variants of domain decomposition methods, such as overlapping Schwarz and one-level FETI, are proposed that are robust to strong coefficient variation. Moreover, we extend the theoretical results for the first time also to the dual-primal variant of FETI.

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.

The Boundary Element Tearing and Interconnecting (BETI) methods have recently been introduced as boundary element counterparts of the well–established Finite Element Tearing and Interconnecting (FETI) methods. In this paper we present inexact data–sparse versions of the BETI methods which avoid the elimination of the primal unknowns and dense matrices. However, instead of symmetric and positive definite systems, we finally have to solve two–fold saddle point problems. The proposed iterative solvers and preconditioners result in almost optimal solvers the complexity of which is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor. Moreover, the solvers are robust with respect to large coefficient jumps. 1