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.

We propose adaptive selection of the coarse space of the BDDC and FETI-DP iterative substructuring methods by adding coarse degrees of freedom (dofs) on faces between substructures constructed using eigenvectors associated with the faces. Provably the minimal number of coarse dofs on the faces is added to decrease the condition number estimate under a target value specified a priori. It is assumed that corner dofs are already sufficient to prevent relative rigid body motions of any two substructures with a common face. It is shown numerically on a 2D elasticity problem that the condition number estimate based on faces is quite indicative of the actual condition number and that the method can select adaptively a hard part of the problem and concentrate computational work there to achieve the target value for the condition number and good convergence of the iterations, at a modest cost.

In this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting (FETI) method and a variant thereof, all-floating FETI. We consider the scalar elliptic equation in a two- or three-dimensional 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 one-level and the all-floating 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 so-called face, edge, and vertex islands in high-contrast 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.

Abstract. BDDC algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with Gauss-Lobatto-Legendre spectral elements in three dimensions. Initially mixed spectral elements are employed to discretize the almost incompressible elasticity system, but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped. Appropriate sets of primal constraints can be associated with the subdomain vertices, edges, and faces so that the resulting BDDC methods have a fast convergence rate independent of the almost incompressibility of the material. In particular, the condition number of the BDDC preconditioned operator is shown to depend only weakly on the polynomial degree n, the ratio H/h of subdomain and element diameters, and the inverse of the inf-sup constants of the subdomains and the underlying mixed formulation, while being scalable, i.e., independent of the number of subdomains and robust, i.e., independent of the Poisson ratio and Young’s modulus of the material considered. These results also apply to the related FETI-DP algorithms defined by the same set of primal constraints. Numerical experiments carried out on parallel computing systems confirm these results. Key words. domain decomposition, BDDC preconditioners, almost incompressible elasticity, mixed spectral elements AMS subject classifications. 65F08, 65N30, 65N35, 65N55

In recent years, Domain Decomposition Methods (DDM) have emerged as advanced solvers in several areas of computational mechanics. In particular, during the last decade, in the area of solid and structural mechanics, they reached a considerable level of advancement and were shown to be more efficient than popular solvers, like advanced sparse direct solvers. The present paper explores the extent of application of the general concept of force-displacement duality in DDM. A general framework for the definition of DDM is set up and it is shown that if the definition of a DDM meets some requirements, then it can lead to one primal and one dual formulation. A number of DDM are included in this setting and its particular implications for each one of them are researched.

Abstract not found

Abstract. A BDDC (balancing domain decomposition by constraints) algorithm is developed for elliptic problems with mortar discretizations for geometrically non-conforming partitions in both two and three spatial dimensions. The coarse component of the preconditioner is defined in terms of one mortar constraint for each edge/face which is an intersection of the boundaries of a pair of subdomains. A condition number bound of the form C maxi (1 + log(Hi/hi)) 3 ¯ is established. In geometrically conforming cases, the bound can be improved to C maxi (1 + log(Hi/hi)) 2 ¯. This estimate is also valid in the geometrically nonconforming case under an additional assumption on the ratio of mesh sizes and jumps of the coefficients. This BDDC preconditioner is also shown to be closely related to the Neumann-Dirichlet preconditioner for the FETI–DP algorithms of [9, 11] and it is shown that the eigenvalues of the BDDC and FETI–DP methods are the same except possibly for an eigenvalue equal to 1. Key words. BDDC, FETI–DP, mortar methods, preconditioner

Numerical solution of linear problems arising from discretization by finite elements is important in many areas of engineering. The matrix of the system is typically large, sparse, and often ill-conditioned. For large problems, iterative methods such as the preconditioned conjugate gradients (PCG) are usually less expensive in terms of memory and computational time. However, their convergence rate deteriorates with growing condition number of the solved

A high-order Galerkin Least-Squares (GLS) finite element discretization is combined with a Balancing Domain Decomposition by Constraints (BDDC) preconditioner and inexact local solvers to provide an efficient solution technique for large-scale, convection-dominated problems. The algorithm is applied to the linear system arising from the discretization of the two-dimensional advection-diffusion equation and Euler equations for compressible, inviscid flow. A Robin-Robin interface condition is extended to the Euler equations using entropy-symmetrized variables. The BDDC method maintains scalability for the high-order discretization of the diffusion-dominated flows, and achieves low iteration count in the advection-dominated regime. The BDDC method based on inexact local solvers with incomplete factorization and p = 1 coarse correction maintains the performance of the exact counterpart for the wide range of the Peclet numbers considered while at significantly reduced memory and computational costs. Key words: Galerkin least-squares, high-order methods, domain decomposition methods, BDDC, preconditioners