Balancing Domain Decomposition by Constraints and Perturbation

Abstract

Abstract. In this paper, we formulate and analyze a perturbed formulation of the balancing domain decomposition by constraints (BDDC) method. We prove that the perturbed BDDC has the same polylogarithmic bound for the condition number as the standard formulation. Two types of properly scaled zero-order perturbations are considered: one uses a mass matrix, and the other uses a Robin-type boundary condition, i.e, a mass matrix on the interface. With perturbation, the wellposedness of the local Neumann problems and the global coarse problem is automatically guaranteed, and coarse degrees of freedom can be defined only for convergence purposes but not well-posedness. This allows a much simpler implementation as no complicated corner selection algorithm is needed. Minimal coarse spaces using only face or edge constraints can also be considered. They are very useful in extreme scale calculations where the coarse problem is usually the bottleneck that can jeopardize scalability. The perturbation also adds extra robustness as the perturbed formulation works even when the constraints fail to eliminate a small number of subdomain rigid body modes from the standard BDDC space. This is extremely important when solving problems on unstructured meshes partitioned by automatic graph partitioners since arbitrary disconnected subdomains are possible. Numerical results are provided to support the theoretical findings.Key words. BDDC, preconditioner, coarse space, parallel solver, scalability AMS subject classifications. 65N55, 65N22, 65F08 DOI. 10.1137/15M10456481. Introduction. The development of highly scalable linear solvers for the solution of large scale linear systems arising from the finite element (FE) discretization of second-order elliptic problems on distributed-memory machines is of great importance in many applications. In this work, we consider nonoverlapping domain decomposition (DD) methods [49], which take advantage of the partition of the FE mesh into submeshes to define effective preconditioners that can exploit large levels of concurrency. In particular, we focus on a variant of the balancing DD by constraints (BDDC) method. The BDDC method was first introduced in 2003 by Dohrmann [18]. It can be regarded as an improved version of the balancing domain decomposition (BDD) method by Mandel [43]. It also has a very close connection with the dual primal finite element tearing and interconnecting (FETI-DP) method [25,24]. In fact, the eigenvalues of the preconditioned systems in the two approaches are almost identical [44,41,15]. The BDDC method is particularly well suited for extreme scale simulations, since it allows for a very aggressive coarsening, the computations at different levels can be computed in parallel, the subdomain problems can be solved inexactly

Balancing domain decomposition by constraints associated with subobjects

“…In addition, even when we drop edge constraints and only use half of the face constraints (vf-min), the number of iterations is still the same except for k = 4. These results confirm the estimate in (5) and indicate that one does not need to impose constraints on all available subobjects to achieve the desired rate of convergence.…”

“…In addition, as BDDC-A is a standard BDDC preconditioner with regular coarse objects, standard weighting operator, and defined on the partition Θ having characteristic size L, the third inequality in (5) is a standard result (see [13,15]). We only need to prove the second inequality in (5). First, we note that w ∈ W can be discontinuous across Γ but is always continuous in each Ω i .…”

Domain Decomposition Methods in Science and Engineering XXIII

“…For detailed mathematical analysis, we refer the reader to Badia and Nguyen (2016). For detailed mathematical analysis, we refer the reader to Badia and Nguyen (2016).…”

“…For a full mathematical treatment with complete analysis and additional numerical experiments, we refer the reader to Badia and Nguyen (2016). For a full mathematical treatment with complete analysis and additional numerical experiments, we refer the reader to Badia and Nguyen (2016).…”

“…For results using a Robin perturbation, we refer to Badia and Nguyen (2016). For results using a Robin perturbation, we refer to Badia and Nguyen (2016).…”