A Nonoverlapping Domain Decomposition Method for Incompressible Stokes Equations with Continuous Pressures

This is the publisher's version, also available electronically from http://epubs.siam.org/doi/abs/10.1137/120861503A nonoverlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible Stokes equations. A continuous finite element space for the pressure is used. In the proposed algorithm, Lagrange multipliers are used to enforce continuity of the velocity component across the subdomain boundary. The continuity of the pressure component is enforced in the primal form, i.e., neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no Lagrange multipliers are needed. After eliminating all velocity variables and the independent subdomain interior parts of the pressures, a symmetric positive semidefinite linear system for the subdomain boundary pressures and the Lagrange multipliers is formed and solved by a preconditioned conjugate gradient method. A lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments demonstrate the convergence rate of the proposed algorithm

Three-level BDDC in three dimensions

This is the published version, also available here: http://dx.doi.org/10.1137/050629902.Balancing domain decomposition by constraints (BDDC) methods are nonoverlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from the discretization of elliptic boundary value problems. Their coarse problems are given in terms of a small number of continuity constraints for each subdomain, which are enforced across the interface. The coarse problem matrix is generated and factored by a direct solver at the beginning of the computation and it can ultimately become a bottleneck if the number of subdomains is very large. In this paper, two three-level BDDC methods are introduced for solving the coarse problem approximately for problems in three dimensions. This is an extension of previous work for the two-dimensional case. Edge constraints are considered in this work since vertex constraints alone, which work well in two dimensions, result in a noncompetitive algorithm in three dimensions. Some new technical tools are then needed in the analysis and this makes the three-dimensional case more complicated. Estimates of the condition numbers are provided for two three-level BDDC methods, and numerical experiments are also discussed

Balancing domain decomposition by constraints and perturbation

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.Peer ReviewedPostprint (published version

A three-level BDDC algorithm for mortar discretizations

This is the published version, also available here: http://dx.doi.org/10.1137/07069081X.In this paper, a three-level balancing domain decomposition by constraints (BDDC) algorithm is developed for the solutions of large sparse algebraic linear systems arising from the mortar discretization of elliptic boundary value problems. The mortar discretization is considered on geometrically nonconforming subdomain partitions. In two-level BDDC algorithms, the coarse problem needs to be solved exactly. However, its size will increase with the increase of the number of the subdomains. To overcome this limitation, the three-level algorithm solves the coarse problem inexactly while a good rate of convergence is maintained. This is an extension of previous work: the three-level BDDC algorithms for standard finite element discretization. Estimates of the condition numbers are provided for the three-level BDDC method, and numerical experiments are also discussed

On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections

In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when using a highly scalable asynchronous parallel implementation where fine and coarse correction computations are overlapped in time. This way, the coarse-grid problem can be fully overlapped by fine-grid computations (which are embarrassingly parallel) in a wide range of cases. Further, we consider inexact solvers to reduce the computational cost/complexity and memory consumption of coarse and local problems and boost the scalability of the solver. Out of our numerical experimentation, we conclude that the BDDC preconditioner is quite insensitive to inexact solvers. In particular, one cycle of algebraic multigrid (AMG) is enough to attain algorithmic scalability. Further, the clear reduction of computing time and memory requirements of inexact solvers compared to sparse direct ones makes possible to scale far beyond state-of-the-art BDDC implementations. Excellent weak scalability results have been obtained with the proposed inexact/overlapped implementation of the two-level BDDC preconditioner, up to 93,312 cores and 20 billion unknowns on JUQUEEN. Further, we have also applied the proposed setting to unstructured meshes and partitions for the pressure Poisson solver in the backward-facing step benchmark domain

Multilevel balancing domain decomposition at extreme scales

In this paper we present a fully distributed, communicator-aware, recursive, and interlevel-overlapped message-passing implementation of the multilevel balancing domain decomposition by constraints (MLBDDC) preconditioner. The implementation highly relies on subcommunicators in order to achieve the desired effect of coarse-grain overlapping of computation and communication, and communication and communication among levels in the hierarchy (namely, interlevel overlapping). Essentially, the main communicator is split into as many nonoverlapping subsets of message-passing interface (MPI) tasks (i.e., MPI subcommunicators) as levels in the hierarchy. Provided that specialized resources (cores and memory) are devoted to each level, a careful rescheduling and mapping of all the computations and communications in the algorithm lets a high degree of overlapping be exploited among levels. All subroutines and associated data structures are expressed recursively, and therefore MLBDDC preconditioners with an arbitrary number of levels can be built while re-using significant and recurrent parts of the codes. This approach leads to excellent weak scalability results as soon as level-1 tasks can fully overlap coarser-levels duties. We provide a model to indicate how to choose the number of levels and coarsening ratios between consecutive levels and determine qualitatively the scalability limits for a given choice. We have carried out a comprehensive weak scalability analysis of the proposed implementation for the three-dimensional Laplacian and linear elasticity problems on structured and unstructured meshes. Excellent weak scalability results have been obtained up to 458,752 IBM BG/Q cores and 1.8 million MPI being, being the first time that exact domain decomposition preconditioners (only based on sparse direct solvers) reach these scales. (An erratum is attached.

A three-level BDDC algorithm for Mortar discretizations

In this paper, a three-level BDDC algorithm is developed for the solutions of large sparse algebraic linear systems arising from the mortar discretization of elliptic boundary value problems. The mortar discretization is considered on geometrically non-conforming subdomain partitions. In two-level BDDC algorithms, the coarse problem needs to be solved exactly. However, its size will increase with the increase of the number of the subdomains. To overcome this limitation, the three-level algorithm solves the coarse problem inexactly while a good rate of convergence is maintained. This is an extension of previous work, the three-level BDDC algorithms for standard finite element discretization. Estimates of the condition numbers are provided for the three-level BDDC method and numerical experiments are also discussed