BDDC Algorithm Research Articles

Robust BDDC algorithms for the Brinkman problem with HDG discretizations

The balancing domain decomposition by constraints methods (BDDC) have been applied to solve the saddle point problem arising from a hybridizable discontinuous Galerkin (HDG) discretization for the Brinkman equations. Edge/face average constraints are enforced across the subdomain interface for each velocity component to ensure that the BDDC preconditioned conjugate gradient (CG) iterations stay in a special subspace, where the reduced system from the original saddle point problem is positive definite. The condition number is proved to be uniformly bounded under both Stokes and Darcy dominant cases with a deluxe scaling for the parameters with jumps only across the subdomain interface. When the parameters are highly discontinuous with large jumps inside each subdomains, additional adaptively chosen primal constraints, obtained by solving local generalized eigenvalue problems, are introduced to control the condition numbers. Numerical experiments confirm the theory.

BDDC algorithms for advection-diffusion problems with HDG discretizations

In this paper, a preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. The preconditioner is the balancing domain decomposition methods (BDDC), one of the most popular nonoverlapping domain decomposition methods. For large viscosity, if the subdomain size is small enough, the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size. The convergence deteriorates when the viscosity decreases. These results are similar to those with the standard finite element discretizations. Numerical results of two examples in two dimensions are provided to confirm the theory.

Adaptive-multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems

In this paper, we are concerned with the weighted plane wave least-squares (PWLS) method for three-dimensional Helmholtz equations, and develop the multi-level adaptive BDDC algorithms for solving the resulting discrete system. In order to form the adaptive coarse components, the local generalized eigenvalue problems for each common face and each common edge are carefully designed. The condition number of the two-level adaptive BDDC preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of faces and edges per subdomain and the number of subdomains sharing a common edge. The efficiency of these algorithms is illustrated on a benchmark problem. The numerical results show the robustness of our two-level adaptive BDDC algorithms with respect to the wave number, the number of subdomains and the mesh size, and illustrate that our multi-level adaptive BDDC algorithm can reduce the scale of the coarse problem and can be used to solve large wave number problems efficiently.

Adaptive BDDC algorithms for the system arising from plane wave discretization of Helmholtz equations

SummaryBalancing domain decomposition by constraints algorithms with adaptive primal constraints are developed in a concise variational framework for the weighted plane wave least‐squares discretization of Helmholtz equations with high and various wave numbers. The unknowns to be solved in this preconditioned system are defined on elements rather than vertices or edges, which are different from the well‐known discretizations such as the classical finite element method. Through choosing suitable “interface” and appropriate primal constraints with complex coefficients and introducing some local techniques, we developed a two‐level adaptive balancing domain decomposition by constraints algorithm for the plane wave least‐squares discretization, and the condition number of the preconditioned system is proved to be bounded above by a user‐defined tolerance and a constant that is only dependent on the maximum number of interfaces per subdomain. A multilevel algorithm is also attempted to resolve the bottleneck in large‐scale coarse problem. Numerical results are carried out to confirm the theoretical results and illustrate the efficiency of the proposed algorithms.

BDDC and FETI-DP preconditioners with adaptive coarse spaces for three-dimensional elliptic problems with oscillatory and high contrast coefficients

BDDC and FETI-DP algorithms are developed for three-dimensional elliptic problems with adaptively enriched coarse components. It is known that these enriched components are necessary in the development of robust preconditioners. To form the adaptive coarse components, carefully designed generalized eigenvalue problems are introduced for each faces and edges, and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance λTOL. Upper bounds for condition numbers of the preconditioned systems are shown to be CλTOL, with the constant C depending only on the maximum number of edges and faces per subdomain, and the maximum number of subdomains sharing an edge. Numerical results are presented to test the robustness of the proposed approach.

BDDC Algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart-Thomas vector fields

A BDDC domain decomposition preconditioner is defined by a coarse component, expressed in terms of primal constraints, a weighted average across the interface between the subdomains, and local components given in terms of solvers of local subdomain problems. BDDC methods for vector field problems discretized with Raviart-Thomas finite elements are introduced. The methods are based on a new type of weighted average and an adaptive selection of primal constraints developed to deal with coefficients with high contrast even inside individual subdomains. For problems with very many subdomains, a third level of the preconditioner is introduced. Under the assumption that the subdomains are all built from elements of a coarse triangulation of the given domain, and that the material parameters are constant in each subdomain, a bound is obtained for the condition number of the preconditioned linear system which is independent of the values and the jumps of these parameters across the interface between the subdomains. Numerical experiments, using the PETSc library, are also presented which support the theory and show the effectiveness of the algorithms even for problems not covered by the theory. Included are also experiments with Brezzi-Douglas-Marini finite element approximations.

A BDDC algorithm with deluxe scaling for $H(curl)$ in two dimensions with irregular subdomains

A BDDC algorithm with deluxe scaling for $H(curl)$ in two dimensions with irregular subdomains

A BDDC Algorithm with Deluxe Scaling for Three‐Dimensional H(curl) Problems

AbstractIn this paper, we present and analyze a BDDC algorithm for a class of elliptic problems in the three‐dimensional H(curl) space. Compared with existing results, our condition number estimate requires fewer assumptions and also involves two fewer powers of log(H/h), making it consistent with optimal estimates for other elliptic problems. Here, H/h is the maximum of Hi/hi over all subdomains, where Hi and hi are the diameter and the smallest element diameter for the subdomain Ωi. The analysis makes use of two recent developments. The first is a new approach to averaging across the subdomain interfaces, while the second is a new technical tool that allows arguments involving trace classes to be avoided. Numerical examples are presented to confirm the theory and demonstrate the importance of the new averaging approach in certain cases. © 2016 Wiley Periodicals, Inc.

A BDDC algorithm for the mortar-type rotated Q 1 FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a BDDC preconditioner for the mortar-type rotated finite element method for second order elliptic partial differential equations with piecewise but discontinuous coefficients. We construct an auxiliary discrete space and build our algorithm on an equivalent auxiliary problem, and we present the BDDC preconditioner based on this constructed discrete space. Meanwhile, in the framework of the standard additive Schwarz methods, we describe this method by a complete variational form. We show that our method has a quasi-optimal convergence behavior, i.e., the condition number of the preconditioned problem is independent of the jumps of the coefficients, and depends only logarithmically on the ratio between the subdomain size and the mesh size. Numerical experiments are presented to confirm our theoretical analysis. MSC:65N55, 65N30.

Adaptive-Multilevel BDDC and its parallel implementation

We combine the adaptive and multilevel approaches to the BDDC and formulate a method which allows an adaptive selection of constraints on each decomposition level. We also present a strategy for the solution of local eigenvalue problems in the adaptive algorithm using the LOBPCG method with a preconditioner based on standard components of the BDDC. The effectiveness of the method is illustrated on several engineering problems. It appears that the Adaptive-Multilevel BDDC algorithm is able to effectively detect troublesome parts on each decomposition level and improve convergence of the method. The developed open-source parallel implementation shows a good scalability as well as applicability to very large problems and core counts.

A two-level nonoverlapping Schwarz algorithm for the Stokes problem: Numerical study

A general framework of a two-level nonoverlapping Schwarz algorithm for the Stokes problem is developed by relaxing average zero condition on pressure unknowns. This framework allows both discontinuous and continuous pressure finite element spaces. The coarse problem is built by algebraic manipulation after selecting appropriate primal unknowns just like in BDDC algorithms. Performance of the suggested algorithm is presented depending on the selection of finite elements and primal unknowns. Under the same set of primal unknowns, the algorithm for the case with discontinuous pressure functions outperforms one with continuous pressure functions. For the two-dimensional Stokes problem, the algorithm with a set of primal unknowns consisting of velocity unknowns at corners, averages of velocity components over common edges, and pressure unknowns at corners presents good scalability when continuous pressure test functions are used. In both two- and three-dimensional Stokes problems, an improvement can be made for the case with continuous pressure test functions by applying the suggested algorithm to the interface problem, which is obtained by eliminating velocity unknowns and pressure unknowns interior to each subdomains.

A Unified Analysis of Balancing Domain Decomposition by Constraints for Discontinuous Galerkin Discretizations

The BDDC algorithm is extended to a large class of discontinuous Galerkin (DG) discretizations of second order elliptic problems. An estimate of $C(1+\log(H/h))^2$ is obtained for the condition number of the preconditioned system where $C$ is a constant independent of $h$ or $H$ or large jumps in the coefficient of the problem. Numerical simulations are presented which confirm the theoretical results. A key component for the development and analysis of the BDDC algorithm is a novel perspective presenting the DG discretization as the sum of elementwise „local” bilinear forms. The elementwise perspective allows for a simple unified analysis of a variety of DG methods and leads naturally to the appropriate choice for the subdomainwise local bilinear forms. Additionally, this new perspective enables a connection to be drawn between the DG discretization and a related continuous finite element discretization to simplify the analysis of the BDDC algorithm.

A three-level BDDC algorithm for a saddle point problem

BDDC algorithms have previously been extended to the saddle point problems arising from mixed formulations of elliptic and incompressible Stokes problems. In these two-level BDDC algorithms, all iterates are required to be in a benign space, a subspace in which the preconditioned operators are positive definite. This requirement can lead to large coarse problems, which have to be generated and factored by a direct solver at the beginning of the computation and they can ultimately become a bottleneck. An additional level is introduced in this paper to solve the coarse problem approximately and to remove this difficulty. This three-level BDDC algorithm keeps all iterates in the benign space and the conjugate gradient methods can therefore be used to accelerate the convergence. This work is an extension of the three-level BDDC methods for standard finite element discretization of elliptic problems and the same rate of convergence is obtained for the mixed formulation of the same problems. Estimate of the condition number for this three-level BDDC methods is provided and numerical experiments are discussed.

FETI-DP, BDDC, and block Cholesky methods

The FETI-DP and BDDC algorithms are reformulated using Block Cholesky factorizations, an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations. Instead of introducing Lagrange multipliers to enforce the coarse level, primal continuity constraints in these algorithms, a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom. With the new formulation of these algorithms, a simplified proof is provided that the spectra of a pair of FETI-DP and BDDC algorithms, with the same set of primal constraints, are essentially the same. Numerical experiments for a two-dimensional Laplace's equation also confirm this result. Copyright © 2005 John Wiley & Sons, Ltd.