Non-overlapping Domain Decomposition Research Articles

The Hermite radial basis function control volume method for multi-zones problems; A non-overlapping domain decomposition algorithm

In this paper a non-overlapping non-iterative multi-domain formulation for the control volume Hermite radial basis functions (CV-HRBF) method is proposed, where the local Hermitian RBF meshless collocation method is used to satisfy a physical matching condition at the sub-domain boundaries. In addition, the robustness of the Hermite interpolation is exploited even further to apply multiple flux continuities for those cases where more than two sub-domains converge in the same point. The algorithm is first validated in one-dimensional advection diffusion problems for which an analytical solution is known. Its accuracy is compared with a classic CV approach and a local radial basis function collocation method (LRBFCM). More general applications in two and three-dimensional domains are then considered. A heat transfer problem in strongly heterogeneous materials, and a groundwater flow problem in presence of geological layers characterised by different hydraulic conductivity, are taken as engineering applications to test the capabilities of the CV-HRBF method to handle multi-zone problems. Finally, the transport of a single species is simulated in a one-dimensional channel consisting of two adjacent zones that feature different Peclet numbers.

Multipliers-free dual-primal domain decomposition methods for nonsymmetric matrices and their numerical testing

The most commonly used nonoverlapping domain decomposition algorithms, such as the FETI-DP and BDDC, require the introduction of discontinuous vector spaces. Most of the works on such methods are based on approaches that originated in Lagrange multipliers formulations. Using a theory of partial differential equations formulated in discontinuous piecewise-defined functions, introduced and developed by Herrera and his collaborators through a long time span, recently the authors have developed an approach to domain decomposition methods in which general problems with prescribed jumps are treated at the discrete level. This yields an elegant and general direct framework that permits analyzing the problems in greater detail. The algorithms derived using it have properties similar to those of well-established methods such as FETI-DP, but, in our experience, they are easier to implement. Also, they yield explicit matrix formulas that unify the different methods. Furthermore, this multipliers-free framework has permitted us to extend such formulas to make them applicable to nonsymmetric matrices. The extension of the unifying matrix formulas to nonsymmetric matrices is the subject matter of the present article. A conspicuous result is that in numerical experiments in 2D and 3D, the MF-DP algorithms for nonsymmetric matrices exhibit an efficiency of the same order as state-of-the-art algorithms for symmetric matrices, such as BDDC, FETI-DP, and MF-DP.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1262-1289, 2011

Domain decomposition methods for the neutron diffusion problem

The neutronic simulation of a nuclear reactor core is performed using the neutron transport equation, and leads to an eigenvalue problem in the steady-state case. Among the deterministic resolution methods, simplified transport ( S P N ) or diffusion approximations are often used. The MINOS solver developed at CEA Saclay uses a mixed dual finite element method for the resolution of these problems, and has shown his efficiency. In order to take into account the heterogeneities of the geometry, a very fine mesh is generally required, and leads to expensive calculations for industrial applications. In order to take advantage of parallel computers, and to reduce the computing time and the local memory requirement, we propose here two domain decomposition methods based on the MINOS solver. The first approach is a component mode synthesis method on overlapping subdomains: several eigenmodes solutions of a local problem on each subdomain are taken as basis functions used for the resolution of the global problem on the whole domain. The second approach is an iterative method based on a non-overlapping domain decomposition with Robin interface conditions. At each iteration, we solve the problem on each subdomain with the interface conditions given by the solutions on the adjacent subdomains estimated at the previous iteration. Numerical results on parallel computers are presented for the diffusion model on realistic 2D and 3D cores.

Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics

Non-overlapping domain decomposition (DD) methods provide efficient algorithms for solving time-harmonic Maxwell equations. It has been shown that the convergence of DD algorithms can be improved significantly by using high order transmission conditions. In this paper, we extend a newly developed second-order transmission condition (SOTC), which involves two second-order transverse derivatives, to facilitate fast convergence in the non-conformal DD algorithms. However, the non-conformal nature of the DD methods introduces an additional technical difficulty, which results in poor convergence in many real-life applications. To mitigate the difficulty, a corner-edge penalty method is proposed and implemented in conjunction with the SOTC to obtain truly robust solver performance. Numerical results verify the analysis and demonstrate the effectiveness of the proposed methods on a few model problems. Finally, drastically improved convergence, compared to the conventional Robin transmission condition, was observed for an electrically large problem of practical interest.

Parallel finite element simulations of incompressible viscous fluid flow by domain decomposition with Lagrange multipliers

A parallel approach to solve three-dimensional viscous incompressible fluid flow problems using discontinuous pressure finite elements and a Lagrange multiplier technique is presented. The strategy is based on non-overlapping domain decomposition methods, and Lagrange multipliers are used to enforce continuity at the boundaries between subdomains. The novelty of the work is the coupled approach for solving the velocity–pressure-Lagrange multiplier algebraic system of the discrete Navier–Stokes equations by a distributed memory parallel ILU (0) preconditioned Krylov method. A penalty function on the interface constraints equations is introduced to avoid the failure of the ILU factorization algorithm. To ensure portability of the code, a message based memory distributed model with MPI is employed. The method has been tested over different benchmark cases such as the lid-driven cavity and pipe flow with unstructured tetrahedral grids. It is found that the partition algorithm and the order of the physical variables are central to parallelization performance. A speed-up in the range of 5–13 is obtained with 16 processors. Finally, the algorithm is tested over an industrial case using up to 128 processors. In considering the literature, the obtained speed-ups on distributed and shared memory computers are found very competitive.

An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media

In this paper, we develop an efficient splitting domain decomposition method (S-DDM) for compressible contamination fluid flows in porous media over multiple block-divided sub-domains by combining the non-overlapping domain decomposition, splitting, linearization and extrapolation techniques. The proposed S-DDM iterative approach divides the large domain into multiple block sub-domains. In each time interval, the S-DDM scheme is applied to solve the water head equation, in which an efficient local multilevel scheme is used for computing the values of water head on the interfaces of sub-domains, and the splitting implicit scheme is used for computing the interior values of water head in sub-domains; and the S-DDM scheme is then proposed to solve the concentration equation by combining the upstream volume technique. Numerical experiments are performed and analyzed to illustrate the efficiency of the S-DDM iterative approach for simulating compressible contamination fluid flows in porous media. The developed method takes the excellent attractive advantages of both the non-overlapping domain decomposition and the splitting technique, and reduces computational complexities, large memory requirements and long computational durations.

An efficient time‐domain method analysis of quasiperiodic structures by a finite‐element tearing and interconnecting algorithm

AbstractBased on the finite‐element approximation and nonoverlapping domain decomposition, an efficient tearing and interconnecting algorithm of the finite‐element time‐domain method is presented. Using edge‐based finite element, the method divides the computation domain into several nonoverlapping subdomains and computes electric fields in each subdomain by solving the vector curl–curl wave equation. Adjacent subdomains are related to each other by Dirichlet and Neumann boundary condition. By use of the repetition of the quasiperiod structures, the method further reduces the memory requirement and CPU time. The time‐dependent formulation employs an unconditional stable time‐integration method based on the Newmark‐beta method. An anisotropic perfectly matched layers absorbing material terminated by the first‐order boundary condition is applied for the truncation of lattices. Numerical results demonstrate that our proposed method is extremely efficient for the time‐domain analysis of the quasiperiodic structures, such as electromagnetic band gap structures.© 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 1072–1078, 2010; Published online in Wiley InterScience (www.interscience. wiley.com). DOI 10.1002/mop.25100

Nonoverlapping Domain Decomposition with Second Order Transmission Condition for the Time-Harmonic Maxwell's Equations

Nonoverlapping domain decomposition methods have been shown to provide efficient iterative algorithms for the solution of the time-harmonic Maxwell's equations. Convergence of the algorithms depends strongly upon the nature of the transmission conditions that communicate information between adjacent subdomains. In this work, we introduce a new second order transmission condition that improves convergence. In contrast to previous high order interface conditions, the new condition uses two second order transverse derivatives to improve the convergence of evanescent modes without deteriorating the convergence of propagating modes. An analysis using a splitting of the field into traverse electric and magnetic components demonstrates the improved convergence provided by the transmission condition. Numerical experiments demonstrate the effectiveness of the algorithm.

Nonoverlapping domain decomposition methods with a simple coarse space for elliptic problems

We propose a substructuring preconditioner for solving three-dimensional 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.

Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations

We introduce nonoverlapping domain decomposition algorithms of Schwarzwaveform relaxation type for the semilinear reaction-diffusionequation. We define linear Robin and secondorder (or Ventcell) transmission conditions between the subdomains, which we prove tolead to a well defined and converging algorithm. We also propose nonlinear transmission conditions. Both types are basedon best approximation problems for the linear equation and provideefficient algorithms, as the numerical results that we present hereshow.

Highly scalable parallel domain decomposition methods with an application to biomechanics

AbstractHighly scalable parallel domain decomposition methods for elliptic partial differential equations are considered with a special emphasis on problems arising in elasticity. The focus of this survey article is on Finite Element Tearing and Interconnecting (FETI) methods, a family of nonoverlapping domain decomposition methods where the continuity between the subdomains, in principle, is enforced by the use of Lagrange multipliers. Exact onelevel and dual‐primal FETI methods as well as related inexact dual‐primal variants are described and theoretical convergence estimates are presented together with numerical results confirming the parallel scalability properties of these methods. New aspects such as a hybrid onelevel FETI/FETI‐DP approach and the behavior of FETI‐DP for anisotropic elasticity problems are presented. Parallel and numerical scalability of the methods for more than 65 000 processor cores of the JUGENE supercomputer is shown. An application of a dual‐primal FETI method to a nontrivial biomechanical problem from nonlinear elasticity, modeling arterial wall stress, is given, showing the robustness of our domain decomposition methods for such problems.

One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems

Non-overlapping domain decomposition (DD) methods with complex first order Robin-type transmission conditions (TCs) provide an efficient iterative solution for Maxwell’s equation. Unfortunately, the first order TCs do not effectively account for some eigenmodes of the system matrix, which limits the scalability of the methods. In this work, we examine two TCs with a second order transverse derivative to improve the method’s performance. A detailed convergence analysis of the two TCs is presented. We then investigate the use of the two second order TCs in non-conformal and non-overlapping one way DD methods. Numerical results illustrate the effectiveness of the proposed methods on some model problems and on several problems of practical interest.

An efficient domain decomposition method for three-dimensional parabolic problems

A non-overlapping domain decomposition algorithm to solve three-dimensional parabolic partial differential equations is presented. It has been shown in this paper that the algorithm is unconditionally stable and efficient. Spectral radii for the interface and interior region are provided. Unlike two-dimensional problem, it has been found out that estimating the values of the points of the interface in three-dimensional problem is no longer negligible.

Implementation of a mortar mixed finite element method using a Multiscale Flux Basis

This paper provides a new implementation of a multiscale mortar mixed finite element method for second order elliptic problems. The algorithm uses non-overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem, which is then solved using an iterative method. The original implementation by Arbogast, Pencheva, Wheeler, and Yotov, Multiscale Model. Simul. 2007, required solving one local Dirichlet problem on each subdomain per interface iteration. We alter this implementation by forming a Multiscale Flux Basis. This basis consists of mortar functions representing the individual flux responses for each mortar degree of freedom, on each subdomain independently. The computation of these basis functions requires solving a fixed number of Dirichlet subdomain problems. Taking linear combinations of the Multiscale Flux Basis functions replaces the need to solve any Dirichlet subdomain problems during the interface iteration. This new implementation yields the same solution as the original implementation, and is computationally more efficient in cases where the number of interface iterations is greater than the number of mortar degrees of freedom per subdomain. The gain in computational efficiency increases with the number of subdomains.

Analysis of augmented three-field macro-hybrid mixed finite element schemes

On the basis of composition duality principles, augmented three-field macro-hybrid mixed variational problems and finite element schemes are analyzed. The compatibility condition adopted here, for compositional dualization, is the coupling operator surjectivity, property that expresses in a general operator sense the Ladysenskaja-Babuska-Brezzi inf-sup condition. Variational macro-hybridization is performed under the assumption of decomposable primal and dual spaces relative to nonoverlapping domain decompositions. Then, through compositional dualization macro-hybrid mixed problems are obtained, with internal boundary dual traces as Lagrange multipliers. Also, “mass” preconditioned augmentation of three-field formulations are derived, stabilizing macro-hybrid mixed finite element schemes and rendering possible speed up of rates of convergence. Dual mixed incompressible Darcy flow problems illustrate the theory throughout the paper.

Parallel Iterative Substructuring in Structural Mechanics

Finite Element Tearing and Interconnecting (FETI) methods are a family of nonoverlapping domain decomposition methods which have been proven to be robust and parallel scalable for a variety of elliptic partial differential equations. Here, an introduction to the classical onelevel FETI methods is given, as well as to the more recent dual-primal FETI methods and some of their variants. With the advent of modern parallel computers with thousands of processors, certain inexact components are needed in these methods to maintain scalability. An introduction to a recent class of inexact dual-primal FETI methods is presented. Scalability results for an elasticity problem using 65 536 processor cores of the JUGENE supercomputer at Forschungszentrum Julich show the potential of these methods. A hyperelastic problem from biomechanics is presented as an application of the methods to nonlinear finite element analysis.

Preconditioned Formulation of FE-BI Equations With Domain Decomposition Method for Calculation of Electromagnetic Scattering From Cavities

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A new formulation of finite element-boundary integral (FE-BI) equations with non-overlapping and non-conforming domain decomposition method (DDM) is proposed to calculate electromagnetic scattering from cavities. The computational domain is divided into two parts: cavity and non-cavity domains. Tetrahedral elements with higher order and lower order vector basis functions are used in the cavity and non-cavity domains, respectively, and they are not conformal on the interface between two sub-domains. The application of Robin transmission condition at the aperture of the cavity relates the fields in the interior and exterior regions of the cavity. In addition, an effective preconditioner is designed to speed up the solution process of iterative solvers. Numerical results obtained show that the proposed method is suitable for this class of problems, and the preconditioner is highly effective to reduce the number of iterations for the solution. </para>

An Explicit-Implicit Predictor-Corrector Domain Decomposition Method for Time Dependent Multi-Dimensional Convection Diffusion Equations

The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms. In this work, we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique, modified upwind differences with explicitimplicit coupling, the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy. Moreover, for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictorcorrector or stabilized schemes. These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments. AMS subject classifications: 65N30, 65F10

Steady filtration problems with seawater intrusion: macro-hybrid primal mixed variational analysis

Macro-hybrid penalized, primal mixed continuous and discrete variational formulations, for steady filtration problems, with seawater intrusion, are studied. This is a mixed version of our previous paper on macro-hybrid penalized approximations (G. Alduncin, J. Esquivel-Avila, and N. Vera-Guzman, Steady filtration problems with seawater intrusion: Macro-hybrid penalized finite element approximations, Int. J. Numer. Meth. Fluids (2005) 49, pp. 935–957). Penalized pressure–velocity mixed variational formulations are introduced for non-overlapping domain decompositions, with vertical interfaces, of sections of coastal aquifers. Well-posedness and stability conditions are established at the continuous and discrete levels. Macro-hybrid primal mixed internal approximations are defined on independent subdomain grids, with transmission conditions imposed in a dual variational sense. Parallel relaxation penalty-duality algorithms are discussed from fixed-point characterizations, for iterative numerical resolution.

An efficient explicit/implicit domain decomposition method for convection‐diffusion equations

AbstractA nonoverlapping domain decomposition method for some time‐dependent convection‐diffusion equations is presented. It combines predictor‐corrector technique, modified upwind differences with explicit/implicit coupling to provide intrinsic parallelism, and unconditional stability while improving the accuracy. Both rigorous mathematical analysis and numerical experiments are carried out to illustrate the stability, accuracy, and parallelism. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010