Substructuring preconditioners for the three fields domain decomposition method

We study a class of preconditioners based on substructuring, for the discrete Steklov-Poincaré operator arising in the three fields formulation of domain decomposition in two dimensions. Under extremely general assumptions on the discretization spaces involved, an upper bound is provided on the condition number of the preconditioned system, which is shown to grow at most as ( and denoting, respectively, the diameter and the discretization mesh-size of the subdomains). Extensive numerical tests--performed on both a plain and a stabilized version of the method--confirm the optimality of such bound.

     

A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems

In recent years, competitive domain-decomposed preconditioned iterative techniques of Krylov-Schwarz type have been developed for nonsymmetric linear elliptic systems. Such systems arise when convection-diffusion-reaction problems from computational fluid dynamics or heat and mass transfer are linearized for iterative solution. Through domain decomposition, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow effective solution on parallel machines. A central question is how to choose these small problems and how to arrange the order of their solution. Different specifications of decomposition and solution order lead to a plethora of algorithms possessing complementary advantages and disadvantages. In this report we compare several methods, including the additive Schwarz algorithm, the classical multiplicative Schwarz algorithm, an accelerated multiplicative Schwarz algorithm, the tile algorithm, the CGK algorithm, the CSPD algorithm, and also the popular global ILU-family of preconditioners, on some nonsymmetric or indefinite elliptic model problems discretized by finite difference methods. The preconditioned problems are solved by the unrestarted GMRES method. A version of the accelerated multiplicative Schwarz method is a consistently good performer.     

Domain decomposition methods for solving the burgers equation

This article presents some results of numerical tests of solving the two-dimensional non-linear unsteady viscous Burgers equation. We have compared the known convergence and parallel performance properties of the additive Schwarz domain decomposition method with or without a coarse grid for the model Poisson problem with those obtained by experiments for the Burgers problem.     

Augmented hybrid finite element method for domain decomposition

We present an Augmented Hybrid Finite Element Method for Domain Decompositon. In this method, finite element approximations are defined independently on each subdomain and do not match at interface. This dows the user to mda local change of design, of meshes on one aubdomain without modifying other subdomains. Optimal reaults are obtained for a second-order model problem.     

Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systems

A variant of balancing domain decomposition method by constraints (BDDC) is proposed for solving a class of indefinite systems of linear equations of the form (K?σ²M)u=f, which arise from solving eigenvalue problems when an inverse shifted method is used and also from the finite element discretization of Helmholtz equations. Here, both K and M are symmetric positive definite. The proposed BDDC method is closely related to the previous dual–primal finite element tearing and interconnecting method (FETI‐DP) for solving this type of problems (Appl. Numer. Math. 2005; 54 :150–166), where a coarse level problem containing certain free‐space solutions of the inherent homogeneous partial differential equation is used in the algorithm to accelerate the convergence. Under the condition that the diameters of the subdomains are small enough, the convergence rate of the proposed algorithm is established, which depends polylogarithmically on the dimension of the individual subdomain problems and which improves with a decrease of the subdomain diameters. These results are supported by numerical experiments of solving a two‐dimensional problem. Copyright © 2009 John Wiley & Sons, Ltd.     

New eighth-order iterative methods for solving nonlinear equations

In this paper, three new families of eighth-order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods. Per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative. This implies that the efficiency index of the developed methods is 1.682, which is optimal according to Kung and Traub’s conjecture [7] for four function evaluations per iteration. Notice that Bi et al.’s method in [2] and [3] are special cases of the developed families of methods. In this study, several new examples of eighth-order methods with efficiency index 1.682 are provided after the development of each family of methods. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

A non-conforming domain decomposition for Raviart-Thomas vector fields in three dimensions

In this paper we are concerned with a domain decomposition method with nonmatching grids for Raviart-Thomas finite elements. In this method, the normal complement of the resulting approximation is not continuous across the interface. To handle such non-conformity, a new matching condition will be introduced. Such matching condition still     

Inexact constraint preconditioners for linear systems arising in interior point methods

Issues of indefinite preconditioning of reduced Newton systems arising in optimization with interior point methods are addressed in this paper. Constraint preconditioners have shown much promise in this context. However, there are situations in which an unfavorable sparsity pattern of Jacobian matrix may adversely affect the preconditioner and make its inverse representation unacceptably dense hence too expensive to be used in practice. A remedy to such situations is proposed in this paper. An approximate constraint preconditioner is considered in which sparse approximation of the Jacobian is used instead of the complete matrix. Spectral analysis of the preconditioned matrix is performed and bounds on its non-unit eigenvalues are provided. Preliminary computational results are encouraging.     

The “Gauss-Seidelization” of iterative methods for solving nonlinear equations in the complex plane

In this paper we introduce a process we have called “Gauss-Seidelization” for solving nonlinear equations. We have used this name because the process is inspired by the well-known Gauss-Seidel method to numerically solve a system of linear equations. Together with some convergence results, we present several numerical experiments in order to emphasize how the Gauss-Seidelization process influences on the dynamical behavior of an iterative method for solving nonlinear equations.     

Real valued iterative methods for solving complex symmetric linear systems

Complex valued systems of equations with a matrix R + 1S where R and S are real valued arise in many applications. A preconditioned iterative solution method is presented when R and S are symmetric positive semi‐definite and at least one of R, S is positive definite. The condition number of the preconditioned matrix is bounded above by 2, so only very few iterations are required. Applications when solving matrix polynomial equation systems, linear systems of ordinary differential equations, and using time‐stepping integration schemes based on Padé approximation for parabolic and hyperbolic problems are also discussed. Numerical comparisons show that the proposed real valued method is much faster than the iterative complex symmetric QMR method. Copyright © 2000 John Wiley & Sons, Ltd.     

The Adomian decomposition method for solving the Boussinesq equation arising in water wave propagation

We will consider the application of the Adomian decomposition method to approximate the solution of the Boussinesq equation. Both the well‐posed and the ill‐posed cases will be considered. The results obtained will be compared to the theoretical solution for single soliton wave. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

A new family of eighth-order iterative methods for solving nonlinear equations

A family of eighth-order iterative methods with four evaluations for the solution of nonlinear equations is presented. Kung and Traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2^n-1. The new family of eighth-order methods agrees with the conjecture of Kung-Traub for the case n=4. Therefore this family of methods has efficiency index equal to 1.682. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

New iterative methods for solving linear systems

In this paper, we introduce some new iterative methods to solve linear systems \(Ax=b\}. We  show that these methods, comparing to the classical Jacobi or Gauss-Seidel method, can be applied to more systems and have faster convergence.     

Node adaptive domain decomposition method by radial basis functions

During the last years, there has been increased interest in developing efficient radial basis function (RBF) algorithms to solve partial differential problems of great scale. In this article, we are interested in solving large PDEs problems, whose solution presents rapid variations. Our main objective is to introduce a RBF dynamical domain decomposition algorithm which simultaneously performs a node adaptive strategy. This algorithm is based on the RBFs unsymmetric collocation setting. Numerical experiments performed with the multiquadric kernel function, for two stationary problems in two dimensions are presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems

A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and estimates for the pressure. In addition, it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces.

     

Differential quadrature domain decomposition method for problems on a triangular domain

The differential quadrature method (DQM) has been studied for years and it has been shown by many researchers that the DQM is an attractive numerical method with high efficiency and accuracy. The conventional DQM is mostly effective for one‐dimensional and multidimensional problems with geometrically regular domains. But to deal with problems on a triangular domain, we will meet difficulties. In this article we will study how to solve problems on a triangular domain by using DQM combined with the domain decomposition method (DDM). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A hybrid domain decomposition method based on aggregation

A new two‐level black‐box preconditioner based on the hybrid domain decomposition technique is proposed and studied. The preconditioner is a combination of an additive Schwarz preconditioner and a special smoother. The smoother removes dependence of the condition number on the number of subdomains and variations of the diffusion coefficient and leaves minor sensitivity to the problem size. The algorithm is parallel and pure algebraic which makes it a convenient framework for the construction parallel black‐box preconditioners on unstructured meshes. Copyright © 2004 John Wiley & Sons, Ltd.     

A new family of iterative methods for solving system of nonlinear algebric equations

Homotopy perturbation method (HPM) is applied to construct a new iterative method for solving system of nonlinear algebric equations. Comparison of the result obtained by the present method with that obtained by revised Adomian decomposition method [Hossein Jafari, Varsha Daftardar-Gejji, Appl. Math. Comput. 175 (2006) 1–7] reveals that the accuracy and fast convergence of the new method.     

An optimization‐based domain decomposition method for the Boussinesq equations

We study an optimization based domain decomposition method for the Boussinesq equations governing natural convection problems. Domain decomposition is cast into a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the interface between solid and fluid subdomains. We showthat solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined. Finite element approximations of the solutions of the optimality system are examined. The domain decomposition method is also reformulated as a nonlinear least‐squares problem and the results of some numerical experiments are given. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 1–25, 2002     

A domain decomposition method using efficient interface-acting preconditioners

The conjugate gradient boundary iteration (CGBI) is a domain decomposition method for symmetric elliptic problems on domains with large aspect ratio. High efficiency is reached by the construction of preconditioners that are acting only on the subdomain interfaces. The theoretical derivation of the method and some numerical results revealing a convergence rate of 0.04-0.1 per iteration step are given in this article. For the solution of the local subdomain problems, both finite element (FE) and spectral Chebyshev methods are considered.