Error reduction for higher derivatives of Chebyshev collocation method using preconditioning and domain decomposition

A new preconditioning method is investigated to reduce the roundoff error in computing derivatives using Chebyshev collocation methods(CCM). Using this preconditioning causes ratio of roundoff error of preconditioning method and CCM becomes small when N gets large. Also for accuracy enhancement of differentiation we use a domain decomposition approach. Error analysis shows that for this domain decomposition method error reduces proportional to the length of subintervals. Numerical results show that using domain decomposition and preconditioning simultaneously, gives super accurate approximate values for first derivative of the function and good approximate values for moderately high derivatives.     

Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition

Summary. Wavelet methods allow to combine high order accuracy, multilevel preconditioning techniques and adaptive approximation, in order to solve efficiently elliptic operator equations. One of the main difficulty in this context is the efficient treatment of non-homogeneous boundary conditions. In this paper, we propose a strategy that allows to append such conditions in the setting of space refinement (i.e. adaptive) discretizations of second order problems. Our method is based on the use of compatible multiscale decompositions for both the domain and its boundary, and on the possibility of characterizing various function spaces from the numerical properties of these decompositions. In particular, this allows the construction of a lifting operator which is stable for a certain range of smoothness classes, and preserves the compression of the solution in the wavelet basis. An explicit construction of the wavelet bases and the lifting is proposed on fairly general domains, based on conforming domain decomposition techniques. Received November 2, 1998 / Published online April 20, 2000     

A coarse–fine‐mesh stabilization for an alternating Schwarz domain decomposition method

Domain decomposition methods can be solved in various ways. In this paper, domain decomposition in strips is used. It is demonstrated that a special version of the Schwarz alternating iteration method coupled with coarse–fine‐mesh stabilization leads to a very efficient solver, which is easy to implement and has a behavior nearly independent of mesh and problem parameters. The novelty of the method is the use of alternating iterations between odd‐ and even‐numbered subdomains and the replacement of the commonly used coarse‐mesh stabilization method with coarse–fine‐mesh stabilization.     

Space-time domain decomposition for parabolic problems

We analyze a space-time domain decomposition iteration, for a model advection diffusion equation in one and two dimensions. The discretization of this iteration is the block red-black variant of the waveform relaxation method, and our analysis provides new convergence results for this scheme. The asymptotic convergence rate is super-linear, and it is governed by the diffusion of the error across the overlap between subdomains. Hence, it depends on both the size of this overlap and the diffusion coefficient in the equation. However it is independent of the number of subdomains, provided the size of the overlap remains fixed. The convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases. The duration of the transient linear regime is proportional to the length of the time window. For advection dominated problems, the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases. Moreover, it is independent of the size of the time window and of the number of subdomains. Numerical calculations illustrate our analysis.     

Variational theory and domain decomposition for nonlocal problems

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented.     

Vector domain decomposition schemes for parabolic equations

A new class of domain decomposition schemes for finding approximate solutions of timedependent problems for partial differential equations is proposed and studied. A boundary value problem for a second-order parabolic equation is used as a model problem. The general approach to the construction of domain decomposition schemes is based on partition of unity. Specifically, a vector problem is set up for solving problems in individual subdomains. Stability conditions for vector regionally additive schemes of first- and second-order accuracy are obtained.     

Balancing domain decomposition for nonconforming plate elements

Summary. In this paper the balancing domain decomposition method is extended to nonconforming plate elements. The condition number of the preconditioned system is shown to be bounded by , where H measures the diameters of the subdomains, h is the mesh size of the triangulation, and the constant C is independent of H, h and the number of subdomains. Received August 14, 1997     

A hybrid graph-genetic method for domain decomposition

A new method is developed for finite element (FE) domain decomposition. This method employs a hybrid graph-genetic algorithm for graph partitioning and correspondingly bisects finite element (FE) meshes.

A weighted incidence graph is first constructed for the FE mesh, denoted by G₀. A coarsening process is then performed using heavy-edge matching. A sequence of such operations is employed in “n” steps, which leads to the formation of G_n with a size suitable for genetic algorithm applications.

Hereafter, G_n is bisected using conventional genetic algorithm. The shortest route tree algorithm is used for the formation of the initial population in genetic algorithm. Then an uncoarsening process is performed and the results are transferred to the graph G_n−1. The initial population for genetic algorithm on G_n−1is constructed from the results of G_n. This process is repeated until G₀ is obtained in the uncoarsening operation. Employing the properties of G₁, the graph G₀ is bisected by the genetic algorithm.     

Algebraic domain decomposition solver for linear elasticity

We generalize the overlapping Schwarz domain decomposition method to problems of linear elasticity. The convergence rate independent of the mesh size, coarse-space size, Korn's constant and essential boundary conditions is proved here. Abstract convergence bounds developed here can be used for an analysis of the method applied to singular perturbations of other elliptic problems.     

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     

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain are given. These procedures use implicit Galerkin method in the subdomains and simple explicit flux calculation on the interdomain boundaries by integral mean method or extrapolation method to predict the inner‐boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the subdomains and across interboundaries. The explicit nature of the flux prediction induces a time‐step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. L²‐norm error estimates are derived for these procedures. Compared with the work of Dawson and Dupont [Math Comp 58 (1992), 21–35], these L²‐norm error estimates avoid the loss of H^?1/2 factor. Experimental results are presented to confirm the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Coercive domain decomposition algorithms for advection-diffusion equations and systems

Two families of non-overlapping coercive domain decomposition methods are proposed for the numerical approximation of advection-dominated advection-diffusion equations and systems. Convergence is proven for both the continuous and the discrete problem. The rate of convergence of the first method is shown to be independent of the total number of degrees of freedom. Several numerical results are presented, showing the efficiency and robustness of the proposed interative algorithms.     

Operator trigonometry of preconditioning,domain decomposition,sparse approximate inverses,successive overrelaxation,minimum residual schemes

This paper extends the relationship between the author's operator trigonometry and convergence rates and other properties of important iterative methods. A new interesting trigonometric preconditioning lemma is given. The general relationship between domain decomposition methods and the operator trigonometry is established. A new basic conceptual link between sparse approximate inverse algorithms and the operator trigonometry is observed. A new underlying fundamental inherent trigonometry of the classical successive over‐relaxation scheme is exposed. Some improved trigonometric interpretations of minimum residual schemes are mentioned. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Parallel Galerkin domain decomposition procedures for wave equation

Parallel Galerkin domain decomposition procedures for wave equation are given. These procedures use implicit method in the sub-domains and simple explicit flux calculation on the inter-boundaries of sub-domains by integral mean method or extrapolation method. Thus, the parallelism can be achieved by these procedures. The explicit nature of the flux prediction induces a time step constraint that is necessary to preserve the stability. L²-norm error estimates are derived for these procedures. Experimental results are presented to confirm the theoretical results.     

A domain decomposition preconditioner for hermite collocation problems

We propose a preconditioning method for linear systems of equations arising from piecewise Hermite bicubic collocation applied to two‐dimensional elliptic PDEs with mixed boundary conditions. We construct an efficient, parallel preconditioner for the GMRES method. The main contribution of the article is a novel interface preconditioner derived in the framework of substructuring and employing a local Hermite collocation discretization for the interface subproblems based on a hybrid fine‐coarse mesh. Interface equations based on this mesh depend only weakly on unknowns associated with subdomains. The effectiveness of the proposed method is highlighted by numerical experiments that cover a variety of problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 135–151, 2003     

The domain decomposition method for parabolic transmission problems

An estimate of the rate of convergence is given for the domain decomposition method for the second-order parabolic transmission problem. A brief discussion of the method and some of its applications are presented.