Domain Decomposition Method For Problems Research Articles

Convergence of the Neumann parallel scheme of the domain decomposition method for problems of frictionless contact between several elastic bodies

Based on the variational formulation and penalty method, we have considered the Neumann parallel scheme of the domain decomposition method for the solution of problems of one-sided contact between three-dimensional elastic bodies. We have shown the existence and uniqueness of a solution of the variational problem with penalty and convergence in the penalty parameter. The convergence of this scheme has been proved, and the optimal value of iteration parameter has been determined.

A domain decomposition method for problems with structural heterogeneities on the interface: Application to a passenger ship

In this article, we extend a domain decomposition method, based on the FETI-DP linear solver, to applications such as passenger ship analysis. More generally, the method is designed for large-scale elastic analysis of a structure which exhibits geometrical and structural heterogeneities, such as plate and stiffener assemblies in presence of structural details. The problem of the structural heterogeneities on the subdomain interfaces, arising from the presence of stiffeners or elastic joints on these interfaces, is addressed. A suited interface connection between subdomains modeled with plate elements in the case of a 3D assembling is proposed and tested. The selection of an efficient preconditioner is presented, and the performances and results are discussed in terms of convergence rate for several examples.

Optimized interface conditions in domain decomposition methods for problems with extreme contrasts in the coefficients

When the coefficients of a problem have jumps of several orders of magnitude and are anisotropic, many preconditioners and domain decomposition methods (DDM) suffer from plateaus in the convergence due to the presence of very small isolated eigenvalues in the spectrum of the preconditioned linear system. One way to improve the preconditioner is to use a linear algebra technique called deflation, or very similarly coarse grid corrections. In both cases, it is necessary to identify and compute, at least approximately, all the eigenvectors corresponding to the “bad” eigenvalues. In the framework of DDM, we propose a way to design interface conditions so that convergence is fast and does not have any plateau. The method relies only on the knowledge of the smallest and largest eigenvalues of an auxiliary matrix. The eigenvectors are not used. The method relies on van der Sluis’ result on a quasi-optimal diagonal preconditioner for a symmetric positive definite matrix. It is then possible to design Robin interface conditions using only one real parameter for the entire interface. By adding a second real parameter and more general interface conditions, it is possible to take into account highly heterogeneous and anisotropic media. Numerical results are given and compared with other approaches.

Efficient domain decomposition methods for elliptic problems arising from flows in heterogeneous porous media

In this paper we study domain decomposition methods for solving some elliptic problem arising from flows in heterogeneous porous media. Due to the multiple scale nature of the elliptic coefficients arising from the heterogeneous formations, the construction of efficient domain decomposition methods for these problems requires a coarse solver which is adaptive to the fine scale features, [4]. We propose the use of a multiscale coarse solver based on a finite volume – finite element formulation. The resulting domain decomposition methods seem to induce a convergence rate nearly independent of the aspect ratio of the extreme permeability values within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the preconditioner through numerical experiments which include problems with multiple scale coefficients, as well as problems with continuous scales.

Differential quadrature domain decomposition method for problems on a triangular domain

AbstractThe 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 Schwarz alternating algorithm for elliptic boundary value problems in an infinite domain with a concave angle

In this paper, we investigate a Schwarz alternating algorithm for elliptic boundary value problems in an infinite domain with a concave angle, which is an overlapping domain decomposition method based on natural boundary reduction (NBR). The algorithm is constructed, and its convergence is discussed. The finite element and natural boundary element method are alternatively applied to solve a bounded subdomain and a typical unbounded subdomain. It has overcome the difficulty met in standard domain decomposition method for problems with unbounded domains. The convergence rate is analyzed in details for a typical domain. Finally, some numerical examples are presented to show the effectiveness of this method.

The hp mortar domain decomposition method for problems in fluid mechanics

AbstractWe consider the incompressible Stokes equations in primal velocity–pressure variables and present the mortar finite element formulation for this problem. The local approximation within each subdomain is designed using divergence stable hp‐mixed finite elements. The velocity is computed in a mortared space while the pressure is not subjected to any constraints across the interfaces. Our computational results show that the mortar finite element method for the Stokes problem satisfies similar rates of convergence as the conforming finite element method, in the presence of various h, p and hp discretizations (including the case of exponential hp convergence over geometric meshes). We also present the numerical results for the Lagrange multiplier when the method is implemented as a mixed method. Copyright © 2002 John Wiley & Sons, Ltd.

An iterative domain decomposition method for problems of mathematical physics. III

An iterative domain decomposition method for problems of mathematical physics. III

Adaptive Pseudo-Spectral Domain Decomposition and the Approximation of Multiple Layers

When Chebyshev pseudo-spectral methods are used with domain decomposition procedures in the numerical solution of partial differential equations, the use of multiple domains can significantly affect the accuracy of the approximation. This is particularly true when the solution exhibits layer type behavior, i.e., there are narrow regions of rapid variation. Accuracy may be enhanced if the interfaces between adjacent subdomains are such that large gradients occur near the interfaces, while accuracy can be degraded if the rapid variations occur in the interior of the subdomains. The use of appropriate mappings within each subdomain can improve the accuracy of the approximation by choosing mappings so that the transformed function is more readily approximated by a low order polynomial. The particular choice of mappings, however, depends critically on whether the solution exhibits boundary layer or interior layer behavior within each subdomain. We analyze the relationship between interface location and mappings required to obtain an efficient approximation of such functions. We compare two strategies, both based on constructing subdomains so that each subdomain contains only one layer. In the first strategy interface locations are chosen so that the rapid variations occur as interior layers and mappings are employed which enhance the resolution of such layers (strategy I for interior). In the second strategy interface locations are chosen so that rapid variations occur as boundary layers and mappings are employed which enhance resolution of boundary layers (strategy B for boundary). Both strategies lead to adaptive domain decomposition procedures based entirely on the locations of the layers. We demonstrate that strategy B offers superior accuracy for a given computational effort and employ this strategy in developing an adaptive domain decomposition method for problems with multiple layers. Both strategies are comparable regarding spectral radii of the resulting matrices, and we conclude that domain decomposition itself cannot result in larger stable time steps when accuracy of the approximation of the layer is considered. The adaptive domain decomposition method is illustrated by the computation of both axisymmetric and cellular flames with a sequential reaction mechanism involving two reaction zones.

Domain Decomposition Methods for Problems with Partial Refinement

In this paper, a flexible mesh refinement strategy for the approximation of solutions of elliptic boundary value problems is considered. The main purpose of the paper is the development of preconditioners for the resulting discrete system of algebraic equations. These techniques lead to efficient computational procedures in serial as well as parallel computing environments. The preconditioners are based on overlapping domain decomposition and involve solving (or preconditioning) subproblems on regular subregions. It is proven that the iteration schemes converge to the discrete solution at a rate which is independent of the mesh parameters in the case of two spatial dimensions. The estimates proved for the iterative convergence rate in three dimensions are somewhat weaker. The results of numerical experiments illustrating the theory are also presented.