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Domain Decomposition (domain + decomposition)
Terms modified by Domain Decomposition Selected AbstractsA least square extrapolation method for the a posteriori error estimate of the incompressible Navier Stokes problemINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2005M. Garbey Abstract A posteriori error estimators are fundamental tools for providing confidence in the numerical computation of PDEs. To date, the main theories of a posteriori estimators have been developed largely in the finite element framework, for either linear elliptic operators or non-linear PDEs in the absence of disparate length scales. On the other hand, there is a strong interest in using grid refinement combined with Richardson extrapolation to produce CFD solutions with improved accuracy and, therefore, a posteriori error estimates. But in practice, the effective order of a numerical method often depends on space location and is not uniform, rendering the Richardson extrapolation method unreliable. We have recently introduced (Garbey, 13th International Conference on Domain Decomposition, Barcelona, 2002; 379,386; Garbey and Shyy, J. Comput. Phys. 2003; 186:1,23) a new method which estimates the order of convergence of a computation as the solution of a least square minimization problem on the residual. This method, called least square extrapolation, introduces a framework facilitating multi-level extrapolation, improves accuracy and provides a posteriori error estimate. This method can accommodate different grid arrangements. The goal of this paper is to investigate the power and limits of this method via incompressible Navier Stokes flow computations. Copyright © 2005 John Wiley & Sons, Ltd. [source] Application of the additive Schwarz method to large scale Poisson problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2004K. M. Singh Abstract This paper presents an application of the additive Schwarz method to large scale Poisson problems on parallel computers. Domain decomposition in rectangular blocks with matching grids on a structured rectangular mesh has been used together with a stepwise approximation to approximate sloping sides and complicated geometric features. A seven-point stencil based on central difference scheme has been used for the discretization of the Laplacian for both interior and boundary grid points, and this results in a symmetric linear algebraic system for any type of boundary conditions. The preconditioned conjugate gradient method has been used as an accelerator for the additive Schwarz method, and three different methods have been assessed for the solution of subdomain problems. Numerical experiments have been performed to determine the most suitable set of subdomain solvers and the optimal accuracy of subdomain solutions; to assess the effect of different decompositions of the problem domain; and to evaluate the parallel performance of the additive Schwarz preconditioner. Application to a practical problem involving complicated geometry is presented which establishes the efficiency and robustness of the method. Copyright © 2004 John Wiley & Sons, Ltd. [source] A Constructive Graphical Model Approach for Knowledge-Based Systems: A Vehicle Monitoring Case StudyCOMPUTATIONAL INTELLIGENCE, Issue 3 2003Y. Xiang Graphical models have been widely applied to uncertain reasoning in knowledge-based systems. For many of the problems tackled, a single graphical model is constructed before individual cases are presented and the model is used to reason about each new case. In this work, we consider a class of problems whose solution requires inference over a very large number of models that are impractical to construct a priori. We conduct a case study in the domain of vehicle monitoring and then generalize the approach taken. We show that the previously held negative belief on the applicability of graphical models to such problems is unjustified. We propose a set of techniques based on domain decomposition, model separation, model approximation, model compilation, and re-analysis to meet the computational challenges imposed by the combinatorial explosion. Experimental results on vehicle monitoring demonstrated good performance at near-real-time. [source] Seine: a dynamic geometry-based shared-space interaction framework for parallel scientific applicationsCONCURRENCY AND COMPUTATION: PRACTICE & EXPERIENCE, Issue 15 2006L. Zhang Abstract While large-scale parallel/distributed simulations are rapidly becoming critical research modalities in academia and industry, their efficient and scalable implementations continue to present many challenges. A key challenge is that the dynamic and complex communication/coordination required by these applications (dependent on the state of the phenomenon being modeled) are determined by the specific numerical formulation, the domain decomposition and/or sub-domain refinement algorithms used, etc. and are known only at runtime. This paper presents Seine, a dynamic geometry-based shared-space interaction framework for scientific applications. The framework provides the flexibility of shared-space-based models and supports extremely dynamic communication/coordination patterns, while still enabling scalable implementations. The design and prototype implementation of Seine are presented. Seine complements and can be used in conjunction with existing parallel programming systems such as MPI and OpenMP. An experimental evaluation using an adaptive multi-block oil-reservoir simulation is used to demonstrate the performance and scalability of applications using Seine. Copyright © 2006 John Wiley & Sons, Ltd. [source] A cache-efficient implementation of the lattice Boltzmann method for the two-dimensional diffusion equationCONCURRENCY AND COMPUTATION: PRACTICE & EXPERIENCE, Issue 14 2004A. C. Velivelli Abstract The lattice Boltzmann method is an important technique for the numerical solution of partial differential equations because it has nearly ideal scalability on parallel computers for many applications. However, to achieve the scalability and speed potential of the lattice Boltzmann technique, the issues of data reusability in cache-based computer architectures must be addressed. Utilizing the two-dimensional diffusion equation, , this paper examines cache optimization for the lattice Boltzmann method in both serial and parallel implementations. In this study, speedups due to cache optimization were found to be 1.9,2.5 for the serial implementation and 3.6,3.8 for the parallel case in which the domain decomposition was optimized for stride-one access. In the parallel non-cached implementation, the method of domain decomposition (horizontal or vertical) used for parallelization did not significantly affect the compute time. In contrast, the cache-based implementation of the lattice Boltzmann method was significantly faster when the domain decomposition was optimized for stride-one access. Additionally, the cache-optimized lattice Boltzmann method in which the domain decomposition was optimized for stride-one access displayed superlinear scalability on all problem sizes as the number of processors was increased. Copyright © 2004 John Wiley & Sons, Ltd. [source] Spectral-element simulations of wave propagation in porous mediaGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2008Christina Morency SUMMARY We present a derivation of the equations describing wave propagation in porous media based upon an averaging technique which accommodates the transition from the microscopic to the macroscopic scale. We demonstrate that the governing macroscopic equations determined by Biot remain valid for media with gradients in porosity. In such media, the well-known expression for the change in porosity, or the change in the fluid content of the pores, acquires two extra terms involving the porosity gradient. One fundamental result of Biot's theory is the prediction of a second compressional wave, often referred to as ,type II' or ,Biot's slow compressional wave', in addition to the classical fast compressional and shear waves. We present a numerical implementation of the Biot equations for 2-D problems based upon the spectral-element method (SEM) that clearly illustrates the existence of these three types of waves as well as their interactions at discontinuities. As in the elastic and acoustic cases, poroelastic wave propagation based upon the SEM involves a diagonal mass matrix, which leads to explicit time integration schemes that are well suited to simulations on parallel computers. Effects associated with physical dispersion and attenuation and frequency-dependent viscous resistance are accommodated based upon a memory variable approach. We perform various benchmarks involving poroelastic wave propagation and acoustic,poroelastic and poroelastic,poroelastic discontinuities, and we discuss the boundary conditions used to deal with these discontinuities based upon domain decomposition. We show potential applications of the method related to wave propagation in compacted sediments, as one encounters in the petroleum industry, and to detect the seismic signature of buried landmines and unexploded ordnance. [source] A 2-D spectral-element method for computing spherical-earth seismograms,II.GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2008Waves in solid, fluid media SUMMARY We portray a dedicated spectral-element method to solve the elastodynamic wave equation upon spherically symmetric earth models at the expense of a 2-D domain. Using this method, 3-D wavefields of arbitrary resolution may be computed to obtain Fréchet sensitivity kernels, especially for diffracted arrivals. The meshing process is presented for varying frequencies in terms of its efficiency as measured by the total number of elements, their spacing variations and stability criteria. We assess the mesh quantitatively by defining these numerical parameters in a general non-dimensionalized form such that comparisons to other grid-based methods are straightforward. Efficient-mesh generation for the PREM example and a minimum-messaging domain decomposition and parallelization strategy lay foundations for waveforms up to frequencies of 1 Hz on moderate PC clusters. The discretization of fluid, solid and respective boundary regions is similar to previous spectral-element implementations, save for a fluid potential formulation that incorporates the density, thereby yielding identical boundary terms on fluid and solid sides. We compare the second-order Newmark time extrapolation scheme with a newly implemented fourth-order symplectic scheme and argue in favour of the latter in cases of propagation over many wavelengths due to drastic accuracy improvements. Various validation examples such as full moment-tensor seismograms, wavefield snapshots, and energy conservation illustrate the favourable behaviour and potential of the method. [source] Asynchronous multi-domain variational integrators for non-linear problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2008Mark Gates Abstract We develop an asynchronous time integration and coupling method with domain decomposition for linear and non-linear problems in mechanics. To ensure stability in the time integration and in coupling between domains, we use variational integrators with local Lagrange multipliers to enforce continuity at the domain interfaces. The asynchronous integrator lets each domain step with its own time step, using a smaller time step where required by stability and accuracy constraints and a larger time step where allowed. We show that in practice the time step is limited by accuracy requirements rather than by stability requirements. Copyright © 2008 John Wiley & Sons, Ltd. [source] Parallel DSMC method using dynamic domain decompositionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2005J.-S. Wu Abstract A general parallel direct simulation Monte Carlo method using unstructured mesh is introduced, which incorporates a multi-level graph-partitioning technique to dynamically decompose the computational domain. The current DSMC method is implemented on an unstructured mesh using particle ray-tracing technique, which takes the advantages of the cell connectivity information. In addition, various strategies applying the stop at rise (SAR) (IEEE Trans Comput 1988; 39:1073,1087) scheme is studied to determine how frequent the domain should be re-decomposed. A high-speed, bottom-driven cavity flow, including small, medium and large problems, based on the number of particles and cells, are simulated. Corresponding analysis of parallel performance is reported on IBM-SP2 parallel machine up to 64 processors. Analysis shows that degree of imbalance among processors with dynamic load balancing is about ,,½ of that without dynamic load balancing. Detailed time analysis shows that degree of imbalance levels off very rapidly at a relatively low value with increasing number of processors when applying dynamic load balancing, which makes the large problem size fairly scalable for processors more than 64. In general, optimal frequency of activating SAR scheme decreases with problem size. At the end, the method is applied to compute two two-dimensional hypersonic flows, a three-dimensional hypersonic flow and a three-dimensional near-continuum twin-jet gas flow to demonstrate its superior computational capability and compare with experimental data and previous simulation data wherever available. Copyright © 2005 John Wiley & Sons, Ltd. [source] A FETI-based multi-time-step coupling method for Newmark schemes in structural dynamicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2004A. Prakash Abstract We present an efficient and accurate multi-time-step coupling method using FETI domain decomposition for structural dynamics. Using this method one can divide a large structural mesh into a number of smaller subdomains, solve the individual subdomains separately and couple the solutions together to obtain the solution to the original problem. The various subdomains can be integrated in time using different time steps and/or different Newmark schemes. This approach will be most effective for very large-scale simulations on complex geometries. Our coupling method builds upon a method previously proposed by Gravouil and Combescure (GC method). We show that for the simplest case when the same time step is used in all subdomains of the mesh our method reduces to the GC method and is unconditionally stable and energy preserving. In addition, we show that our method possesses these desirable properties for general multi-time-step cases too unlike the GC method which is dissipative. Greater computational efficiency is also achieved through our method by limiting the computation of interface forces to the largest time step as opposed to the smallest time step in the GC method. Copyright © 2004 John Wiley & Sons, Ltd. [source] LayTracks: a new approach to automated geometry adaptive quadrilateral mesh generation using medial axis transformINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2004W. R. Quadros Abstract A new mesh generation algorithm called ,LayTracks', to automatically generate an all quad mesh that is adapted to the variation of geometric feature size in the domain is described. LayTracks combines the merits of two popular direct techniques for quadrilateral mesh generation,quad meshing by decomposition and advancing front quad meshing. While the MAT has been used for the domain decomposition before, this is the first attempt to use the MAT, for the robust subdivision of a complex domain into a well defined sub-domain called ,Tracks', for terminating the advancing front of the mesh elements without complex interference checks and to use radius function for providing sizing function for adaptive meshing. The process of subdivision of a domain is analogous to, formation of railway tracks by laying rails on the ground. Each rail starts from a node on the boundary and propagates towards the medial axis (MA) and then from the MA towards the boundary. Quadrilateral elements are then obtained by placing nodes on these rails and connecting them inside each track, formed by adjacent rails. The algorithm has been implemented and tested on some typical geometries and the quality of the output mesh obtained are presented. Extension of this technique to all hexahedral meshing is discussed. Copyright © 2004 John Wiley & Sons, Ltd. [source] A-scalability and an integrated computational technology and framework for non-linear structural dynamics.INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2003Part 1: Theoretical developments, parallel formulations Abstract For large-scale problems and large processor counts, the accuracy and efficiency with reduced solution times and attaining optimal parallel scalability of the entire transient duration of the simulation for general non-linear structural dynamics problems poses many computational challenges. For transient analysis, explicit time operators readily inherit algorithmic scalability and consequently enable parallel scalability. However, the key issues concerning parallel simulations via implicit time operators within the framework and encompassing the class of linear multistep methods include the totality of the following considerations to foster the proposed notion of A-scalability: (a) selection of robust scalable optimal time discretized operators that foster stabilized non-linear dynamic implicit computations both in terms of convergence and the number of non-linear iterations for completion of large-scale analysis of the highly non-linear dynamic responses, (b) selecting an appropriate scalable spatial domain decomposition method for solving the resulting linearized system of equations during the implicit phase of the non-linear computations, (c) scalable implementation models and solver technology for the interface and coarse problems for attaining parallel scalability of the computations, and (d) scalable parallel graph partitioning techniques. These latter issues related to parallel implicit formulations are of interest and focus in this paper. The former involving parallel explicit formulations are also a natural subset of the present framework and have been addressed previously in Reference 1 (Advances in Engineering Software 2000; 31: 639,647). In the present context, of the key issues, although a particular aspect or a solver as related to the spatial domain decomposition may be designed to be numerically scalable, the totality of the aforementioned issues simultaneously play an important and integral role to attain A-scalability of the parallel formulations for the entire transient duration of the simulation and is desirable for transient problems. As such, the theoretical developments of the parallel formulations are first detailed in Part 1 of this paper, and the subsequent practical applications and performance results of general non-linear structural dynamics problems are described in Part 2 of this paper to foster the proposed notion of A-scalability. Copyright © 2003 John Wiley & Sons, Ltd. [source] Parallel computation of a highly nonlinear Boussinesq equation model through domain decompositionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2005Khairil Irfan Sitanggang Abstract Implementations of the Boussinesq wave model to calculate free surface wave evolution in large basins are, in general, computationally very expensive, requiring huge amounts of CPU time and memory. For large scale problems, it is either not affordable or practical to run on a single PC. To facilitate such extensive computations, a parallel Boussinesq wave model is developed using the domain decomposition technique in conjunction with the message passing interface (MPI). The published and well-tested numerical scheme used by the serial model, a high-order finite difference method, is identical to that employed in the parallel model. Parallelization of the tridiagonal matrix systems included in the serial scheme is the most challenging aspect of the work, and is accomplished using a parallel matrix solver combined with an efficient data transfer scheme. Numerical tests on a distributed-memory super-computer show that the performance of the current parallel model in simulating wave evolution is very satisfactory. A linear speedup is gained as the number of processors increases. These tests showed that the CPU time efficiency of the model is about 75,90%. Copyright © 2005 John Wiley & Sons, Ltd. [source] Microfluidics simulation using adaptive unstructured gridsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2004Jacob Waltz Abstract A methodology for microfluidics simulation is presented. The methodology solves the three-dimensional incompressible Navier,Stokes equations with an adaptive unstructured Finite Element method. A semi-implicit Fractional Step procedure is used for time integration. The entire methodology has been parallelized for shared-memory architectures via an algebraic domain decomposition. Results from both verification problems and prototypical applications are included. Copyright © 2004 John Wiley & Sons, Ltd. [source] Parallel adaptive refinement for unsteady flow calculations on 3D unstructured grids,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2004Jacob Waltz Abstract A parallel adaptive refinement algorithm for three-dimensional unstructured grids is presented. The algorithm is based on an hierarchical h -refinement/derefinement scheme for tetrahedral elements. The algorithm has been fully parallelized for shared-memory platforms via a domain decomposition of the mesh at the algebraic level. The effectiveness of the procedure is demonstrated with applications which involve unsteady compressible fluid flow. A parallel speedup study of the algorithm also is included. Published in 2004 by John Wiley & Sons, Ltd. [source] Numerical approximation of optimal control of unsteady flows using SQP and time decompositionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2004S. S. RavindranArticle first published online: 1 APR 200 Abstract In this paper, we present numerical approximations of optimal control of unsteady flow problems using sequential quadratic programming method (SQP) and time domain decomposition. The SQP method is considered superior due to its fast convergence and its ability to take advantage of existing numerical techniques for fluid flow problems. It iteratively solves a sequence of linear quadratic optimal control problems converging to the solution of the non-linear optimal control problem. The solution to the linear quadratic problem is characterized by the Karush,Kuhn,Tucker (KKT) optimality system which in the present context is a formidable system to solve. As a remedy various time domain decompositions, inexact SQP implementations and block iterative methods to solve the KKT systems are examined. Numerical results are presented showing the efficiency and feasibility of the algorithms. Copyright © 2004 John Wiley & Sons, Ltd. [source] A new parallelization strategy for solving time-dependent 3D Maxwell equations using a high-order accurate compact implicit scheme,INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 5 2006Eugene Kashdan Abstract With progress in computer technology there has been renewed interest in a time-dependent approach to solving Maxwell equations. The commonly used Yee algorithm (an explicit central difference scheme for approximation of spatial derivatives coupled with the Leapfrog scheme for approximation of temporal derivatives) yields only a second-order of accuracy. On the other hand, an increasing number of industrial applications, especially in optic and microwave technology, demands high-order accurate numerical modelling. The standard way to increase accuracy of the finite difference scheme without increasing the differential stencil is to replace a 2nd-order accurate explicit scheme for approximation of spatial derivatives with the 4th-order accurate compact implicit scheme. In general, such a replacement requires additional memory resources and slows the computations. However, the curl-based form of Maxwell equations allows us to construct an effective parallel algorithm with the alternating domain decomposition (ADD) minimizing the communication time. We present a new parallel approach to the solution of three-dimensional time-dependent Maxwell equations and provide a theoretical and experimental analysis of its performance. Copyright © 2006 John Wiley & Sons, Ltd. [source] Parallel Algorithm for Cell Dynamics Simulation of Block CopolymersMACROMOLECULAR THEORY AND SIMULATIONS, Issue 9 2007Xiaohu Guo Abstract Cell dynamics simulation (CDS) is a very promising approach to model dynamic processes in block copolymer systems at the mesoscale level. It is difficult to implement a real time and experimental-scale simulation with traditional serial algorithms because of the expensive computation. A parallel, spatial decomposition-based algorithm for large-scale CDS is proposed. With the efficient strategy of domain decomposition and the fast method of neighbouring points location, we greatly reduce the calculating and communicating cost. The numerical results indicate that the proposed parallel algorithm can provide an efficient procedure for computer simulation of block copolymer systems of experimental size. [source] Application of algebraic domain decomposition combined with Krylov subspace iterative methods to solve 3D vector finite element equationsMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 3 2007X. W. Ping Abstract In this paper, a parallel algorithm based on MPI (Message Passing Interface) parallel computing library for the finite element method is presented to analyze three-dimensional electromagnetic devices. The algebraic domain decomposition method is used in the algorithm. The original problem is decomposed into several subproblems according to its features. Each of them is allocated to one process in one computation node and solved independently with a direct method. The data are exchanged by communication between adjacent subdomains with overlapped data based on the MPI network. The example of its application is given. Numerical shows that the proposed algorithm can get excellent performance vs. price ratio and can save much memory and CPU time than sequential computing. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 686,692, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22247 [source] Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficientsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2002N. S. Bakhvalov Abstract We prove extension theorems in the norms described by Stokes and Lamé operators for the three-dimensional case with periodic boundary conditions. For the Lamé equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e. for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem. Extension theorems serve as an important tool in many applications, e.g. in domain decomposition and fictitious domain methods, and in analysis of finite element methods. We consider an application of established extension theorems to an efficient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coefficients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coefficient problem. Such preconditioner allows the use of well-known fast algorithms for preconditioning. Under some natural assumptions on smoothness and topological properties of subdomains with small coefficients, we prove convergence of the simplest Richardson method uniform in the jump of coefficients. For the Lamé equations, the convergence is also uniform in the incompressible limit. Our preliminary numerical results for two-dimensional diffusion problems show fast convergence uniform in the jump and in the mesh size parameter. Copyright © 2002 John Wiley & Sons, Ltd. [source] A fractional splitting algorithm for nonoverlapping domain decomposition for parabolic problemNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2002Daoud S. Daoud Abstract In this article we study the convergence of the nonoverlapping domain decomposition for solving large linear system arising from semi-discretization of two-dimensional initial value problem with homogeneous boundary conditions and solved by implicit time stepping using first and two alternatives of second-order FS-methods. The interface values along the artificial boundary condition line are found using explicit forward Euler's method for the first-order FS-method, and for the second-order FS-method to use extrapolation procedure for each spatial variable individually. The solution by the nonoverlapping domain decomposition with FS-method is applicable to problems that requires the solution on nonuniform meshes for each spatial variable, which will enable us to use different time-stepping over different subdomains and with the possibility of extension to three-dimensional problem. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 609,624, 2002 [source] General theory of domain decomposition: Indirect methodsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2002Ismael Herrera Abstract According to a general theory of domain decomposition methods (DDM), recently proposed by Herrera, DDM may be classified into two broad categories: direct and indirect (or Trefftz-Herrera methods). This article is devoted to formulate systematically indirect methods and apply them to differential equations in several dimensions. They have interest since they subsume some of the best-known formulations of domain decomposition methods, such as those based on the application of Steklov-Poincaré operators. Trefftz-Herrera approach is based on a special kind of Green's formulas applicable to discontinuous functions, and one of their essential features is the use of weighting functions which yield information, about the sought solution, at the internal boundary of the domain decomposition exclusively. A special class of Sobolev spaces is introduced in which boundary value problems with prescribed jumps at the internal boundary are formulated. Green's formulas applicable in such Sobolev spaces, which contain discontinuous functions, are established and from them the general framework for indirect methods is derived. Guidelines for the construction of the special kind of test functions are then supplied and, as an illustration, the method is applied to elliptic problems in several dimensions. A nonstandard method of collocation is derived in this manner, which possesses significant advantages over more standard procedures. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 296,322, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10008 [source] General theory of domain decomposition: Beyond Schwarz methodsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2001Ismael Herrera Abstract Recently, Herrera presented a general theory of domain decomposition methods (DDM). This article is part of a line of research devoted to its further development and applications. According to it, DDM are classified into direct and indirect, which in turn can be subdivided into overlapping and nonoverlapping. Some articles dealing with general aspects of the theory and with indirect (Trefftz,Herrera) methods have been published. In the present article, a very general direct-overlapping method, which subsumes Schwarz methods, is introduced. Also, this direct-overlapping method is quite suitable for parallel implementation. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 495,517, 2001 [source] An efficient domain-decomposition pseudo-spectral method for solving elliptic differential equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2008N. Mai-Duy Abstract In this paper, a new numerical scheme based on non-overlapping domain decompositions and integrated Chebyshev approximations for solving elliptic differential equations (DEs) is presented. The distinguishing feature of the present scheme is that it achieves a Cp continuous solution across the interfaces (p is the order of the DE). Several test problems are employed to verify the method. The obtained results indicate that the achievement of higher-order smoothness leads to a significant improvement in accuracy. Copyright © 2007 John Wiley & Sons, Ltd. [source] Steady filtration problems with seawater intrusion: macro-hybrid penalized finite element approximationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2005Gonzalo Alduncin Abstract Macro-hybrid penalized finite element approximations are studied for steady filtration problems with seawater intrusion. On the basis of nonoverlapping domain decompositions with vertical interfaces, sections of coastal aquifers are decomposed into subsystems with simpler geometries and small scales, interconnected via transmission conditions of pressure and flux continuity. Corresponding local penalized formulations are derived from the global penalized variational formulation of the two-free boundary flow problem, with continuity transmission conditions modelled variationally in a dual sense. Then, macro-hybrid finite element approximations are derived for the system, defined on independent subdomain grids. Parallel relaxation penalty-duality algorithms are proposed from fixed-point problem characterizations. Numerical experiments exemplify the macro-hybrid penalized theory, showing a good agreement with previous primal conforming penalized finite element approximations (Comput. Methods Appl. Mech. Engng. 2000; 190:609,624). Copyright © 2005 John Wiley & Sons, Ltd. [source] Numerical approximation of optimal control of unsteady flows using SQP and time decompositionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2004S. S. RavindranArticle first published online: 1 APR 200 Abstract In this paper, we present numerical approximations of optimal control of unsteady flow problems using sequential quadratic programming method (SQP) and time domain decomposition. The SQP method is considered superior due to its fast convergence and its ability to take advantage of existing numerical techniques for fluid flow problems. It iteratively solves a sequence of linear quadratic optimal control problems converging to the solution of the non-linear optimal control problem. The solution to the linear quadratic problem is characterized by the Karush,Kuhn,Tucker (KKT) optimality system which in the present context is a formidable system to solve. As a remedy various time domain decompositions, inexact SQP implementations and block iterative methods to solve the KKT systems are examined. Numerical results are presented showing the efficiency and feasibility of the algorithms. Copyright © 2004 John Wiley & Sons, Ltd. [source] | ||||||||||||||||