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The Robust Multigrid Technique
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Book Synopsis The Robust Multigrid Technique by : Sergey I. Martynenko
Download or read book The Robust Multigrid Technique written by Sergey I. Martynenko and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-09-25 with total page 211 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a detailed description of a robust pseudomultigrid algorithm for solving (initial-)boundary value problems on structured grids in a black-box manner. To overcome the problem of robustness, the presented Robust Multigrid Technique (RMT) is based on the application of the essential multigrid principle in a single grid algorithm. It results in an extremely simple, very robust and highly parallel solver with close-to-optimal algorithmic complexity and the least number of problem-dependent components. Topics covered include an introduction to the mathematical principles of multigrid methods, a detailed description of RMT, results of convergence analysis and complexity, possible expansion on unstructured grids, numerical experiments and a brief description of multigrid software, parallel RMT and estimations of speed-up and efficiency of the parallel multigrid algorithms, and finally applications of RMT for the numerical solution of the incompressible Navier Stokes equations. Potential readers are graduate students and researchers working in applied and numerical mathematics as well as multigrid practitioners and software programmers. ContentsIntroduction to multigridRobust multigrid techniqueParallel multigrid methodsApplications of multigrid methods in computational fluid dynamics
Book Synopsis Towards Robust Algebraic Multigrid Methods for Nonsymmetric Problems by : James Lottes
Download or read book Towards Robust Algebraic Multigrid Methods for Nonsymmetric Problems written by James Lottes and published by Springer. This book was released on 2017-03-24 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis presents a rigorous, abstract analysis of multigrid methods for positive nonsymmetric problems, particularly suited to algebraic multigrid, with a completely new approach to nonsymmetry which is based on a new concept of absolute value for nonsymmetric operators. Multigrid, and in particular algebraic multigrid, has become an indispensable tool for the solution of discretizations of partial differential equations. While used in both the symmetric and nonsymmetric cases, the theory for the nonsymmetric case has lagged substantially behind that for the symmetric case. This thesis closes some of this gap, presenting a major and highly original contribution to an important problem of computational science. The new approach to nonsymmetry will be of interest to anyone working on the analysis of discretizations of nonsymmetric operators, even outside the context of multigrid. The presentation of the convergence theory may interest even those only concerned with the symmetric case, as it sheds some new light on and extends existing results.
Book Synopsis The Robust Multigrid Technique by : Sergey I. Martynenko
Download or read book The Robust Multigrid Technique written by Sergey I. Martynenko and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-09-25 with total page 211 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a detailed description of a robust pseudomultigrid algorithm for solving (initial-)boundary value problems on structured grids in a black-box manner. To overcome the problem of robustness, the presented Robust Multigrid Technique (RMT) is based on the application of the essential multigrid principle in a single grid algorithm. It results in an extremely simple, very robust and highly parallel solver with close-to-optimal algorithmic complexity and the least number of problem-dependent components. Topics covered include an introduction to the mathematical principles of multigrid methods, a detailed description of RMT, results of convergence analysis and complexity, possible expansion on unstructured grids, numerical experiments and a brief description of multigrid software, parallel RMT and estimations of speed-up and efficiency of the parallel multigrid algorithms, and finally applications of RMT for the numerical solution of the incompressible Navier Stokes equations. Potential readers are graduate students and researchers working in applied and numerical mathematics as well as multigrid practitioners and software programmers. Contents Introduction to multigrid Robust multigrid technique Parallel multigrid methods Applications of multigrid methods in computational fluid dynamics
Book Synopsis Multi-Grid Methods and Applications by : Wolfgang Hackbusch
Download or read book Multi-Grid Methods and Applications written by Wolfgang Hackbusch and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multi-grid methods are the most efficient tools for solving elliptic boundary value problems. The reader finds here an elementary introduction to multi-grid algorithms as well as a comprehensive convergence analysis. One section describes special applications (convection-diffusion equations, singular perturbation problems, eigenvalue problems, etc.). The book also contains a complete presentation of the multi-grid method of the second kind, which has important applications to integral equations (e.g. the "panel method") and to numerous other problems. Readers with a practical interest in multi-grid methods will benefit from this book as well as readers with a more theoretical interest.
Book Synopsis A Multigrid Tutorial by : William L. Briggs
Download or read book A Multigrid Tutorial written by William L. Briggs and published by SIAM. This book was released on 2000-07-01 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Book Synopsis Multigrid Techniques by : Achi Brandt
Download or read book Multigrid Techniques written by Achi Brandt and published by SIAM. This book was released on 2011-01-01 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: This classic text presents the best practices of developing multigrid solvers for large-scale computational problems in science and engineering. By representing a problem at multiple scales and employing suitable interscale interactions, multigrid avoids slowdown due to stiffness and reduces the computational cost of classical algorithms by orders of magnitude. Starting from simple examples, this book guides the reader through practical stages for developing reliable multigrid solvers, methodically supported by accurate performance predictors. The revised edition presents discretization and fast solution of linear and nonlinear partial differential systems; treatment of boundary conditions, global constraints and singularities; grid adaptation, high-order approximations, and system design optimization; applications to fluid dynamics, from simple models to advanced systems; new quantitative performance predictors, a MATLAB sample code, and more. Readers will also gain access to the Multigrid Guide 2.0 Web site, where updates and new developments will be continually posted, including a chapter on Algebraic Multigrid.
Book Synopsis Multigrid Finite Element Methods for Electromagnetic Field Modeling by : Yu Zhu
Download or read book Multigrid Finite Element Methods for Electromagnetic Field Modeling written by Yu Zhu and published by John Wiley & Sons. This book was released on 2006-03-10 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first comprehensive monograph that features state-of-the-art multigrid methods for enhancing the modeling versatility, numerical robustness, and computational efficiency of one of the most popular classes of numerical electromagnetic field modeling methods: the method of finite elements. The focus of the publication is the development of robust preconditioners for the iterative solution of electromagnetic field boundary value problems (BVPs) discretized by means of finite methods. Specifically, the authors set forth their own successful attempts to utilize concepts from multigrid and multilevel methods for the effective preconditioning of matrices resulting from the approximation of electromagnetic BVPs using finite methods. Following the authors' careful explanations and step-by-step instruction, readers can duplicate the authors' results and take advantage of today's state-of-the-art multigrid/multilevel preconditioners for finite element-based iterative electromagnetic field solvers. Among the highlights of coverage are: * Application of multigrid, multilevel, and hybrid multigrid/multilevel preconditioners to electromagnetic scattering and radiation problems * Broadband, robust numerical modeling of passive microwave components and circuits * Robust, finite element-based modal analysis of electromagnetic waveguides and cavities * Application of Krylov subspace-based methodologies for reduced-order macromodeling of electromagnetic devices and systems * Finite element modeling of electromagnetic waves in periodic structures The authors provide more than thirty detailed algorithms alongside pseudo-codes to assist readers with practical computer implementation. In addition, each chapter includes an applications section with helpful numerical examples that validate the authors' methodologies and demonstrate their computational efficiency and robustness. This groundbreaking book, with its coverage of an exciting new enabling computer-aided design technology, is an essential reference for computer programmers, designers, and engineers, as well as graduate students in engineering and applied physics.
Book Synopsis Introduction to Numerical Geodynamic Modelling by : Taras Gerya
Download or read book Introduction to Numerical Geodynamic Modelling written by Taras Gerya and published by Cambridge University Press. This book was released on 2010 with total page 359 pages. Available in PDF, EPUB and Kindle. Book excerpt: This user-friendly reference for students and researchers presents the basic mathematical theory, before introducing modelling of key geodynamic processes.
Book Synopsis Robust Multigrid Algorithms for the Incompressible Navier-Stokes Equations by : Ruben S. Montero
Download or read book Robust Multigrid Algorithms for the Incompressible Navier-Stokes Equations written by Ruben S. Montero and published by . This book was released on 2000 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: Anisotropies occur naturally in CFD where the simulation of small scale physical phenomena, such as boundary layers at high Reynolds numbers, causes the grid to be highly stretched leading to a slow down in convergence of multigrid methods. Several approaches aimed at making multigrid a robust solver have been proposed and analyzed in literature using the scalar diffusion equation. However, they have been rarely applied to solving more complicated models, like the incompressible Navier-Stokes equations. This paper contains the first published numerical results of the behavior of two popular robust multigrid approaches (alternating-plane smoothers combined with standard coarsening and plane implicit smoothers combined with semi-coarsening) for solving the 3-D incompressible Navier-Stokes equations in the simulation of the driven cavity and a boundary layer over a flat plate on a stretched grid. The discrete operator is obtained using a staggered-grid arrangement of variables with a finite volume technique and second-order accuracy is achieved using defect correction within the multigrid cycle. Grid size, grid stretching and Reynolds number are the factors considered in evaluating the robustness of the multigrid methods. Both approaches yield large increases in convergence rates over cell-implicit smoothers on stretched grids. The combination of plane implicit smoothers and semi-coarsening was found to be fully robust in the fiat plate simulation up to Reynolds numbers 10(exp 6) and the best alternative in the driven cavity simulation for Reynolds numbers above 10(exp 3). The alternating-plane approach exhibits a better behavior for lower Reynolds numbers (below to 10(exp 3) in the driven cavity simulation. A parallel variant of the smoother, tri-plane ordering, presents a good trade-off between convergence and parallel properties.
Book Synopsis Robust Algebraic Multilevel Methods and Algorithms by : Johannes Kraus
Download or read book Robust Algebraic Multilevel Methods and Algorithms written by Johannes Kraus and published by Walter de Gruyter. This book was released on 2009 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with algorithms for the solution of linear systems of algebraic equations with large-scale sparse matrices, with a focus on problems that are obtained after discretization of partial differential equations using finite element methods. The authors provide a systematic presentation of the recent advances in robust algebraic multilevel methods and algorithms, e.g., the preconditioned conjugate gradient method, algebraic multilevel iteration (AMLI) preconditioners, the classical algebraic multigrid (AMG) method and its recent modifications, namely AMG using element interpolation (AMGe) and AMG based on smoothed aggregation. The first six chapters can serve as a short introductory course on the theory of AMLI methods and algorithms. The next part of the monograph is devoted to more advanced topics, including the description of new generation AMG methods, AMLI methods for discontinuous Galerkin systems, looking-free algorithms for coupled problems etc., ending with important practical issues of implementation and challenging applications. This second part is addressed to some more experienced students and practitioners and can be used to complete a more advanced course on robust AMLI and AMG methods and their efficient application. This book is intended for mathematicians, engineers, natural scientists etc.
Download or read book Multigrid Methods written by W. Hackbusch and published by Springer. This book was released on 2006-11-15 with total page 664 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis An Introduction to Multigrid Methods by : Pieter Wesseling
Download or read book An Introduction to Multigrid Methods written by Pieter Wesseling and published by R.T. Edwards, Inc.. This book was released on 2004 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduces the principles, techniques, applications and literature of multigrid methods. Aimed at an audience with non-mathematical but computing-intensive disciplines and basic knowledge of analysis, partial differential equations and numerical mathematics, it is packed with helpful exercises, examples and illustrations.
Book Synopsis multigrid methods by : Stephen F. Mccormick
Download or read book multigrid methods written by Stephen F. Mccormick and published by CRC Press. This book was released on 2020-08-12 with total page 668 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of research papers on a wide variety of multigrid topics, including applications, computation and theory. It represents proceedings of the Third Copper Mountain Conference on Multigrid Methods, which was held at Copper Mountain, Colorado.
Book Synopsis Multigrid Methods IV by : P.W. Hemker
Download or read book Multigrid Methods IV written by P.W. Hemker and published by Birkhäuser. This book was released on 2012-12-06 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains a selection from the papers presented at the Fourth European Multigrid Conference, held in Amsterdam, July 6-9,1993. There were 78 registered participants from 14 different countries, and 56 presentations were given. The preceding conferences in this series were held in Cologne (1981, 1985) and in Bonn (1990). Also at the other side of the Atlantic special multigrid conferences are held regularly, at intervals of two years, always in Copper Mountain, Colorado, US. The Sixth Copper Mountain Conference on Multigrid Methods took place in April, 1993. Circumstances prevented us from putting a larger time interval between the Copper and Amsterdam meetings. The next European meeting is planned in 1996, a year later than the next Copper Meeting. When the first multigrid conference was held in 1981 there was no doubt about the usefulness of a conference dedicated specially to multigrid, because multigrid was a new and relatively unexplored subject, still in a pioneering stage, and pursued by specialists. The past twenty years have shown a rapid growth in theoretical understanding, useful applications and widespread acceptance of multi grid in the applied disciplines. Hence, one might ask whether there is still a need today for conferences specially dedicated to multigrid. The general consensus is that the answer is affirmative. New issues have arisen that are best addressed or need also be addressed from a special multigrid point of view.
Book Synopsis Numerical Methods for Black-Box Software in Computational Continuum Mechanics by : Sergey I. Martynenko
Download or read book Numerical Methods for Black-Box Software in Computational Continuum Mechanics written by Sergey I. Martynenko and published by Walter de Gruyter GmbH & Co KG. This book was released on 2023-10-23 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: The organization of the material is presented as follows: This introductory chapter I represents a theoretical analysis of the computational algorithms for a numerical solution of the basic equations in continuum mechanics. In this chapter, the general requirements for computational grids, discretization, and iterative methods for black-box software are examined. Finally, a concept of a two-grid algorithm for (de-)coupled solving multidimensional non-linear (initial-)boundary value problems in continuum mechanics (multiphysics simulation) in complex domains is presented. Chapter II contains descriptions of the sequential Robust Multigrid Technique which is developed as a general-purpose solver in black-box codes. This chapter presents the main components of the Robust Multigrid Technique (RMT) used in the two-grid algorithm (Chapter I) to compute the auxiliary (structured) grid correction. This includes the generation of multigrid structures, computation of index mapping, and integral evaluation. Finite volume discretization on the multigrid structures will be explained by studying a 1D linear model problem. In addition, the algorithmic complexity of RMT and black-box optimization of the problem-dependent components of RMT are analysed. Chapter III provides a description of parallel RMT. This chapter introduces parallel RMT-based algorithms for solving the boundary value problems and initial-boundary value problems in unified manner. Section 1 presents a comparative analysis of the parallel RMT and the sequential V-cycle. Sections 2 and 3 present a geometric and an algebraic parallelism of RMT, i.e. parallelization of the smoothing iterations on the coarse and the levels. A parallel multigrid cycle will be considered in Section 4. A parallel RMT for the time-dependent problems is given in Section 5. Finally, the basic properties of parallel RMT will be summarized in Section 6. Theoretical aspects of the used algorithms for solving multidimensional problems are discussed in Chapters IV. This chapter contains the theoretical aspects of the algorithms used for the numerical solving of the resulting system of linear algebraic equations obtained from discrete multidimensional (initial-)boundary value problems.
Book Synopsis Mesh Methods for Boundary-Value Problems and Applications by : Ildar B. Badriev
Download or read book Mesh Methods for Boundary-Value Problems and Applications written by Ildar B. Badriev and published by Springer Nature. This book was released on 2022-09-14 with total page 607 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gathers papers presented at the 13th International Conference on Mesh Methods for Boundary-Value Problems and Applications, which was held in Kazan, Russia, in October 2020. The papers address the following topics: the theory of mesh methods for boundary-value problems in mathematical physics; non-linear mathematical models in mechanics and physics; algorithms for solving variational inequalities; computing science; and educational systems. Given its scope, the book is chiefly intended for students in the fields of mathematical modeling science and engineering. However, it will also benefit scientists and graduate students interested in these fields.
Book Synopsis Robust Multi-Grid Methods by : Wolfgang Hackbusch
Download or read book Robust Multi-Grid Methods written by Wolfgang Hackbusch and published by Springer-Verlag. This book was released on 2013-07-01 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt:
