Adaptive Finite Element Methods for Multiscale Partial Differential Equations
Author | : Achim Nonnenmacher |
Publisher | : |
Total Pages | : 195 |
Release | : 2011 |
ISBN-10 | : OCLC:759842315 |
ISBN-13 | : |
Rating | : 4/5 (15 Downloads) |
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Author | : Achim Nonnenmacher |
Publisher | : |
Total Pages | : 195 |
Release | : 2011 |
ISBN-10 | : OCLC:759842315 |
ISBN-13 | : |
Rating | : 4/5 (15 Downloads) |
Author | : Silvia Bertoluzza |
Publisher | : Springer |
Total Pages | : 324 |
Release | : 2012-01-06 |
ISBN-10 | : 9783642240799 |
ISBN-13 | : 3642240798 |
Rating | : 4/5 (99 Downloads) |
This book is a collection of lecture notes for the CIME course on "Multiscale and Adaptivity: Modeling, Numerics and Applications," held in Cetraro (Italy), in July 2009. Complex systems arise in several physical, chemical, and biological processes, in which length and time scales may span several orders of magnitude. Traditionally, scientists have focused on methods that are particularly applicable in only one regime, and knowledge of the system on one scale has been transferred to another scale only indirectly. Even with modern computer power, the complexity of such systems precludes their being treated directly with traditional tools, and new mathematical and computational instruments have had to be developed to tackle such problems. The outstanding and internationally renowned lecturers, coming from different areas of Applied Mathematics, have themselves contributed in an essential way to the development of the theory and techniques that constituted the subjects of the courses.
Author | : Zhangxin Chen |
Publisher | : Springer Science & Business Media |
Total Pages | : 415 |
Release | : 2005-06-23 |
ISBN-10 | : 9783540240785 |
ISBN-13 | : 3540240780 |
Rating | : 4/5 (85 Downloads) |
Introduce every concept in the simplest setting and to maintain a level of treatment that is as rigorous as possible without being unnecessarily abstract. Contains unique recent developments of various finite elements such as nonconforming, mixed, discontinuous, characteristic, and adaptive finite elements, along with their applications. Describes unique recent applications of finite element methods to important fields such as multiphase flows in porous media and semiconductor modelling. Treats the three major types of partial differential equations, i.e., elliptic, parabolic, and hyperbolic equations.
Author | : Yalchin Efendiev |
Publisher | : Springer Science & Business Media |
Total Pages | : 242 |
Release | : 2009-01-10 |
ISBN-10 | : 9780387094960 |
ISBN-13 | : 0387094962 |
Rating | : 4/5 (60 Downloads) |
The aim of this monograph is to describe the main concepts and recent - vances in multiscale ?nite element methods. This monograph is intended for thebroaderaudienceincludingengineers,appliedscientists,andforthosewho are interested in multiscale simulations. The book is intended for graduate students in applied mathematics and those interested in multiscale compu- tions. It combines a practical introduction, numerical results, and analysis of multiscale ?nite element methods. Due to the page limitation, the material has been condensed. Each chapter of the book starts with an introduction and description of the proposed methods and motivating examples. Some new techniques are introduced using formal arguments that are justi?ed later in the last chapter. Numerical examples demonstrating the signi?cance of the proposed methods are presented in each chapter following the description of the methods. In the last chapter, we analyze a few representative cases with the objective of demonstrating the main error sources and the convergence of the proposed methods. A brief outline of the book is as follows. The ?rst chapter gives a general introductiontomultiscalemethodsandanoutlineofeachchapter.Thesecond chapter discusses the main idea of the multiscale ?nite element method and its extensions. This chapter also gives an overview of multiscale ?nite element methods and other related methods. The third chapter discusses the ext- sion of multiscale ?nite element methods to nonlinear problems. The fourth chapter focuses on multiscale methods that use limited global information.
Author | : Ronald DeVore |
Publisher | : Springer Science & Business Media |
Total Pages | : 671 |
Release | : 2009-09-16 |
ISBN-10 | : 9783642034138 |
ISBN-13 | : 3642034136 |
Rating | : 4/5 (38 Downloads) |
The book of invited articles offers a collection of high-quality papers in selected and highly topical areas of Applied and Numerical Mathematics and Approximation Theory which have some connection to Wolfgang Dahmen's scientific work. On the occasion of his 60th birthday, leading experts have contributed survey and research papers in the areas of Nonlinear Approximation Theory, Numerical Analysis of Partial Differential and Integral Equations, Computer-Aided Geometric Design, and Learning Theory. The main focus and common theme of all the articles in this volume is the mathematics building the foundation for most efficient numerical algorithms for simulating complex phenomena.
Author | : Wolfgang Bangerth |
Publisher | : Birkhäuser |
Total Pages | : 216 |
Release | : 2013-11-11 |
ISBN-10 | : 9783034876056 |
ISBN-13 | : 303487605X |
Rating | : 4/5 (56 Downloads) |
These Lecture Notes have been compiled from the material presented by the second author in a lecture series ('Nachdiplomvorlesung') at the Department of Mathematics of the ETH Zurich during the summer term 2002. Concepts of 'self adaptivity' in the numerical solution of differential equations are discussed with emphasis on Galerkin finite element methods. The key issues are a posteriori er ror estimation and automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method (or shortly D WR method) for goal-oriented error estimation is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. The basics of the DWR method and various of its applications are described in the following survey articles: R. Rannacher [114], Error control in finite element computations. In: Proc. of Summer School Error Control and Adaptivity in Scientific Computing (H. Bulgak and C. Zenger, eds), pp. 247-278. Kluwer Academic Publishers, 1998. M. Braack and R. Rannacher [42], Adaptive finite element methods for low Mach-number flows with chemical reactions.
Author | : Wenbin Liu |
Publisher | : Alpha Science International Limited |
Total Pages | : 197 |
Release | : 2012 |
ISBN-10 | : 1842657151 |
ISBN-13 | : 9781842657157 |
Rating | : 4/5 (51 Downloads) |
Summary: "This book emphasizes the discussions of some unique issues from the adaptive finite element approximation of optimal control. The main idea used in the approximation error analysis (both a priori and a posteriori) is to first combine convex analysis and interpolation error estimations of suitable interpolators, which much depend on the structure of the control constraints, to derive the error estimates for the control via the variational inequalities in the optimality conditions, and then to apply the standard techniques to derive the error estimates for the state equations. The need, the framework and the techniques of using multi adaptive meshes in developing efficient numerical algorithms for optimal control have been emphasized throughout the book. The book starts from several typical examples of optimal control problems and then discusses existence and optimality conditions for some optimal control problems. It is believed that these discussions are especially useful for the researchers and students who first entered this area. Then the finite element approximation schemes for several typical optimal control problems are set up, their a priori and a posteriori error estimates are derived following the main idea mentioned, and their computational methods are studied."-- Publisher website, viewed 13th July, 2012.
Author | : Wolfgang Dahmen |
Publisher | : Elsevier |
Total Pages | : 587 |
Release | : 1997-08-13 |
ISBN-10 | : 9780080537146 |
ISBN-13 | : 0080537146 |
Rating | : 4/5 (46 Downloads) |
This latest volume in the Wavelets Analysis and Its Applications Series provides significant and up-to-date insights into recent developments in the field of wavelet constructions in connection with partial differential equations. Specialists in numerical applications and engineers in a variety of fields will find Multiscale Wavelet for Partial Differential Equations to be a valuable resource. - Covers important areas of computational mechanics such as elasticity and computational fluid dynamics - Includes a clear study of turbulence modeling - Contains recent research on multiresolution analyses with operator-adapted wavelet discretizations - Presents well-documented numerical experiments connected with the development of algorithms, useful in specific applications
Author | : Christian A. Möller |
Publisher | : Logos Verlag Berlin GmbH |
Total Pages | : 259 |
Release | : 2011 |
ISBN-10 | : 9783832528157 |
ISBN-13 | : 3832528156 |
Rating | : 4/5 (57 Downloads) |
Adaptivity is a crucial tool in state-of-the-art scientific computing. However, its theoretical foundations are only understood partially and are subject of current research. This self-contained work provides theoretical basics on partial differential equations and finite element discretizations before focusing on adaptive finite element methods for time dependent problems. In this context, aspects of temporal adaptivity and error control are considered in particular. Based on the gained insights, a specific adaptive algorithm is designed and analyzed thoroughly. Most importantly, it is proven that the presented adaptive method terminates within any demanded error tolerance. Moreover, the developed algorithm is analyzed from a numerical point of view and its performance is compared to well-known standard methods. Finally, it is applied to the real-life problem of concrete carbonation, where two different discretizations are compared.
Author | : Stefan F. D'Heedene |
Publisher | : |
Total Pages | : 142 |
Release | : 2005 |
ISBN-10 | : OCLC:60654119 |
ISBN-13 | : |
Rating | : 4/5 (19 Downloads) |
The Finite Element Method (FEM) is a widely popular method for the numerical solution of Partial Differential Equations (PDE), on multi-dimensional unstructured meshes. Lagrangian finite elements, which preserve C0 continuity with interpolating piecewise-polynomial shape functions, are a common choice for second-order PDEs. Conventional single-scale methods often have difficulty in efficiently capturing fine-scale behavior (e.g. singularities or transients), without resorting to a prohibitively large number of variables. This can be done more effectively with a multi-scale method, such as the Hierarchical Basis (HB) method. However, the HB FEM generally yields a multi-resolution stiffness matrix that is coupled across scales. We propose a powerful generalization of the Hierarchical Basis: a second-generation wavelet basis, spanning a Lagrangian finite element space of any given polynomial order. Unlike first-generation wavelets, second-generation wavelets can be constructed on any multi-dimensional unstructured mesh. Instead of limiting ourselves to the choice of primitive wavelets, effectively HB detail functions, we can tailor the wavelets to gain additional qualities. In particular, we propose to customize our wavelets to the problem's operator. For any given linear elliptic second-order PDE, and within a Lagrangian FE space of any given order, we can construct a basis of compactly supported wavelets that are orthogonal to the coarser basis functions with respect to the weak form of the PDE. We expose the connection between the wavelet's vanishing moment properties and the requirements for operator-orthogonality in multiple dimensions. We give examples in which we successfully eliminate all scale-coupling in the problem's multi-resolution stiffness matrix. Consequently, details can be added locally to a coarser solution without having to re-compute the coarser solution. This quality can be exploited in the adaptive solution of a wide range of problems. By using an adaptive operator-customized wavelet basis, we achieve an optimal solution speed for problems with concentrated local errors. We illustrate this with the computation of a two-dimensional Green's Function on a bounded domain. We also apply our adaptive solution technique to speed up barrier option valuation, governed by a multi-dimensional diffusion-convection-reaction PDE with varying coefficients.