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| Author | : R. B. Kellogg |
| Publisher | : |
| Total Pages | : |
| Release | : 1981 |
| Genre | : |
| ISBN | : |
| Author | : Jens M. Melenk |
| Publisher | : Springer |
| Total Pages | : 331 |
| Release | : 2004-10-19 |
| Genre | : Mathematics |
| ISBN | : 354045781X |
Many partial differential equations arising in practice are parameter-dependent problems that are of singularly perturbed type. Prominent examples include plate and shell models for small thickness in solid mechanics, convection-diffusion problems in fluid mechanics, and equations arising in semi-conductor device modelling. Common features of these problems are layers and, in the case of non-smooth geometries, corner singularities. Mesh design principles for the efficient approximation of both features by the hp-version of the finite element method (hp-FEM) are proposed in this volume. For a class of singularly perturbed problems on polygonal domains, robust exponential convergence of the hp-FEM based on these mesh design principles is established rigorously.
| Author | : Gung-Min Gie |
| Publisher | : Springer |
| Total Pages | : 412 |
| Release | : 2018-11-21 |
| Genre | : Mathematics |
| ISBN | : 3030006387 |
Singular perturbations occur when a small coefficient affects the highest order derivatives in a system of partial differential equations. From the physical point of view singular perturbations generate in the system under consideration thin layers located often but not always at the boundary of the domains that are called boundary layers or internal layers if the layer is located inside the domain. Important physical phenomena occur in boundary layers. The most common boundary layers appear in fluid mechanics, e.g., the flow of air around an airfoil or a whole airplane, or the flow of air around a car. Also in many instances in geophysical fluid mechanics, like the interface of air and earth, or air and ocean. This self-contained monograph is devoted to the study of certain classes of singular perturbation problems mostly related to thermic, fluid mechanics and optics and where mostly elliptic or parabolic equations in a bounded domain are considered. This book is a fairly unique resource regarding the rigorous mathematical treatment of boundary layer problems. The explicit methodology developed in this book extends in many different directions the concept of correctors initially introduced by J. L. Lions, and in particular the lower- and higher-order error estimates of asymptotic expansions are obtained in the setting of functional analysis. The review of differential geometry and treatment of boundary layers in a curved domain is an additional strength of this book. In the context of fluid mechanics, the outstanding open problem of the vanishing viscosity limit of the Navier-Stokes equations is investigated in this book and solved for a number of particular, but physically relevant cases. This book will serve as a unique resource for those studying singular perturbations and boundary layer problems at the advanced graduate level in mathematics or applied mathematics and may be useful for practitioners in other related fields in science and engineering such as aerodynamics, fluid mechanics, geophysical fluid mechanics, acoustics and optics.
| Author | : Hans-Görg Roos |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 364 |
| Release | : 2013-06-29 |
| Genre | : Mathematics |
| ISBN | : 3662032066 |
The analysis of singular perturbed differential equations began early in this century, when approximate solutions were constructed from asymptotic ex pansions. (Preliminary attempts appear in the nineteenth century [vD94].) This technique has flourished since the mid-1960s. Its principal ideas and methods are described in several textbooks. Nevertheless, asymptotic ex pansions may be impossible to construct or may fail to simplify the given problem; then numerical approximations are often the only option. The systematic study of numerical methods for singular perturbation problems started somewhat later - in the 1970s. While the research frontier has been steadily pushed back, the exposition of new developments in the analysis of numerical methods has been neglected. Perhaps the only example of a textbook that concentrates on this analysis is [DMS80], which collects various results for ordinary differential equations, but many methods and techniques that are relevant today (especially for partial differential equa tions) were developed after 1980.Thus contemporary researchers must comb the literature to acquaint themselves with earlier work. Our purposes in writing this introductory book are twofold. First, we aim to present a structured account of recent ideas in the numerical analysis of singularly perturbed differential equations. Second, this important area has many open problems and we hope that our book will stimulate further investigations.Our choice of topics is inevitably personal and reflects our own main interests.
| Author | : Kenneth I. Gross |
| Publisher | : SIAM |
| Total Pages | : 262 |
| Release | : 1984-01-01 |
| Genre | : Technology & Engineering |
| ISBN | : 9780898711998 |
| Author | : |
| Publisher | : |
| Total Pages | : 1214 |
| Release | : 1983 |
| Genre | : Engineering mathematics |
| ISBN | : |
| Author | : Feng Kang |
| Publisher | : M.E. Sharpe |
| Total Pages | : 1056 |
| Release | : 1983 |
| Genre | : Mathematics |
| ISBN | : |
| Author | : S. Selberherr |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 308 |
| Release | : 2012-12-06 |
| Genre | : Technology & Engineering |
| ISBN | : 3709187524 |
The invention of semiconductor devices is a fairly recent one, considering classical time scales in human life. The bipolar transistor was announced in 1947, and the MOS transistor, in a practically usable manner, was demonstrated in 1960. From these beginnings the semiconductor device field has grown rapidly. The first integrated circuits, which contained just a few devices, became commercially available in the early 1960s. Immediately thereafter an evolution has taken place so that today, less than 25 years later, the manufacture of integrated circuits with over 400.000 devices per single chip is possible. Coincident with the growth in semiconductor device development, the literature concerning semiconductor device and technology issues has literally exploded. In the last decade about 50.000 papers have been published on these subjects. The advent of so called Very-Large-Scale-Integration (VLSI) has certainly revealed the need for a better understanding of basic device behavior. The miniaturization of the single transistor, which is the major prerequisite for VLSI, nearly led to a breakdown of the classical models of semiconductor devices.
| Author | : |
| Publisher | : |
| Total Pages | : 640 |
| Release | : 2009 |
| Genre | : Differentiable dynamical systems |
| ISBN | : |
| Author | : O. Axelsson |
| Publisher | : Academic Press |
| Total Pages | : 453 |
| Release | : 2014-05-10 |
| Genre | : Mathematics |
| ISBN | : 1483260569 |
Finite Element Solution of Boundary Value Problems: Theory and Computation provides an introduction to both the theoretical and computational aspects of the finite element method for solving boundary value problems for partial differential equations. This book is composed of seven chapters and begins with surveys of the two kinds of preconditioning techniques, one based on the symmetric successive overrelaxation iterative method for solving a system of equations and a form of incomplete factorization. The subsequent chapters deal with the concepts from functional analysis of boundary value problems. These topics are followed by discussions of the Ritz method, which minimizes the quadratic functional associated with a given boundary value problem over some finite-dimensional subspace of the original space of functions. Other chapters are devoted to direct methods, including Gaussian elimination and related methods, for solving a system of linear algebraic equations. The final chapter continues the analysis of preconditioned conjugate gradient methods, concentrating on applications to finite element problems. This chapter also looks into the techniques for reducing rounding errors in the iterative solution of finite element equations. This book will be of value to advanced undergraduates and graduates in the areas of numerical analysis, mathematics, and computer science, as well as for theoretically inclined workers in engineering and the physical sciences.
