[PDF] Error Estimates For Semidiscrete Finite Element Methods For Parabolic Integro Differential Equations eBook

Author : Chuanmiao Chen
Publisher : World Scientific
Page : 294 pages
File Size : 13,7 MB
Release : 1998
Category : Mathematics
ISBN : 9789810232634

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Recently, there has appeared a new type of evaluating partial differential equations with Volterra integral operators in various practical areas. Such equations possess new physical and mathematical properties. This monograph systematically discusses application of the finite element methods to numerical solution of integrodifferential equations. It will be useful for numerical analysts, mathematicians, physicists and engineers. Advanced undergraduates and graduate students should also find it beneficial.

Author : Chuan Miao Chen
Publisher : World Scientific
Page : 291 pages
File Size : 44,11 MB
Release : 1998-02-28
Category : Mathematics
ISBN : 9814496952

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Recently, there has appeared a new type of evaluating partial differential equations with Volterra integral operators in various practical areas. Such equations possess new physical and mathematical properties. This monograph systematically discusses application of the finite element methods to numerical solution of integrodifferential equations. It will be useful for numerical analysts, mathematicians, physicists and engineers. Advanced undergraduates and graduate students should also find it beneficial.

Author : Rüdiger Verfürth
Publisher : OUP Oxford
Page : 573 pages
File Size : 36,7 MB
Release : 2013-04-18
Category : Mathematics
ISBN : 019166877X

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Self-adaptive discretization methods are now an indispensable tool for the numerical solution of partial differential equations that arise from physical and technical applications. The aim is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools in achieving this goal are a posteriori error estimates which give global and local information on the error of the numerical solution and which can easily be computed from the given numerical solution and the data of the differential equation. This book reviews the most frequently used a posteriori error estimation techniques and applies them to a broad class of linear and nonlinear elliptic and parabolic equations. Although there are various approaches to adaptivity and a posteriori error estimation, they are all based on a few common principles. The main aim of the book is to elaborate these basic principles and to give guidelines for developing adaptive schemes for new problems. Chapters 1 and 2 are quite elementary and present various error indicators and their use for mesh adaptation in the framework of a simple model problem. The basic principles are introduced using a minimal amount of notations and techniques providing a complete overview for the non-specialist. Chapters 4-6 on the other hand are more advanced and present a posteriori error estimates within a general framework using the technical tools collected in Chapter 3. Most sections close with a bibliographical remark which indicates the historical development and hints at further results.

Author : Zhong-ci Shi
Publisher : World Scientific
Page : 210 pages
File Size : 45,22 MB
Release : 1993-10-25
Category :
ISBN : 9814553263

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Traditional quantum theory has a very rigid structure, making it difficult to accommodate new properties emerging from novel systems. This book presents a flexible and unified theory for physical systems, from micro and macro quantum to classical. This is achieved by incorporating superselection rules and maximal symmetric operators into the theory. The resulting theory is applicable to classical, microscopic quantum and non-orthodox mixed quantum systems of which macroscopic quantum systems are examples. A unified formalism also greatly facilitates the discussion of interactions between these systems. A scheme of quantization by parts is introduced, based on the mathematics of selfadjoint and maximal symmetric extensions of symmetric operators, to describe point interactions. The results are applied to treat superconducting quantum circuits in various configurations.This book also discusses various topics of interest such as the asymptotic treatment of quantum state preparation and quantum measurement, local observables and local values, Schrödinger's cat states in superconducting systems, and a path space formulation of quantum mechanics.This self-contained book is complete with a review of relevant geometric and operator theories, for example, vector fields and operators, symmetric operators and their maximal symmetric extensions, direct integrals of Hilbert spaces and operators./a

Author : Dieter Sirch
Publisher : Logos Verlag Berlin GmbH
Page : 166 pages
File Size : 38,74 MB
Release : 2010
Category : Mathematics
ISBN : 3832525572

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Subject of this work is the analysis of numerical methods for the solution of optimal control problems governed by elliptic partial differential equations. Such problems arise, if one does not only want to simulate technical or physical processes but also wants to optimize them with the help of one or more influence variables. In many practical applications these influence variables, so called controls, cannot be chosen arbitrarily, but have to fulfill certain inequality constraints. The numerical treatment of such control constrained optimal control problems requires a discretization of the underlying infinite dimensional function spaces. To guarantee the quality of the numerical solution one has to estimate and to quantify the resulting approximation errors. In this thesis a priori error estimates for finite element discretizations are proved in case of corners or edges in the underlying domain and nonsmooth coefficients in the partial differential equation. These facts influence the regularity properties of the solution and require adapted meshes to get optimal convergence rates. Isotropic and anisotropic refinement strategies are given and error estimates in polygonal and prismatic domains are proved. The theoretical results are confirmed by numerical tests.

Author : Schatz
Publisher : Birkhäuser
Page : 268 pages
File Size : 41,48 MB
Release : 2013-03-09
Category : Science
ISBN : 3034876300

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These are the lecture notes of the seminar "Mathematische Theorie der finiten Element und Randelementmethoden" organized by the "Deutsche Mathematiker-Vereinigung" and held in Dusseldorf from 07. - 14. of June 1987. Finite element methods and the closely related boundary element methods nowadays belong to the standard routines for the computation of solutions to boundary and initial boundary value problems of partial differential equations with many applications as e.g. in elasticity and thermoelasticity, fluid mechanics, acoustics, electromagnetics, scatter ing and diffusion. These methods also stimulated the development of corresponding mathematical numerical analysis. I was very happy that A. Schatz and V. Thomee generously joined the adventure of the seminar and not only gave stimulating lectures but also spent so much time for personal discussion with all the participants. The seminar as well as these notes consist of three parts: 1. An Analysis of the Finite Element Method for Second Order Elliptic Boundary Value Problems by A. H. Schatz. II. On Finite Elements for Parabolic Problems by V. Thomee. III. I30undary Element Methods for Elliptic Problems by \V. L. Wendland. The prerequisites for reading this book are basic knowledge in partial differential equations (including pseudo-differential operators) and in numerical analysis. It was not our intention to present a comprehensive account of the research in this field, but rather to give an introduction and overview to the three different topics which shed some light on recent research.

Author : Vidar Thomee
Publisher : Springer Science & Business Media
Page : 376 pages
File Size : 14,68 MB
Release : 2007-06-25
Category : Mathematics
ISBN : 3540331220

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This book provides insight into the mathematics of Galerkin finite element method as applied to parabolic equations. The revised second edition has been influenced by recent progress in application of semigroup theory to stability and error analysis, particulatly in maximum-norm. Two new chapters have also been added, dealing with problems in polygonal, particularly noncovex, spatial domains, and with time discretization based on using Laplace transformation and quadrature.

Author : Wolfgang Bangerth
Publisher : Birkhäuser
Page : 216 pages
File Size : 29,62 MB
Release : 2013-11-11
Category : Mathematics
ISBN : 303487605X

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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.