Author: Christian A. Möller Publisher: Logos Verlag Berlin GmbH ISBN: 3832528156 Category : Mathematics Languages : en Pages : 259
Book Description
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: Vidar Thomee Publisher: Springer Science & Business Media ISBN: 3662033593 Category : Mathematics Languages : en Pages : 310
Book Description
My purpose in this monograph is to present an essentially self-contained account of the mathematical theory of Galerkin finite element methods as applied to parabolic partial differential equations. The emphases and selection of topics reflects my own involvement in the field over the past 25 years, and my ambition has been to stress ideas and methods of analysis rather than to describe the most general and farreaching results possible. Since the formulation and analysis of Galerkin finite element methods for parabolic problems are generally based on ideas and results from the corresponding theory for stationary elliptic problems, such material is often included in the presentation. The basis of this work is my earlier text entitled Galerkin Finite Element Methods for Parabolic Problems, Springer Lecture Notes in Mathematics, No. 1054, from 1984. This has been out of print for several years, and I have felt a need and been encouraged by colleagues and friends to publish an updated version. In doing so I have included most of the contents of the 14 chapters of the earlier work in an updated and revised form, and added four new chapters, on semigroup methods, on multistep schemes, on incomplete iterative solution of the linear algebraic systems at the time levels, and on semilinear equations. The old chapters on fully discrete methods have been reworked by first treating the time discretization of an abstract differential equation in a Hilbert space setting, and the chapter on the discontinuous Galerkin method has been completely rewritten.
Author: Slimane Adjerid Publisher: ISBN: Category : Languages : en Pages : 39
Book Description
Adaptive finite element methods are given for solving initial boundary value problems for vector systems of parabolic partial differential equations in one- and two-space dimensions. One-dimension systems are discretized using piecewise linear finite element approximations in space and a backward difference code for stiff ordinary differential systems in time. A spatial error estimate is calculated using piecewise quadratic approximations that employ nodal superconvergence to increase computational efficiency. This error estimate is used to move and refine the finite element mesh in order to equidistribute a measure of the total spatial error and to satisfy a prescribed error tolerance. Ordinary differential equations for the spatial error estimate and the mesh motion are integrated in time using the same backward difference software that is used to determine the finite element solution. Two-dimension systems are discretized using piecewise bilinear finite element approximations in space and backward difference software in time. A spatial error estimate is calculated using piecewise cubic approximations that take advantage of nodal superconvergence. This error estimate is used to locally refine a stationary finite element mesh in order to satisfy a prescribed spatial error tolerance.
Author: Jens Lang Publisher: Springer Science & Business Media ISBN: 3662044846 Category : Computers Languages : en Pages : 161
Book Description
Nowadays there is an increasing emphasis on all aspects of adaptively gener ating a grid that evolves with the solution of a PDE. Another challenge is to develop efficient higher-order one-step integration methods which can handle very stiff equations and which allow us to accommodate a spatial grid in each time step without any specific difficulties. In this monograph a combination of both error-controlled grid refinement and one-step methods of Rosenbrock-type is presented. It is my intention to impart the beauty and complexity found in the theoretical investigation of the adaptive algorithm proposed here, in its realization and in solving non-trivial complex problems. I hope that this method will find many more interesting applications. Berlin-Dahlem, May 2000 Jens Lang Acknowledgements I have looked forward to writing this section since it is a pleasure for me to thank all friends who made this work possible and provided valuable input. I would like to express my gratitude to Peter Deuflhard for giving me the oppor tunity to work in the field of Scientific Computing. I have benefited immensly from his help to get the right perspectives, and from his continuous encourage ment and support over several years. He certainly will forgive me the use of Rosenbrock methods rather than extrapolation methods to integrate in time.
Author: Thomas Apel Publisher: Springer ISBN: 3030142442 Category : Mathematics Languages : en Pages : 428
Book Description
Finite element methods are the most popular methods for solving partial differential equations numerically, and despite having a history of more than 50 years, there is still active research on their analysis, application and extension. This book features overview papers and original research articles from participants of the 30th Chemnitz Finite Element Symposium, which itself has a 40-year history. Covering topics including numerical methods for equations with fractional partial derivatives; isogeometric analysis and other novel discretization methods, like space-time finite elements and boundary elements; analysis of a posteriori error estimates and adaptive methods; enhancement of efficient solvers of the resulting systems of equations, discretization methods for partial differential equations on surfaces; and methods adapted to applications in solid and fluid mechanics, it offers readers insights into the latest results.
Author: Ulrich Langer Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110548488 Category : Mathematics Languages : en Pages : 261
Book Description
This volume provides an introduction to modern space-time discretization methods such as finite and boundary elements and isogeometric analysis for time-dependent initial-boundary value problems of parabolic and hyperbolic type. Particular focus is given on stable formulations, error estimates, adaptivity in space and time, efficient solution algorithms, parallelization of the solution pipeline, and applications in science and engineering.
Author: Leszek Demkowicz Publisher: CRC Press ISBN: 1420011693 Category : Mathematics Languages : en Pages : 437
Book Description
With a focus on 1D and 2D problems, the first volume of Computing with hp-ADAPTIVE FINITE ELEMENTS prepared readers for the concepts and logic governing 3D code and implementation. Taking the next step in hp technology, Volume II Frontiers: Three-Dimensional Elliptic and Maxwell Problems with Applications presents the theoretical foundations of the
Author: Claes Johnson Publisher: Courier Corporation ISBN: 0486131599 Category : Mathematics Languages : en Pages : 290
Book Description
An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science- and engineering-related specialties.This text encompasses all varieties of the basic linear partial differential equations, including elliptic, parabolic and hyperbolic problems, as well as stationary and time-dependent problems. Additional topics include finite element methods for integral equations, an introduction to nonlinear problems, and considerations of unique developments of finite element techniques related to parabolic problems, including methods for automatic time step control. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.
Author: Magdalena Salazar-Palma Publisher: Artech House Publishers ISBN: Category : Mathematics Languages : en Pages : 824
Book Description
Ensure the accuracy of your results when applying the Finite Element Method (FEM) to electromagnetic and antenna problems with this self-contained reference. It provides you with a solid understanding of the method, describes its key elements and numerical techniques, and identifies various approaches to using the FEM in solving real-world microwave field problems.
Author: Christian A. Möller
Author: Vidar Thomee
Author: Vidar Thomée
Author: Slimane Adjerid
Author: Jens Lang
Author: Thomas Apel
Author: Ulrich Langer
Author: Leszek Demkowicz
Author: Claes Johnson
Author: Magdalena Salazar-Palma