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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: 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: Peter K. Moore Publisher: ISBN: Category : Languages : en Pages : 30
Book Description
An adaptive local refinement element method for solving vector systems of parabolic partial differential equations in two-space dimensions and time. The algorithm uses the finite element Galerkin method in space and backward Euler temporal integration. Each time step obtains an estimate of the error on each element, group the elements whose error violates a user prescribed tolerance, form new local grids, and solve the problem again on each of the new grids. We discuss several aspects of the algorithm including the necessary data structures, the error estimation technique, and the determination of initial and boundary conditions at coarse-fine mesh interfaces. Finally, several examples demonstrate the viability of this approach. Keywords: Finite element methods, Adaptive methods, Overlapping grids, Local refinement, Parabolic systems.
Author: George D. Byrne Publisher: World Scientific ISBN: 9789810205577 Category : Mathematics Languages : en Pages : 226
Book Description
Ordinary differential equations (ODEs), differential-algebraic equations (DAEs) and partial differential equations (PDEs) are among the forms of mathematics most widely used in science and engineering. Each of these equation types is a focal point for international collaboration and research. This book contains papers by recognized numerical analysts who have made important contributions to the solution of differential systems in the context of realistic applications, and who now report the latest results of their work in numerical methods and software for ODEs/DAEs/PDEs. The papers address parallelization and vectorization of numerical methods, the numerical solution of ODEs/DAEs/PDEs, and the use of these numerical methods in realistic scientific and engineering applications.
Author: Publisher: ISBN: Category : Aeronautics Languages : en Pages : 456
Book Description
Lists citations with abstracts for aerospace related reports obtained from world wide sources and announces documents that have recently been entered into the NASA Scientific and Technical Information Database.
Author: Zhangxin Chen Publisher: World Scientific Publishing Company ISBN: 9813100656 Category : Mathematics Languages : en Pages : 348
Book Description
This Finite Element Method offers a fundamental and practical introduction to the finite element method, its variants, and their applications in engineering. Every concept is introduced in the simplest possible setting, while maintaining a level of treatment that is as rigorous as possible without being unnecessarily abstract. Various finite elements in one, two, and three space dimensions are introduced, and their applications to elliptic, parabolic, hyperbolic, and nonlinear equations and to solid mechanics, fluid mechanics, and porous media flow problems are addressed. The variants include the control volume, multipoint flux approximation, nonconforming, mixed, discontinuous, characteristic, adaptive, and multiscale finite element methods. Illustrative computer programs in Fortran and C++ are described. An extensive set of exercises are provided in each chapter. This book serves as a text a for one-semester course for upper-level undergraduates and beginning graduate students and as a professional reference for engineers, mathematicians, and scientists.
Author: Zhangxin Chen Publisher: Springer Science & Business Media ISBN: 3540280782 Category : Science Languages : en Pages : 415
Book Description
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: Wolfgang Bangerth Publisher: Birkhäuser ISBN: 303487605X Category : Mathematics Languages : en Pages : 216
Book Description
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: A, Vande Wouwer Publisher: CRC Press ISBN: 1420035614 Category : Mathematics Languages : en Pages : 432
Book Description
The general Method of Lines (MOL) procedure provides a flexible format for the solution of all the major classes of partial differential equations (PDEs) and is particularly well suited to evolutionary, nonlinear wave PDEs. Despite its utility, however, there are relatively few texts that explore it at a more advanced level and reflect the method's current state of development. Written by distinguished researchers in the field, Adaptive Method of Lines reflects the diversity of techniques and applications related to the MOL. Most of its chapters focus on a particular application but also provide a discussion of underlying philosophy and technique. Particular attention is paid to the concept of both temporal and spatial adaptivity in solving time-dependent PDEs. Many important ideas and methods are introduced, including moving grids and grid refinement, static and dynamic gridding, the equidistribution principle and the concept of a monitor function, the minimization of a functional, and the moving finite element method. Applications addressed include shallow water flow, combustion and flame propagation, transport in porous media, gas dynamics, chemical engineering processes, solitary waves, and magnetohydrodynamics. As the first advanced text to represent the modern era of the method of lines, this monograph offers an outstanding opportunity to discover new concepts, learn new techniques, and explore a wide range of applications.
Author: Slimane Adjerid
Author: Slimane Adjerid
Author: Peter K. Moore
Author: George D. Byrne
Author: 
Author: Zhangxin Chen
Author: Zhangxin Chen
Author: 
Author: Wolfgang Bangerth
Author: A, Vande Wouwer
Author: J. H. Bolstad