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Author: Dirk Broersen Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
"In this thesis, Discontinuous Petrov-Galerkin (DPG) finite element methods are developed for convection-diffusion equations. In particular, this thesis focuses on the use of optimal test spaces. A convection-diffusion equation is a singularly perturbed problem. That is, the nature of the problem changes when the diffusion term vanishes, which makes it challenging to solve numerically for small diffusion values, i.e. when convection dominates. Standard finite element methods give very unsatisfactory results, producing approximations that exhibit spurious oscillations and other nonphysical behavior. Recently, a class of finite element methods has been developed, in which optimal test spaces are used. These spaces guarantee that one gets the best approximation from the trial space in which the solution is sought. The methods are examples of least-squares methods, with the special property that one can choose the norm in which the residual is minimized. This freedom of choice allows us to control the norm in which the best approximation is obtained. The new approach in this thesis is that the variational formulation associated with the convection-diffusion problem also gives a well-posed variational formulation of the limit convection problem if the diffusion term vanishes. This is necessary in order to retain stability, and to make sure that the computational cost does not grow, when the diffusion term decreases. Special attention is paid to the transport problem which, besides being the limit problem for vanishing diffusion, also has other applications. A new method is introduced that outperforms existing methods in convergence rates, but also in reducing the smearing of discontinuities of solutions. The theory developed in this thesis is illustrated by various numerical results."--Samenvatting auteur.
Author: Dirk Broersen Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
"In this thesis, Discontinuous Petrov-Galerkin (DPG) finite element methods are developed for convection-diffusion equations. In particular, this thesis focuses on the use of optimal test spaces. A convection-diffusion equation is a singularly perturbed problem. That is, the nature of the problem changes when the diffusion term vanishes, which makes it challenging to solve numerically for small diffusion values, i.e. when convection dominates. Standard finite element methods give very unsatisfactory results, producing approximations that exhibit spurious oscillations and other nonphysical behavior. Recently, a class of finite element methods has been developed, in which optimal test spaces are used. These spaces guarantee that one gets the best approximation from the trial space in which the solution is sought. The methods are examples of least-squares methods, with the special property that one can choose the norm in which the residual is minimized. This freedom of choice allows us to control the norm in which the best approximation is obtained. The new approach in this thesis is that the variational formulation associated with the convection-diffusion problem also gives a well-posed variational formulation of the limit convection problem if the diffusion term vanishes. This is necessary in order to retain stability, and to make sure that the computational cost does not grow, when the diffusion term decreases. Special attention is paid to the transport problem which, besides being the limit problem for vanishing diffusion, also has other applications. A new method is introduced that outperforms existing methods in convergence rates, but also in reducing the smearing of discontinuities of solutions. The theory developed in this thesis is illustrated by various numerical results."--Samenvatting auteur.
Author: K.W. Morton Publisher: CRC Press ISBN: 1351359673 Category : Mathematics Languages : en Pages : 385
Book Description
Accurate modeling of the interaction between convective and diffusive processes is one of the most common challenges in the numerical approximation of partial differential equations. This is partly due to the fact that numerical algorithms, and the techniques used for their analysis, tend to be very different in the two limiting cases of elliptic and hyperbolic equations. Many different ideas and approaches have been proposed in widely differing contexts to resolve the difficulties of exponential fitting, compact differencing, number upwinding, artificial viscosity, streamline diffusion, Petrov-Galerkin and evolution Galerkin being some examples from the main fields of finite difference and finite element methods. The main aim of this volume is to draw together all these ideas and see how they overlap and differ. The reader is provided with a useful and wide ranging source of algorithmic concepts and techniques of analysis. The material presented has been drawn both from theoretically oriented literature on finite differences, finite volume and finite element methods and also from accounts of practical, large-scale computing, particularly in the field of computational fluid dynamics.
Author: Gabriel R. Barrenechea Publisher: Springer ISBN: 3319416405 Category : Computers Languages : en Pages : 443
Book Description
This volume contains contributed survey papers from the main speakers at the LMS/EPSRC Symposium “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”. This meeting took place in July 8-16, 2014, and its main purpose was to gather specialists in emerging areas of numerical PDEs, and explore the connections between the different approaches. The type of contributions ranges from the theoretical foundations of these new techniques, to the applications of them, to new general frameworks and unified approaches that can cover one, or more than one, of these emerging techniques.
Author: Martin Stynes Publisher: American Mathematical Soc. ISBN: 1470448688 Category : Mathematics Languages : en Pages : 168
Book Description
Many physical problems involve diffusive and convective (transport) processes. When diffusion dominates convection, standard numerical methods work satisfactorily. But when convection dominates diffusion, the standard methods become unstable, and special techniques are needed to compute accurate numerical approximations of the unknown solution. This convection-dominated regime is the focus of the book. After discussing at length the nature of solutions to convection-dominated convection-diffusion problems, the authors motivate and design numerical methods that are particularly suited to this class of problems. At first they examine finite-difference methods for two-point boundary value problems, as their analysis requires little theoretical background. Upwinding, artificial diffusion, uniformly convergent methods, and Shishkin meshes are some of the topics presented. Throughout, the authors are concerned with the accuracy of solutions when the diffusion coefficient is close to zero. Later in the book they concentrate on finite element methods for problems posed in one and two dimensions. This lucid yet thorough account of convection-dominated convection-diffusion problems and how to solve them numerically is meant for beginning graduate students, and it includes a large number of exercises. An up-to-date bibliography provides the reader with further reading.
Author: Vít Dolejší Publisher: Springer ISBN: 3319192671 Category : Mathematics Languages : en Pages : 575
Book Description
The subject of the book is the mathematical theory of the discontinuous Galerkin method (DGM), which is a relatively new technique for the numerical solution of partial differential equations. The book is concerned with the DGM developed for elliptic and parabolic equations and its applications to the numerical simulation of compressible flow. It deals with the theoretical as well as practical aspects of the DGM and treats the basic concepts and ideas of the DGM, as well as the latest significant findings and achievements in this area. The main benefit for readers and the book’s uniqueness lie in the fact that it is sufficiently detailed, extensive and mathematically precise, while at the same time providing a comprehensible guide through a wide spectrum of discontinuous Galerkin techniques and a survey of the latest efficient, accurate and robust discontinuous Galerkin schemes for the solution of compressible flow.
Author: Daniel Hayes Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
We present new results on both saddle point least-squares and Petrov-Galerkin methods applied to reaction-diffusion and convection-diffusion equations. In the reaction-diffusion equation, we provide a preconditioner which proves to be efficient and easily implementable in practice. For the convection-diffusion equation, we present an optimal trial norm for both a saddle point least-squares reformulation, as well as an up-winding Petrov-Galerkin method. For the Petrov-Galerkin method in one dimension, we make a connection to a standard streamline upwind Petrov-Galerkin method. We introduce optimal trial norms that enable robust stability analysis of the discretization approach. Furthermore, our discretization methods provide efficient and trustworthy means of approximation. For the two dimensional convection-diffusion equation, we present three methods of discretization. These methods are natural extensions of ideas of the one dimensional problems leading to stabilization and accurate approximation. Numerical results are included to support the theoretical aspects, as well as to give motivation for directions of further research and development.
Author: Pavel B. Bochev Publisher: Springer Science & Business Media ISBN: 0387689222 Category : Mathematics Languages : en Pages : 669
Book Description
Since their emergence, finite element methods have taken a place as one of the most versatile and powerful methodologies for the approximate numerical solution of Partial Differential Equations. These methods are used in incompressible fluid flow, heat, transfer, and other problems. This book provides researchers and practitioners with a concise guide to the theory and practice of least-square finite element methods, their strengths and weaknesses, established successes, and open problems.
Author: Hans-Görg Roos Publisher: Springer Science & Business Media ISBN: 3540344675 Category : Mathematics Languages : en Pages : 599
Book Description
This new edition incorporates new developments in numerical methods for singularly perturbed differential equations, focusing on linear convection-diffusion equations and on nonlinear flow problems that appear in computational fluid dynamics.
Author: Dirk Broersen
Author: Dirk Broersen
Author: K.W. Morton
Author: Bernardo Cockburn
Author: Gabriel R. Barrenechea
Author: Martin Stynes
Author: American Society of Mechanical Engineers. Applied Mechanics Division
Author: Vít Dolejší
Author: Daniel Hayes
Author: Pavel B. Bochev
Author: Hans-Görg Roos