A Posteriori Error Estimation for Hybridized Mixed and Discontinuous Galerkin Methods PDF Download
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Author: Johannes Neher
Publisher: Logos Verlag Berlin GmbH
ISBN: 3832530886
Category : Mathematics
Languages : en
Pages : 106
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Book Description
There is a variety of finite element based methods applicable to the discretization of second order elliptic boundary value problems in mixed form. However, it is expensive to solve the resulting discrete linear system due to its size and its algebraic structure. Hybridization serves as a tool to circumvent these difficulties. Furthermore hybridization is an elegant concept to establish connections among various finite element methods. In this work connections between the methods and their hybridized counterparts are established after showing the link between three different formulations of the elliptic model problem. The main part of the work contains the development of a reliable a posteriori error estimator, which is applicable to all of the methods above. This estimator is the key ingredient of an adaptive numerical approximation of the original boundary value problem. Finally, a number of numerical tests is discussed in order to exhibit the performance of the adaptive hybridized methods.
Author: Johannes Neher
Publisher: Logos Verlag Berlin GmbH
ISBN: 3832530886
Category : Mathematics
Languages : en
Pages : 106
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Book Description
There is a variety of finite element based methods applicable to the discretization of second order elliptic boundary value problems in mixed form. However, it is expensive to solve the resulting discrete linear system due to its size and its algebraic structure. Hybridization serves as a tool to circumvent these difficulties. Furthermore hybridization is an elegant concept to establish connections among various finite element methods. In this work connections between the methods and their hybridized counterparts are established after showing the link between three different formulations of the elliptic model problem. The main part of the work contains the development of a reliable a posteriori error estimator, which is applicable to all of the methods above. This estimator is the key ingredient of an adaptive numerical approximation of the original boundary value problem. Finally, a number of numerical tests is discussed in order to exhibit the performance of the adaptive hybridized methods.
Author: Christian Nolde
Publisher: Logos Verlag Berlin GmbH
ISBN: 3832544534
Category : Mathematics
Languages : en
Pages : 88
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Book Description
The use of rigorous numerical methods to approach problems which can not be solved using standard methods (yet) has increased signifiantly in recent years. In this book, riogorous a-posteriori methods are used to study the time evolution of a surface growth model, given by a fourth order semi-linear parabolic partial differential equation, where standard methods fail to verify global uniqueness and smoothness of solutions. Based on an arbitrary numerical approximation, a-posteriori error-analysis is applied in order to prevent a blow up analytically. This is a method that in a similar way also applies to the three dimensional Navier-Stokes equations. The main idea consists of energy-estimates for the error between solution and approximation that yields a scalar differential equation controlling the norm of the error with coefficients depending solely on the numerical data. This allows the solution of the differential equation to be bounded using only numerical data. A key technical tool is a rigorous eigenvalue bound for the nonlinear operator linearized around the numerical approximation. The presented method succeeds to show global uniqueness for relatively large initial conditions, which is demonstrated in many numerical examples.
Author: Martin Jesenko
Publisher: Logos Verlag Berlin GmbH
ISBN: 383254478X
Category : Mathematics
Languages : en
Pages : 145
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Book Description
In the calculus of variations, the goal is to explore extrema of a given integral functional. From origins of the problem, it might be expected that the functional can be adequately simplified by neglecting some small quantities. A way to rigorously justify such an approximation is the Γ-convergence that ensures convergence of corresponding (global) extrema. The main motivation of this work is to investigate properties of doubly indexed integral functionals that Γ-converge for one index fixed. In other words, for two possible approximations we would like to determine whether we may perform them consecutively and if they commute. Our examples are taken from material science with homogenization being one of these two processes. In the first part we are considering a setting related to the elastic regime. However, our assumptions are fairly general and allow for applications in different areas. The second part is devoted to problems in the Hencky plasticity. They are considerably different due to special growth properties of the density.
Author: Robert M. Kirby
Publisher: Springer
ISBN: 3319198009
Category : Computers
Languages : en
Pages : 530
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Book Description
The book contains a selection of high quality papers, chosen among the best presentations during the International Conference on Spectral and High-Order Methods (2014), and provides an overview of the depth and breadth of the activities within this important research area. The carefully reviewed selection of papers will provide the reader with a snapshot of the state-of-the-art and help initiate new research directions through the extensive biography.
Author: Andrea Cangiani
Publisher: Springer
ISBN: 3319676733
Category : Mathematics
Languages : en
Pages : 131
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Book Description
Over the last few decades discontinuous Galerkin finite element methods (DGFEMs) have been witnessed tremendous interest as a computational framework for the numerical solution of partial differential equations. Their success is due to their extreme versatility in the design of the underlying meshes and local basis functions, while retaining key features of both (classical) finite element and finite volume methods. Somewhat surprisingly, DGFEMs on general tessellations consisting of polygonal (in 2D) or polyhedral (in 3D) element shapes have received little attention within the literature, despite the potential computational advantages. This volume introduces the basic principles of hp-version (i.e., locally varying mesh-size and polynomial order) DGFEMs over meshes consisting of polygonal or polyhedral element shapes, presents their error analysis, and includes an extensive collection of numerical experiments. The extreme flexibility provided by the locally variable elemen t-shapes, element-sizes, and element-orders is shown to deliver substantial computational gains in several practical scenarios.
Author: Alexandre Ern
Publisher: Springer Nature
ISBN: 3030569233
Category : Mathematics
Languages : en
Pages : 491
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Book Description
This book is the second volume of a three-part textbook suitable for graduate coursework, professional engineering and academic research. It is also appropriate for graduate flipped classes. Each volume is divided into short chapters. Each chapter can be covered in one teaching unit and includes exercises as well as solutions available from a dedicated website. The salient ideas can be addressed during lecture, with the rest of the content assigned as reading material. To engage the reader, the text combines examples, basic ideas, rigorous proofs, and pointers to the literature to enhance scientific literacy. Volume II is divided into 32 chapters plus one appendix. The first part of the volume focuses on the approximation of elliptic and mixed PDEs, beginning with fundamental results on well-posed weak formulations and their approximation by the Galerkin method. The material covered includes key results such as the BNB theorem based on inf-sup conditions, Céa's and Strang's lemmas, and the duality argument by Aubin and Nitsche. Important implementation aspects regarding quadratures, linear algebra, and assembling are also covered. The remainder of Volume II focuses on PDEs where a coercivity property is available. It investigates conforming and nonconforming approximation techniques (Galerkin, boundary penalty, Crouzeix—Raviart, discontinuous Galerkin, hybrid high-order methods). These techniques are applied to elliptic PDEs (diffusion, elasticity, the Helmholtz problem, Maxwell's equations), eigenvalue problems for elliptic PDEs, and PDEs in mixed form (Darcy and Stokes flows). Finally, the appendix addresses fundamental results on the surjectivity, bijectivity, and coercivity of linear operators in Banach spaces.
Author: Shukai Du
Publisher: Springer Nature
ISBN: 3030272303
Category : Mathematics
Languages : en
Pages : 124
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Book Description
This monograph requires basic knowledge of the variational theory of elliptic PDE and the techniques used for the analysis of the Finite Element Method. However, all the tools for the analysis of FEM (scaling arguments, finite dimensional estimates in the reference configuration, Piola transforms) are carefully introduced before being used, so that the reader does not need to go over longforgotten textbooks. Readers include: computational mathematicians, numerical analysts, engineers and scientists interested in new and computationally competitive Discontinuous Galerkin methods. The intended audience includes graduate students in computational mathematics, physics, and engineering, since the prerequisites are quite basic for a second year graduate student who has already taken a non necessarily advanced class in the Finite Element method.
Author: Gabriel R. Barrenechea
Publisher: Springer
ISBN: 3319416405
Category : Computers
Languages : en
Pages : 433
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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: Daniele Antonio Di Pietro
Publisher: Springer
ISBN: 3319946765
Category : Mathematics
Languages : en
Pages : 312
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Book Description
This volume gathers contributions from participants of the Introductory School and the IHP thematic quarter on Numerical Methods for PDE, held in 2016 in Cargese (Corsica) and Paris, providing an opportunity to disseminate the latest results and envisage fresh challenges in traditional and new application fields. Numerical analysis applied to the approximate solution of PDEs is a key discipline in applied mathematics, and over the last few years, several new paradigms have appeared, leading to entire new families of discretization methods and solution algorithms. This book is intended for researchers in the field.
Author: Bernardo Cockburn
Publisher: Springer Science & Business Media
ISBN: 3642597211
Category : Mathematics
Languages : en
Pages : 468
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Book Description
A class of finite element methods, the Discontinuous Galerkin Methods (DGM), has been under rapid development recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simula tion, turbomachinery, turbulent flows, materials processing, MHD and plasma simulations, and image processing. While there has been a lot of interest from mathematicians, physicists and engineers in DGM, only scattered information is available and there has been no prior effort in organizing and publishing the existing volume of knowledge on this subject. In May 24-26, 1999 we organized in Newport (Rhode Island, USA), the first international symposium on DGM with equal emphasis on the theory, numerical implementation, and applications. Eighteen invited speakers, lead ers in the field, and thirty-two contributors presented various aspects and addressed open issues on DGM. In this volume we include forty-nine papers presented in the Symposium as well as a survey paper written by the organiz ers. All papers were peer-reviewed. A summary of these papers is included in the survey paper, which also provides a historical perspective of the evolution of DGM and its relation to other numerical methods. We hope this volume will become a major reference in this topic. It is intended for students and researchers who work in theory and application of numerical solution of convection dominated partial differential equations. The papers were written with the assumption that the reader has some knowledge of classical finite elements and finite volume methods.
