Adaptive Discontinuous Galerkin Finite Element Methods For Second And Fourth Order Elliptic Partial Differential Equations

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Adaptive Discontinuous Galerkin Finite Element Methods for Second and Fourth Order Elliptic Partial Differential Equations

Author : Michael Authur Saum
Publisher :
ISBN 13 :
Total Pages : 221 pages
Book Rating : 4.:/5 (714 download)

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Book Synopsis Adaptive Discontinuous Galerkin Finite Element Methods for Second and Fourth Order Elliptic Partial Differential Equations by : Michael Authur Saum

Download or read book Adaptive Discontinuous Galerkin Finite Element Methods for Second and Fourth Order Elliptic Partial Differential Equations written by Michael Authur Saum and published by . This book was released on 2006 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: A unified mathematical and computational framework for implementation of an adaptive discontinuous Galerkin (DG) finite element method (FEM) is developed using the symmetric interior penalty formulation to obtain numerical approximations to solutions of second and fourth order elliptic partial deferential equations. The DG-FEM formulation implemented allows for h-adaptivity and has the capability to work with linear, quadratic, cubic, and quartic polynomials on triangular elements in two dimensions. Two different formulations of DG are implemented based on how fluxes are represented on interior edges and comparisons are made. Explicit representations of two a posteriori error estimators, a residual based type and a "local" based type, are extended to include both Dirichlet and Neumann type boundary conditions on bounded domains. New list-based approaches to data management in an adaptive computational environment are introduced in an effort to utilize computational resources in an efficient and flexible manner.

Adaptive Discontinuous Galerkin Methods for Fourth Order Problems

Author : Juha Mikael Virtanen
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (757 download)

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Book Synopsis Adaptive Discontinuous Galerkin Methods for Fourth Order Problems by : Juha Mikael Virtanen

Download or read book Adaptive Discontinuous Galerkin Methods for Fourth Order Problems written by Juha Mikael Virtanen and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This work is concerned with the derivation of adaptive methods for discontinuous Galerkin approximations of linear fourth order elliptic and parabolic partial differential equations. Adaptive methods are usually based on a posteriori error estimates. To this end, a new residual-based a posteriori error estimator for discontinuous Galerkin approximations to the biharmonic equation with essential boundary conditions is presented. The estimator is shown to be both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm, under minimal regularity assumptions. The reliability bound is based on a new recovery operator, which maps discontinuous finite element spaces to conforming finite element spaces (of two polynomial degrees higher), consisting of triangular or quadrilateral Hsieh-Clough-Tocher macroelements. The efficiency bound is based on bubble function techniques. The performance of the estimator within an h-adaptive mesh refinement procedure is validated through a series of numerical examples, verifying also its asymptotic exactness. Some remarks on the question of proof of convergence of adaptive algorithms for discontinuous Galerkin for fourth order elliptic problems are also presented. Furthermore, we derive a new energy-norm a posteriori error bound for an implicit Euler time-stepping method combined with spatial discontinuous Galerkin scheme for linear fourth order parabolic problems. A key tool in the analysis is the elliptic reconstruction technique. A new challenge, compared to the case of conforming finite element methods for parabolic problems, is the control of the evolution of the error due to non-conformity. Based on the error estimators, we derive an adaptive numerical method and discuss its practical implementation and illustrate its performance in a series of numerical experiments.

Adaptive Spline Finite Element Methods for Fourth Order Elliptic Problems

Author : Ibrahim Al Balushi
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (125 download)

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Book Synopsis Adaptive Spline Finite Element Methods for Fourth Order Elliptic Problems by : Ibrahim Al Balushi

Download or read book Adaptive Spline Finite Element Methods for Fourth Order Elliptic Problems written by Ibrahim Al Balushi and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: "This thesis concentrates on the error analysis of B-spline based finite-element methods for three fourth-order elliptic partial differential equations subject to essential boundary conditions. The first being the biharmonic equation with square-integrable right-hand side and the second and third are models for quasi-geostrophic equations (QGE) simulating large-scale wind-driven oceanic currents. The goal of this thesis is two-fold. On one hand, we derive and analyze error estimators for the purpose of adaptive h-refinement. The earliest effort was concerned with the linear Stommel-Munk. We note that a second-order treatment has been done in 2009 by Juntunen and Stenberg where the analysis hinges on a so-called saturation assumption to relate the numerical error with the discrete error between two refinements. We carry out a similar analysis for the fourth-order PDE. In the nonlinear SQGE we perform the error analysis without a saturation assumption making this work novel in two ways: The treatment requires dealing with the nonlinear convective term and the reliability proofs are saturation-assumption free. The second goal of this thesis is concerned with the convergence and optimality of Nitschetype adaptive methods for the biharmonic equation. Such a study for general second order elliptic order equations has been extensively studied when essential boundary conditions are prescribed into the discrete space. The first convergence proof for the Poisson problem was given by D ̈orfler in 1996 and improved on by Morin, Nochetto, and Siebert in 2000 where some stringent conditions on the domain partitions were removed. Those ideas were soon to be extended to general second order linear elliptic problems by Mekchay and Nochetto, and finally a convergence analysis in a Hilbert space setting was given by Morin, Siebert and Veeser. The first analysis of convergence rates and quasi-optimality for the Poisson problem is pioneered by Binev, Dahmen and DeVore in 2004 and also by Stevenson where he removed an artificial coarsening step. Those ideas were applied to symmetric second order linear elliptic problems by Casc ́on, Kreuzer, Nochetto and Siebert and further generalized by Feischl, Führer and Praetorius to non-symmetric linear problems as well as to strongly monotone nonlinear operators. We add that all aforementioned literature consider boundary condition conforming finite-element spaces in that those discrete spaces satisfy the boundary conditions. For completeness, we do the same for the biharmonic problem. As far as non-conforming methods are concerned, to the best of our knowledge, no such study has been made for Nitsche’s method before the appearance of our work, not even for the Poisson problem. The closest situation we have is that of discontinuous Galerkin methods for symmetric second order elliptic problems which we draw our inspiration from. The convergence and quasi-optimality of discontinuous Galerkin methods was studied by Bonito, Andrea and Nochetto in 2010"--

Galerkin Finite Element Methods for Parabolic Problems

Author : Vidar Thomee
Publisher : Springer Science & Business Media
ISBN 13 : 3662033593
Total Pages : 310 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Galerkin Finite Element Methods for Parabolic Problems by : Vidar Thomee

Download or read book Galerkin Finite Element Methods for Parabolic Problems written by Vidar Thomee and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations

Author : Xiaobing Feng
Publisher : Springer Science & Business Media
ISBN 13 : 3319018183
Total Pages : 289 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations by : Xiaobing Feng

Download or read book Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations written by Xiaobing Feng and published by Springer Science & Business Media. This book was released on 2013-11-08 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: The field of discontinuous Galerkin finite element methods has attracted considerable recent attention from scholars in the applied sciences and engineering. This volume brings together scholars working in this area, each representing a particular theme or direction of current research. Derived from the 2012 Barrett Lectures at the University of Tennessee, the papers reflect the state of the field today and point toward possibilities for future inquiry. The longer survey lectures, delivered by Franco Brezzi and Chi-Wang Shu, respectively, focus on theoretical aspects of discontinuous Galerkin methods for elliptic and evolution problems. Other papers apply DG methods to cases involving radiative transport equations, error estimates, and time-discrete higher order ALE functions, among other areas. Combining focused case studies with longer sections of expository discussion, this book will be an indispensable reference for researchers and students working with discontinuous Galerkin finite element methods and its applications.

Adaptive Finite Element Methods for Differential Equations

Author : Wolfgang Bangerth
Publisher : Birkhäuser
ISBN 13 : 303487605X
Total Pages : 216 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Adaptive Finite Element Methods for Differential Equations by : Wolfgang Bangerth

Download or read book Adaptive Finite Element Methods for Differential Equations written by Wolfgang Bangerth and published by Birkhäuser. This book was released on 2013-11-11 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Adaptive Discontinuous Galerkin Finite Element Methods

Author : Haihang You
Publisher :
ISBN 13 :
Total Pages : 112 pages
Book Rating : 4.:/5 (557 download)

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Book Synopsis Adaptive Discontinuous Galerkin Finite Element Methods by : Haihang You

Download or read book Adaptive Discontinuous Galerkin Finite Element Methods written by Haihang You and published by . This book was released on 2009 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Discontinuous Galerkin Method is one variant of the Finite Element Methods for solving partial differential equations, which was first introduced by Reed and Hill in 1970's [27]. Discontinuous Galerkin Method (DGFEM) differs from the standard Galerkin FEM that continuity constraints are not imposed on the inter-element boundaries. It results in a solution which is composed of totally piecewise discontinuous functions. The absence of continuity constraints on the inter-element boundaries implies that DG method has a great deal of flexibility at the cost of increasing the number of degrees of freedom. This flexibility is the source of many but not all of the advantages of the DGFEM method over the Continuous Galerkin (CGFEM) method that uses spaces of continuous piecewise polynomial functions and other "less standard" methods such as nonconforming methods. As DGFEM method leads to bigger system to solve, theoretical and practical approaches to speed it up are our main focus in this dissertation. This research aims at designing and building an adaptive discontinuous Galerkin finite element method to solve partial differential equations with fast time for desired accuracy on modern architecture.

Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations

Author : Xiaobing Feng
Publisher :
ISBN 13 : 9783319018195
Total Pages : 292 pages
Book Rating : 4.0/5 (181 download)

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Book Synopsis Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations by : Xiaobing Feng

Download or read book Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations written by Xiaobing Feng and published by . This book was released on 2013-11-30 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations

Download Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations PDF Online Free

Author : Thomas Lee Lewis
Publisher :
ISBN 13 :
Total Pages : 305 pages
Book Rating : 4.:/5 (12 download)

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Book Synopsis Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations by : Thomas Lee Lewis

Download or read book Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations written by Thomas Lee Lewis and published by . This book was released on 2013 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: The dissertation focuses on numerically approximating viscosity solutions to second order fully nonlinear partial differential equations (PDEs). The primary goals of the dissertation are to develop, analyze, and implement a finite difference (FD) framework, a local discontinuous Galerkin (LDG) framework, and an interior penalty discontinuous Galerkin (IPDG) framework for directly approximating viscosity solutions of fully nonlinear second order elliptic PDE problems with Dirichlet boundary conditions. The developed frameworks are also extended to fully nonlinear second order parabolic PDEs. All of the proposed direct methods are tested using Monge-Ampere problems and Hamilton-Jacobi-Bellman (HJB) problems. Due to the significance of HJB problems in relation to stochastic optimal control, an indirect methodology for approximating HJB problems that takes advantage of the inherent structure of HJB equations is also developed. First, a FD framework is developed that guarantees convergence to viscosity solutions when certain properties concerning admissibility, stability, consistency, and monotonicity are satisfied. The key concepts introduced are numerical operators, numerical moments, and generalized monotonicity. One class of FD methods that fulfills the framework provides a direct realization of the vanishing moment method for approximating second order fully nonlinear PDEs. Next, the emphasis is on extending the FD framework using DG methodologies. In particular, some nonstandard LDG and IPDG methods that utilize key concepts from the FD framework are formulated. Benefits of the DG methodologies over the FD methodology include the ability to handle more complicated domains, more freedom in the design of meshes, higher potential for adaptivity, and the ability to use high order elements as a means for increased accuracy. Last, a class of indirect methods for approximating HJB equations using the vanishing moment method paired with a splitting formulation of the HJB problem is developed and tested numerically. The proposed methodology is well-suited for both continuous and discontinuous Galerkin methods, and it complements the direct methods developed in the dissertation.

Galerkin Finite Element Methods for Parabolic Problems

Author : Vidar Thomée
Publisher : Springer Science & Business Media
ISBN 13 : 9783540632368
Total Pages : 320 pages
Book Rating : 4.6/5 (323 download)

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Book Synopsis Galerkin Finite Element Methods for Parabolic Problems by : Vidar Thomée

Download or read book Galerkin Finite Element Methods for Parabolic Problems written by Vidar Thomée and published by Springer Science & Business Media. This book was released on 2010 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Finite Elements II

Author : Alexandre Ern
Publisher : Springer Nature
ISBN 13 : 3030569233
Total Pages : 491 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Finite Elements II by : Alexandre Ern

Download or read book Finite Elements II written by Alexandre Ern and published by Springer Nature. This book was released on 2021-04-22 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Discontinuous Galerkin Methods

Author : Bernardo Cockburn
Publisher : Springer Science & Business Media
ISBN 13 : 3642597211
Total Pages : 468 pages
Book Rating : 4.6/5 (425 download)

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Book Synopsis Discontinuous Galerkin Methods by : Bernardo Cockburn

Download or read book Discontinuous Galerkin Methods written by Bernardo Cockburn and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Nodal Discontinuous Galerkin Methods

Author : Jan S. Hesthaven
Publisher : Springer Science & Business Media
ISBN 13 : 0387720650
Total Pages : 507 pages
Book Rating : 4.3/5 (877 download)

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Book Synopsis Nodal Discontinuous Galerkin Methods by : Jan S. Hesthaven

Download or read book Nodal Discontinuous Galerkin Methods written by Jan S. Hesthaven and published by Springer Science & Business Media. This book was released on 2007-12-18 with total page 507 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. It covers all key theoretical results, including an overview of relevant results from approximation theory, convergence theory for numerical PDE’s, and orthogonal polynomials. Through embedded Matlab codes, coverage discusses and implements the algorithms for a number of classic systems of PDE’s: Maxwell’s equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations.

Finite Element Methods and Their Applications

Author : Zhangxin Chen
Publisher : Springer Science & Business Media
ISBN 13 : 3540240780
Total Pages : 415 pages
Book Rating : 4.5/5 (42 download)

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Book Synopsis Finite Element Methods and Their Applications by : Zhangxin Chen

Download or read book Finite Element Methods and Their Applications written by Zhangxin Chen and published by Springer Science & Business Media. This book was released on 2005-06-23 with total page 415 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

Author : National Aeronautics and Space Adm Nasa
Publisher :
ISBN 13 : 9781724110190
Total Pages : 26 pages
Book Rating : 4.1/5 (11 download)

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Book Synopsis Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives by : National Aeronautics and Space Adm Nasa

Download or read book Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives written by National Aeronautics and Space Adm Nasa and published by . This book was released on 2018-09-27 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh. Yan, Jue and Shu, Chi-Wang and Bushnell, Dennis M. (Technical Monitor) Langley Research Center NASA/CR-2002-211959, NAS 1.26:211959, ICASE-2002-42...

Discontinuous Galerkin Finite Element Methods for (non)conservative Partial Differential Equations

Author : Sander Rhebergen
Publisher :
ISBN 13 : 9789036529648
Total Pages : 171 pages
Book Rating : 4.5/5 (296 download)

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Book Synopsis Discontinuous Galerkin Finite Element Methods for (non)conservative Partial Differential Equations by : Sander Rhebergen

Download or read book Discontinuous Galerkin Finite Element Methods for (non)conservative Partial Differential Equations written by Sander Rhebergen and published by . This book was released on 2010 with total page 171 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Adaptive Finite Elements in the Discretization of Parabolic Problems

Author : Christian A. Möller
Publisher : Logos Verlag Berlin GmbH
ISBN 13 : 3832528156
Total Pages : 259 pages
Book Rating : 4.8/5 (325 download)

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Book Synopsis Adaptive Finite Elements in the Discretization of Parabolic Problems by : Christian A. Möller

Download or read book Adaptive Finite Elements in the Discretization of Parabolic Problems written by Christian A. Möller and published by Logos Verlag Berlin GmbH. This book was released on 2011 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.