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Adaptive Discontinuous Galerkin Finite Element Methods
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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.
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.
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.
Book Synopsis Adaptive Discontinuous Galerkin Finite Element Methods for the Compressible Euler Equations by : Ralf Hartmann
Download or read book Adaptive Discontinuous Galerkin Finite Element Methods for the Compressible Euler Equations written by Ralf Hartmann and published by . This book was released on 2001 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Adaptive Discontinuous Galerkin Finite Element Methods for a Diffuse Interface Model of Biological Growth by : Andreas Costa Aristotelous
Download or read book Adaptive Discontinuous Galerkin Finite Element Methods for a Diffuse Interface Model of Biological Growth written by Andreas Costa Aristotelous and published by . This book was released on 2011 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: This PhD dissertation concentrates on the development and application of adaptive Discontinuous Galerkin Finite Element (DG-FE) methods for the numerical solution of a Cahn-Hilliard-type diffuse interface model for biological growth. Models of this type have become popular for studying cancerous tumor progression in vivo. The work in this dissertation advances the state-of-the-art in the following ways: To our knowledge the work here contains the first primitive-variable, completely discontinuous numerical implementations of a 2D scheme for the Cahn-Hilliard equation as well as a diffuse interface model of cancer growth. We provide numerical evidence that the schemes above are convergent, with the optimal order. The efficiency of the numerical algorithms depends largely on the implementation of fast solvers for the systems of equations resulting from the DG-FE discretizations. We have developed such capabilities based on multigrid and sparse direct solver techniques. We demonstrate proof-of-concept regarding the implementation of a practical spatially adaptive meshing algorithm for the numerical schemes just mentioned and the effective use of a very simple, but powerful, marking strategy based on an inverse estimate. We demonstrate proof-of-concept for a novel simplified diffuse interface model of tumor growth. This model is essentially the Cahn-Hilliard equation with an added source term that is specialized for the context of cancerous tumor progression. We devise and analyze a mixed DG-FE scheme of convex splitting (CS) type for the Cahn-Hilliard equation in any space dimension. Specifically, we prove that our scheme is unconditionally energy stable and unconditionally uniquely solvable. Likewise, we devise and analyze a CS, mixed DG-FE scheme for our diffuse interface cancer model. This scheme is energy stable for any (positive) time step size and for any (positive) space step size that is sufficiently small.
Book Synopsis hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes by : Andrea Cangiani
Download or read book hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes written by Andrea Cangiani and published by Springer. This book was released on 2017-11-27 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws by : Ralf Hartmann
Download or read book Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws written by Ralf Hartmann and published by . This book was released on 2001 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Hp-adaptive Discontinuous Galerkin Finite Element Method for Accurate Configurational Force Brittle Crack Propagation by : Robert Edward Bird
Download or read book A Hp-adaptive Discontinuous Galerkin Finite Element Method for Accurate Configurational Force Brittle Crack Propagation written by Robert Edward Bird and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Hp-Adaptive Time-discontinuous Galerkin Finite Element Methods for Second-order Hyperbolic Systems by : Sethuramalingam Subbarayalu
Download or read book Hp-Adaptive Time-discontinuous Galerkin Finite Element Methods for Second-order Hyperbolic Systems written by Sethuramalingam Subbarayalu and published by . This book was released on 2004 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Adaptive Discontinuous Petrov-Galerkin Finite-element-methods by : Friederike Hellwig
Download or read book Adaptive Discontinuous Petrov-Galerkin Finite-element-methods written by Friederike Hellwig and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Adaptive Discontinuous Galerkin Methods for Non-linear Reactive Flows by : Murat Uzunca
Download or read book Adaptive Discontinuous Galerkin Methods for Non-linear Reactive Flows written by Murat Uzunca and published by Birkhäuser. This book was released on 2016-05-17 with total page 111 pages. Available in PDF, EPUB and Kindle. Book excerpt: The focus of this monograph is the development of space-time adaptive methods to solve the convection/reaction dominated non-stationary semi-linear advection diffusion reaction (ADR) equations with internal/boundary layers in an accurate and efficient way. After introducing the ADR equations and discontinuous Galerkin discretization, robust residual-based a posteriori error estimators in space and time are derived. The elliptic reconstruction technique is then utilized to derive the a posteriori error bounds for the fully discrete system and to obtain optimal orders of convergence.As coupled surface and subsurface flow over large space and time scales is described by (ADR) equation the methods described in this book are of high importance in many areas of Geosciences including oil and gas recovery, groundwater contamination and sustainable use of groundwater resources, storing greenhouse gases or radioactive waste in the subsurface.
Book Synopsis A High-order, Adaptive, Discontinuous Galerkin Finite Element Method for the Reynolds-Averaged Navier-Stokes Equations by : Todd A. Oliver
Download or read book A High-order, Adaptive, Discontinuous Galerkin Finite Element Method for the Reynolds-Averaged Navier-Stokes Equations written by Todd A. Oliver and published by . This book was released on 2008 with total page 182 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis presents high-order, discontinuous Galerkin (DG) discretizations of the Reynolds-Averaged Navier-Stokes (RANS) equations and an output-based error estimation and mesh adaptation algorithm for these discretizations. In particular, DG discretizations of the RANS equations with the Spalart-Allmaras (SA) turbulence model are examined. The dual consistency of multiple DG discretizations of the RANS-SA system is analyzed. The approach of simply weighting gradient dependent source terms by a test function and integrating is shown to be dual inconsistent. A dual consistency correction for this discretization is derived. The analysis also demonstrates that discretizations based on the popular mixed formulation, where dependence on the state gradient is handled by introducing additional state variables, are generally asymptotically dual consistent. Numerical results are presented to confirm the results of the analysis. The output error estimation and output-based adaptation algorithms used here are extensions of methods previously developed in the finite volume and finite element communities. In particular, the methods are extended for application on the curved, highly anisotropic meshes required for boundary conforming, high-order RANS simulations. Two methods for generating such curved meshes are demonstrated. One relies on a user-defined global mapping of the physical domain to a straight meshing domain. The other uses a linear elasticity node movement scheme to add curvature to an initially linear mesh. Finally, to improve the robustness of the adaptation process, an "unsteady" algorithm, where the mesh is adapted at each time step, is presented. The goal of the unsteady procedure is to allow mesh adaptation prior to converging a steady state solution, not to obtain a time-accurate solution of an unsteady problem. Thus, an estimate of the error due to spatial discretization in the output of interest averaged over the current time step is developed. This error estimate is then used to drive an h-adaptation algorithm. Adaptation results demonstrate that the high-order discretizations are more efficient than the second-order method in terms of degrees of freedom required to achieve a desired error tolerance. Furthermore, using the unsteady adaptation process, adaptive RANS simulations may be started from extremely coarse meshes, significantly decreasing the mesh generation burden to the user.
Book Synopsis Applications of H- and P-adaptive Discontinuous Galerkin Finite Element Methods to the Rotating Disk Electrode Problem in Electrochemistry by : Daniel Okuonghae
Download or read book Applications of H- and P-adaptive Discontinuous Galerkin Finite Element Methods to the Rotating Disk Electrode Problem in Electrochemistry written by Daniel Okuonghae and published by . This book was released on 2004 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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 Springer Science & Business Media. This book was released on 2003-01-23 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual 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. At the end of each chapter some exercises are posed in order to assist the interested reader in better understanding the concepts presented. Solutions and accompanying remarks are given in the Appendix.
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.
Book Synopsis High-Order Methods for Computational Physics by : Timothy J. Barth
Download or read book High-Order Methods for Computational Physics written by Timothy J. Barth and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining chal lenges facing the field of computational fluid dynamics. In structural me chanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the com putation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order ac curacy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence sug gests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Cen ter. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18,1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25,1998 at the NASA Ames Research Center in the United States.
Book Synopsis Lectures on Elliptic Boundary Value Problems by : Shmuel Agmon
Download or read book Lectures on Elliptic Boundary Value Problems written by Shmuel Agmon and published by American Mathematical Soc.. This book was released on 2010-02-03 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, which is a new edition of a book originally published in 1965, presents an introduction to the theory of higher-order elliptic boundary value problems. The book contains a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher-order elliptic boundary value problems. It also contains a study of spectral properties of operators associated with elliptic boundary value problems. Weyl's law on the asymptotic distribution of eigenvalues is studied in great generality.
