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

Author: Xiaobing Feng

Publisher: Springer Science & Business Media

Published: 2013-11-08

Total Pages: 289

ISBN-13: 3319018183

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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.

Galerkin Finite Element Methods for Parabolic Problems
Galerkin Finite Element Methods for Parabolic Problems

Author: Vidar Thomee

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 310

ISBN-13: 3662033593

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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.

Nodal Discontinuous Galerkin Methods
Nodal Discontinuous Galerkin Methods

Author: Jan S. Hesthaven

Publisher: Springer Science & Business Media

Published: 2007-12-18

Total Pages: 507

ISBN-13: 0387720650

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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.

Comparision of Continuous and Discontinuous Galerkin Finite Element Methods for Parabolic Partial Differential Equations with Implicit Time Stepping
Comparision of Continuous and Discontinuous Galerkin Finite Element Methods for Parabolic Partial Differential Equations with Implicit Time Stepping

Author: Garret Dan Vo

Publisher:

Published: 2012

Total Pages: 208

ISBN-13:

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A number of different discretization techniques and algorithms have been developed for approximating the solution of parabolic partial differential equations. A standard approach, especially for applications that involve complex geometries, is the classic continuous Galerkin finite element method. This approach has a strong theoretical foundation and has been widely and successfully applied to this category of differential equations. One challenging sub-category of problems, however, are equations that include an advection term that is large relative to the second-order, diffusive term. For these advection dominated problems, the continuous Galerkin finite element method can become unstable and yield highly inaccurate results. An alternative to the continuous Galerkin finite element method is the discontinuous Galerkin finite element method, and, through the use of a numerical flux term used in deriving the weak form, the discontinuous approach has the potential to be much more stable in highly advective problems. However, the discontinuous Galerkin finite element method also has significantly more degrees-of-freedom due to the replication of nodes along element edges and vertices. The work presented here compares the computational cost, stability, and accuracy (when possible) of continuous and discontinuous Galerkin finite element methods for four different test problems, including the advection-diffusion equation, viscous Burgers' equation, and the Turing pattern formation equation system. The comparison is performed using as much shared code as possible between the two algorithms and direct, iterative, and multilevel linear solvers. The results show that, for implicit time stepping, the continuous Galerkin finite element method is typically 5-20 times less computationally expensive than the discontinuous Galerkin finite element method using the same finite element mesh and element order. However, the discontinuous Galerkin finite element method is significantly more stable than the continuous Galerkin finite element method for advection dominated problems and is able to obtain accurate approximate solutions for cases where the classic, un-stabilized continuous Galerkin finite element method fails.

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

Author: Thomas Lee Lewis

Publisher:

Published: 2013

Total Pages: 305

ISBN-13:

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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.

Mathematical Aspects of Finite Elements in Partial Differential Equations
Mathematical Aspects of Finite Elements in Partial Differential Equations

Author: Carl de Boor

Publisher: Academic Press

Published: 2014-05-10

Total Pages: 431

ISBN-13: 1483268071

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Mathematical Aspects of Finite Elements in Partial Differential Equations addresses the mathematical questions raised by the use of finite elements in the numerical solution of partial differential equations. This book covers a variety of topics, including finite element method, hyperbolic partial differential equation, and problems with interfaces. Organized into 13 chapters, this book begins with an overview of the class of finite element subspaces with numerical examples. This text then presents as models the Dirichlet problem for the potential and bipotential operator and discusses the question of non-conforming elements using the classical Ritz- and least-squares-method. Other chapters consider some error estimates for the Galerkin problem by such energy considerations. This book discusses as well the spatial discretization of problem and presents the Galerkin method for ordinary differential equations using polynomials of degree k. The final chapter deals with the continuous-time Galerkin method for the heat equation. This book is a valuable resource for mathematicians.

Finite Element Methods and Their Applications
Finite Element Methods and Their Applications

Author: Zhangxin Chen

Publisher: Springer Science & Business Media

Published: 2005-06-23

Total Pages: 415

ISBN-13: 3540240780

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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.