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

Adaptive Discontinuous Galerkin Finite Element Methods for Second and Fourth Order Elliptic Partial Differential Equations
Author: Michael Authur Saum
Publisher:
Total Pages: 221
Release: 2006
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ISBN:
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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

Adaptive Discontinuous Galerkin Methods for Fourth Order Problems
Author: Juha Mikael Virtanen
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Total Pages:
Release: 2010
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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

Adaptive Spline Finite Element Methods for Fourth Order Elliptic Problems
Author: Ibrahim Al Balushi
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Total Pages:
Release: 2021
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ISBN:
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"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

Galerkin Finite Element Methods for Parabolic Problems
Author: Vidar Thomee
Publisher: Springer Science & Business Media
Total Pages: 310
Release: 2013-04-17
Genre: Mathematics
ISBN: 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.

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
Total Pages: 289
Release: 2013-11-08
Genre: Mathematics
ISBN: 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.

Adaptive Finite Element Methods for Differential Equations

Adaptive Finite Element Methods for Differential Equations
Author: Wolfgang Bangerth
Publisher: Birkhäuser
Total Pages: 216
Release: 2013-11-11
Genre: Mathematics
ISBN: 303487605X
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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

Adaptive Discontinuous Galerkin Finite Element Methods
Author: Haihang You
Publisher:
Total Pages: 112
Release: 2009
Genre:
ISBN:
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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.

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:
Total Pages: 305
Release: 2013
Genre: Differential equations, Partial
ISBN:
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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.

Finite Element Methods and Their Applications

Finite Element Methods and Their Applications
Author: Zhangxin Chen
Publisher: Springer Science & Business Media
Total Pages: 415
Release: 2005-06-23
Genre: Science
ISBN: 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.