1st Edition

Computing with hp-ADAPTIVE FINITE ELEMENTS Volume 1 One and Two Dimensional Elliptic and Maxwell Problems

By Leszek Demkowicz Copyright 2007
    426 Pages 114 B/W Illustrations
    by Chapman & Hall

    Offering the only existing finite element (FE) codes for Maxwell equations that support hp refinements on irregular meshes, Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1. One- and Two-Dimensional Elliptic and Maxwell Problems presents 1D and 2D codes and automatic hp adaptivity. This self-contained source discusses the theory and implementation of hp-adaptive FE methods, focusing on projection-based interpolation and the corresponding hp-adaptive strategy.

    The book is split into three parts, progressing from simple to more advanced problems. Part I examines the hp elements for the standard 1D model elliptic problem. The author develops the variational formulation and explains the construction of FE basis functions. The book then introduces the 1D code (1Dhp) and automatic hp adaptivity. This first part ends with a study of a 1D wave propagation problem. In Part II, the book proceeds to 2D elliptic problems, discussing two model problems that are slightly beyond standard-level examples: 3D axisymmetric antenna problem for Maxwell equations (example of a complex-valued, indefinite problem) and 2D elasticity (example of an elliptic system). The author concludes with a presentation on infinite elements - one of the possible tools to solve exterior boundary-value problems. Part III focuses on 2D time-harmonic Maxwell equations. The book explains the construction of the hp edge elements and the fundamental de Rham diagram for the whole family of hp discretizations. Next, it explores the differences between the elliptic and Maxwell versions of the 2D code, including automatic hp adaptivity. Finally, the book presents 2D exterior (radiation and scattering) problems and sample solutions using coupled hp finite/infinite elements.

    In Computing with hp-ADAPTIVE FINITE ELEMENTS, the information provided, including many unpublished details, aids in solving elliptic and Maxwell problems.

    1D PROBLEMS
    1D Model Elliptic Problem
    A Two-Point Boundary Value Problem
    Algebraic Structure of the Variational Formulation
    Equivalence with a Minimization Problem
    Sobolev Space H1(0, l)
    Well Posedness of the Variational BVP
    Examples from Mechanics and Physics
    The Case with "Pure Neumann" BCs
    Exercises

    Galerkin Method
    Finite Dimensional Approximation of the VBVP
    Elementary Convergence Analysis
    Comments
    Exercises

    1D hp Finite Element Method
    1D hp Discretization
    Assembling Element Matrices into Global Matrices
    Computing the Element Matrices
    Accounting for the Dirichlet BC
    Summary
    Assignment 1: A Dry Run
    Exercises

    1D hp Code
    Setting up the 1D hp Code
    Fundamentals
    Graphics
    Element Routine
    Assignment 2: Writing Your Own Processor
    Exercises

    Mesh Refinements in 1D
    The h-Extension Operator. Constrained Approximation Coefficients
    Projection-Based Interpolation in 1D
    Supporting Mesh Refinements
    Data-Structure-Supporting Routines
    Programming Bells and Whistles
    Interpolation Error Estimates
    Convergence
    Assignment 3: Studying Convergence
    Definition of a Finite Element
    Exercises

    Automatic hp Adaptivity in 1D
    The hp Algorithm
    Supporting the Optimal Mesh Selection
    Exponential Convergence. Comparing with h Adaptivity
    Discussion of the hp Algorithm
    Algebraic Complexity and Reliability of the Algorithm
    Exercises

    Wave Propagation Problems
    Convergence Analysis for Noncoercive Problems
    Wave Propagation Problems
    Asymptotic Optimality of the Galerkin Method
    Dispersion Error Analysis
    Exercises

    2D ELLIPTIC PROBLEMS
    2D Elliptic Boundary-Value Problem
    Classical Formulation
    Variational (Weak) Formulation
    Algebraic Structure of the Variational Formulation
    Equivalence with a Minimization Problem
    Examples from Mechanics and Physics
    Exercises

    Sobolev Spaces
    Sobolev Space H1(O)
    Sobolev Spaces of an Arbitrary Order
    Density and Embedding Theorems
    Trace Theorem
    Well Posedness of the Variational BVP
    Exercises

    2D hp Finite Element Method on Regular Meshes
    Quadrilateral Master Element
    Triangular Master Element
    Parametric Element
    Finite Element Space. Construction of Basis Functions
    Calculation of Element Matrices
    Modified Element. Imposing Dirichlet Boundary Conditions
    Postprocessing- Local Access to Element d.o.f
    Projection-Based Interpolation
    Exercises

    2D hp Code
    Getting Started
    Data Structure in FORTRAN 90
    Fundamentals
    The Element Routine
    Modified Element. Imposing Dirichlet Boundary Conditions
    Assignment 4: Assembly of Global Matrices
    The Case with "Pure Neumann" Boundary Conditions

    Geometric Modeling and Mesh Generation
    Manifold Representation
    Construction of Compatible Parametrizations
    Implicit Parametrization of a Rectangle
    Input File Preparation
    Initial Mesh Generation

    The hp Finite Element Method on h-Refined Meshes
    Introduction. The h Refinements
    1-Irregular Mesh Refinement Algorithm
    Data Structure in Fortran 90 (Continued)
    Constrained Approximation for C0 Discretizations
    Reconstructing Element Nodal Connectivities
    Determining Neighbors for Midedge Nodes
    Additional Comments

    Automatic hp Adaptivity in 2D
    The Main Idea
    The 2D hp Algorithm
    Example: L-Shape Domain Problem
    Example: 2D "Shock" Problem
    Additional Remarks

    Examples of Applications
    A "Battery Problem"
    Linear Elasticity
    An Axisymmetric Maxwell Problem
    Exercises

    Exterior Boundary-Value Problems
    Variational Formulation. Infinite Element Discretization
    Selection of IE Radial Shape Functions
    Implementation
    Calculation of Echo Area
    Numerical Experiments
    Comments
    Exercises

    2D MAXWELL PROBLEMS
    2D Maxwell Equations
    Introduction to Maxwell's Equation
    Variational Formulation
    Exercises

    Edge Elements and the de Rham Diagram
    Exact Sequences
    Projection-Based Interpolation
    De Rham Diagram
    Shape Functions
    Exercises

    2D Maxwell Code
    Directories. Data Structure
    The Element Routine
    Constrained Approximation. Modified Element
    Setting up a Maxwell Problem
    Exercises

    hp Adaptivity for Maxwell Equations
    Projection-Based Interpolation Revisited
    The hp Mesh Optimization Algorithm
    Example: The Screen Problem

    Exterior Maxwell Boundary-Value Problems
    Variational Formulation
    Infinite Element Discretization in 3D
    Infinite Element Discretization in 2D
    Stability
    Implementation
    Numerical Experiments
    Exercises

    A Quick Summary and Outlook

    Appendix
    Bibliography
    Index

    Biography

    Leszek Demkowicz

    "This book is valuable both for mathematicians and researchers working in finite element methods . . . Instructors who have been using the classical textbooks to teach finite element methods might find this book a worthy successor."

    – Tsu-Fen Chen, in Mathematical Reviews, 2007k

    "It is very well suited for advanced students of mathematics, engineering as well as computer science. In my opinion it is an excellent resource and guide for everybody working on hp- adaptive FEM."

    – Alexander Düster, University of Munich, in ZAMM- Journal of Applied Mathematics and Mechanics, 2007, Vol. 87, No. 7