Computing with hp-ADAPTIVE FINITE ELEMENTS : Volume 1 One and Two Dimensional Elliptic and Maxwell Problems book cover
1st Edition

Volume 1 One and Two Dimensional Elliptic and Maxwell Problems

ISBN 9781584886716
Published October 25, 2006 by Chapman and Hall/CRC
398 Pages 114 B/W Illustrations

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Book Description

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In Computing with hp-ADAPTIVE FINITE ELEMENTS, the information provided, including many unpublished details, aids in solving elliptic and Maxwell problems.

Table of Contents

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

Galerkin Method
Finite Dimensional Approximation of the VBVP
Elementary Convergence Analysis

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

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

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
Assignment 3: Studying Convergence
Definition of a Finite Element

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

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

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

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

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

2D hp Code
Getting Started
Data Structure in FORTRAN 90
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

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

2D Maxwell Equations
Introduction to Maxwell's Equation
Variational Formulation

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

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

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
Numerical Experiments

A Quick Summary and Outlook


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"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