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

Computing with hp-ADAPTIVE FINITE ELEMENTS

Volume 1 One and Two Dimensional Elliptic and Maxwell Problems, 1st Edition

By Leszek Demkowicz

Chapman and Hall/CRC

398 pages | 114 B/W Illus.

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pub: 2006-10-25
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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In Computing with hp-ADAPTIVE FINITE ELEMENTS, the information provided, including many unpublished details, aids in solving elliptic and Maxwell problems.

Reviews

"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

Table of Contents

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

About the Series

Chapman & Hall/CRC Applied Mathematics & Nonlinear Science

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT021000
MATHEMATICS / Number Systems
SCI041000
SCIENCE / Mechanics / General