4.3 Draft − 5/28/04 © 2004 J.E. Akin 1ContentsPreface ..................................................................................................................... iiiNotation .................................................................................................................. iv1. Introduction ..................................................................................................... 11.1 Finite Element Methods ........................................................................ 11.2 Capabilities of FEA ............................................................................... 31.3 Outline of Finite Element Procedures ................................................... 71.4 Assembly into the System Equations .................................................... 121.5 Error Concepts ...................................................................................... 211.6 Exercises ............................................................................................... 221.7 Bibliography ......................................................................................... 242. Mathematical Preliminaries ........................................................................... 262.1 Introduction ........................................................................................... 262.2 Linear Spaces and Norms ..................................................................... 282.3 Sobolev Norms*................................................................................... 292.4 Dual Problems, Self-Adjointness .......................................................... 292.5 Weighted Residuals ............................................................................... 312.6 Boundary Conditions Terms ................................................................. 352.7 Adding More Unknowns ....................................................................... 392.8 Numerical Integration ........................................................................... 392.9 Integration By Parts .............................................................................. 412.10 Finite Element Model Problem .............................................................. 412.11 Continuous Nodal Flux Recovery ......................................................... 562.12 A One-Dimensional Example Error Analysis ....................................... 592.13 General Boundary Condition Choices ................................................... 672.14 General Matrix Partitions ....................................................................... 692.15 Elliptic Boundary Value Problems ........................................................ 702.16 Initial Value Problems ........................................................................... 772.17 Equivalent Forms*................................................................................ 822.18 Exercises ............................................................................................... 854.3 Draft − 5/28/04© 2004 J.E. Akin iii J. E. Akin2.19 Bibliography ......................................................................................... 893. Element Interpolation and Local Coordinates3.1 Introduction ........................................................................................... 913.2 Linear Interpolation .............................................................................. 913.3 Quadratic Interpolation ......................................................................... 953.4 Lagrange Interpolation .......................................................................... 963.5 Hermitian Interpolation ......................................................................... 973.6 Hierarchal Interpolation ........................................................................ 1003.7 Space-Time Interpolation*................................................................... 1053.8 Nodally Exact Interpolations*.............................................................. 1053.9 Interpolation Error*.............................................................................. 1063.10 Gradient Estimates*............................................................................. 1093.11 Exercises ............................................................................................... 1123.12 Bibliography ......................................................................................... 1144. One-Dimensional Integration ........................................................................ 1154.1 Introduction ........................................................................................... 1154.2 Local Coordinate Jacobian .................................................................... 1154.3 Exact Polynomial Integration*............................................................. 1164.4 Numerical Integration ........................................................................... 1184.5 Variable Jacobians ................................................................................. 1224.6 Exercises ............................................................................................... 1254.7 Bibliography ......................................................................................... 1255. Error Estimation for Elliptic Problems ........................................................ 1265.1 Introduction ........................................................................................... 1265.2 Error Estimates ...................................................................................... 1305.3 Hierarchical Error Indicator .................................................................. 1315.4 Flux Balancing Methods ....................................................................... 1355.5 Element Adaptivity ............................................................................... 1375.6 H Adaptivity .......................................................................................... 1385.7 P Adaptivity .......................................................................................... 1385.8 HP Adaptivity ....................................................................................... 1395.9 Exercises