An Analysis of the Finite Element Method Second Edition 2008 Gilbert Strang / George J. Fix INTRODUCTION TO THE SECOND EDITION FOREWORD TO THE 1997 EDITION PREFACE 1 AN INTRODUCTION TO THE THEORY 1.1 The Basic Ideas 1.2 A Two-point Boundary-value Problem 1.3 The Variational Form of the Problem 1.4 Finite Dierence Approximations 1.5 The Ritz Method and Linear Elements 1.6 The Error with Linear Elements 1.7 The Finite Element Method in One Dimension 1.8 Boundary Value Problems in Two Dimensions 1.9 Triangular and Rectangular Elements 1.10 Element Matrices in Two-dimensional Problems 2 A SUMMARY OF THE THEORY 2.1 Basis Functions for the Finite Element Spaces S^h 2.2 Rates of Convergence 2.3 Galerkin's Method, Collocation, and the Mixed Method 2.4 Systems of Equations; Shell Problems; Variations on the Finite Element Method 3 APPROXIMATION 3.1 Pointwise Approximation 3.2 Mean-square Approximation 3.3 Curved Elements and Isoparametric Transformations 3.4 Error Estimates 4 VARIATIONAL CRIMES 4.1 Violations of the Rayleigh-Ritz Code 4.2 Non-conforming Elements 4.3 Numerical Integration 4.4 Approximation of Domain and Boundary Conditions 5 STABILITY 5.1 Independence of the Basis 5.2 The Condition Number 6 EIGENVALUE PROBLEMS 6.1 Variational Formulation and the Min-max Principle 6.2 Some Elementary Examples 6.3 Eigenvalue and Eigenfunction Errors 6.4 Computational Techniques 7 INITIAL-VALUE PROBLEMS 7.1 The Galerkin-Crank-Nicolson Method for the Heat Equation 7.2 Stability and Convergence in Parabolic Problems 7.3 Hyperbolic Equations 8 SINGULARITIES 8.1 Corners and Interfaces 8.2 Singular Functions 8.3 Errors in the Presence of Singularities 8.4 Experimental Results BIBLIOGRAPHY INDEX OF NOTATIONS INDEX PART 2 (2008) 9 FINITE ELEMENTS IN ONE DIMENSION 9.1 Weak Form with Linear Trial Functions 9.2 Cubic Splines and Fourth Order Equations 10 THE FINITE ELEMENT METHOD IN 2D AND 3D 11 ERRORS IN PROJECTIONS AND EIGENVALUES 12 MIXED FINITE ELEMENTS: VELOCITY AND PRESSURE APPENDIX A: DISCONTINUOUS GALERKIN METHODS APPENDIX B: FAST POISSON SOLVERS INDEX FOR CHAPTERS 9-12, APPENDICES A,B