Intro; Preface; Second Preface; Contents; List of Exercises and Problems; 1 Quick Overview of the Finite Element Method; 2 Function Approximation by Global Functions; 2.1 Approximation of Vectors; 2.1.1 Approximation of Planar Vectors; 2.1.2 Approximation of General Vectors; 2.2 Approximation Principles; 2.2.1 The Least Squares Method; 2.2.2 The Projection (or Galerkin) Method; 2.2.3 Example of Linear Approximation; 2.2.4 Implementation of the Least Squares Method; 2.2.5 Perfect Approximation; 2.2.6 The Regression Method; 2.3 Orthogonal Basis Functions; 2.3.1 Ill-Conditioning

2.3.2 Fourier Series2.3.3 Orthogonal Basis Functions; 2.3.4 Numerical Computations; 2.4 Interpolation; 2.4.1 The Interpolation (or Collocation) Principle; 2.4.2 Lagrange Polynomials; 2.4.3 Bernstein Polynomials; 2.5 Approximation Properties and Convergence Rates; 2.6 Approximation of Functions in Higher Dimensions; 2.6.1 2D Basis Functions as Tensor Products of 1D Functions; 2.6.2 Example on Polynomial Basis in 2D; 2.6.3 Implementation; 2.6.4 Extension to 3D; 2.7 Exercises; Problem 2.1: Linear Algebra Refresher; Problem 2.2: Approximate a Three-Dimensional Vector in a Plane

Problem 2.3: Approximate a Parabola by a SineProblem 2.4: Approximate the Exponential Function by Power Functions; Problem 2.5: Approximate the Sine Function by Power Functions; Problem 2.6: Approximate a Steep Function by Sines; Problem 2.7: Approximate a Steep Function by Sines with Boundary Adjustment; Exercise 2.8: Fourier Series as a Least Squares Approximation; Problem 2.9: Approximate a Steep Function by Lagrange Polynomials; Problem 2.10: Approximate a Steep Function by Lagrange Polynomials and Regression; 3 Function Approximation by Finite Elements; 3.1 Finite Element Basis Functions

3.1.1 Elements and Nodes3.1.2 The Basis Functions; 3.1.3 Example on Quadratic Finite Element Functions; 3.1.4 Example on Linear Finite Element Functions; 3.1.5 Example on Cubic Finite Element Functions; 3.1.6 Calculating the Linear System; 3.1.7 Assembly of Elementwise Computations; 3.1.8 Mapping to a Reference Element; 3.1.9 Example on Integration over a Reference Element; 3.2 Implementation; 3.2.1 Integration; 3.2.2 Linear System Assembly and Solution; 3.2.3 Example on Computing Symbolic Approximations; 3.2.4 Using Interpolation Instead of Least Squares

3.2.5 Example on Computing Numerical Approximations3.2.6 The Structure of the Coefficient Matrix; 3.2.7 Applications; 3.2.8 Sparse Matrix Storage and Solution; 3.3 Comparison of Finite Elements and Finite Differences; 3.3.1 Finite Difference Approximation of Given Functions; 3.3.2 Interpretation of a Finite Element Approximation in Terms of Finite Difference Operators; 3.3.3 Making Finite Elements Behave as Finite Differences; 3.4 A Generalized Element Concept; 3.4.1 Cells, Vertices, and Degrees of Freedom; 3.4.2 Extended Finite Element Concept; 3.4.3 Implementation