$$ \newcommand{\half}{\frac{1}{2}} \newcommand{\halfi}{{1/2}} \newcommand{\tp}{\thinspace .} \newcommand{\x}{\boldsymbol{x}} \newcommand{\X}{\boldsymbol{X}} \renewcommand{\u}{\boldsymbol{u}} \renewcommand{\v}{\boldsymbol{v}} \newcommand{\w}{\boldsymbol{w}} \newcommand{\e}{\boldsymbol{e}} \newcommand{\f}{\boldsymbol{f}} \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\Iz}{\mathcal{I}_z} \newcommand{\If}{\mathcal{I}_s} % for FEM \newcommand{\Ifd}{{I_d}} % for FEM \newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}} \newcommand{\sequencej}[1]{\left\{ {#1}_j \right\}_{j\in\If}} \newcommand{\basphi}{\varphi} \newcommand{\baspsi}{\psi} \newcommand{\refphi}{\tilde\basphi} \newcommand{\psib}{\boldsymbol{\psi}} \newcommand{\xno}[1]{x_{#1}} \newcommand{\Xno}[1]{X_{(#1)}} \newcommand{\xdno}[1]{\boldsymbol{x}_{#1}} \newcommand{\dX}{\, \mathrm{d}X} \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\Real}{\mathbb{R}} \newcommand{\Integer}{\mathbb{Z}} $$

Approximation of functions

Hans Petter Langtangen [1, 2]

[1] Center for Biomedical Computing, Simula Research Laboratory
[2] Department of Informatics, University of Oslo

2016


PRELIMINARY VERSION

Table of contents

Approximation of vectors
      Approximation of planar vectors
      Approximation of general vectors
Approximation of functions
      The least squares method
      The projection (or Galerkin) method
      Example: linear approximation
      Implementation of the least squares method
      Perfect approximation
      Ill-conditioning
      Fourier series
      Orthogonal basis functions
      Numerical computations
      The interpolation (or collocation) method
      The regression method
      Lagrange polynomials
Finite element basis functions
      Elements and nodes
      The basis functions
      Example on piecewise quadratic finite element functions
      Example on piecewise linear finite element functions
      Example on piecewise cubic finite element basis functions
      Calculating the linear system
      Assembly of elementwise computations
      Mapping to a reference element
      Example: Integration over a reference element
Implementation
      Integration
      Linear system assembly and solution
      Example on computing symbolic approximations
      Using interpolation instead of least squares
      Example on computing numerical approximations
      The structure of the coefficient matrix
      Applications
      Sparse matrix storage and solution
Comparison of finite element and finite difference approximations
      Finite difference approximation of given functions
      Finite difference interpretation of a finite element approximation
      Making finite elements behave as finite differences
A generalized element concept
      Cells, vertices, and degrees of freedom
      Extended finite element concept
      Implementation
      Computing the error of the approximation
      Example: Cubic Hermite polynomials
Numerical integration
      Newton-Cotes rules
      Gauss-Legendre rules with optimized points
Approximation of functions in 2D
      2D basis functions as tensor products of 1D functions
      Example: Polynomial basis in 2D
      Implementation
      Extension to 3D
Finite elements in 2D and 3D
      Basis functions over triangles in the physical domain
      Basis functions over triangles in the reference cell
      Affine mapping of the reference cell
      Isoparametric mapping of the reference cell
      Computing integrals
Exercises
      Problem 1: Linear algebra refresher
      Problem 2: Approximate a three-dimensional vector in a plane
      Problem 3: Approximate a parabola by a sine
      Problem 4: Approximate the exponential function by power functions
      Problem 5: Approximate the sine function by power functions
      Problem 6: Approximate a steep function by sines
      Problem 7: Approximate a steep function by sines with boundary adjustment
      Exercise 8: Fourier series as a least squares approximation
      Problem 9: Approximate a steep function by Lagrange polynomials
      Problem 10: Approximate a steep function by Lagrange polynomials and regression
      Problem 11: Define nodes and elements
      Problem 12: Define vertices, cells, and dof maps
      Problem 13: Construct matrix sparsity patterns
      Problem 14: Perform symbolic finite element computations
      Problem 15: Approximate a steep function by P1 and P2 elements
      Problem 16: Approximate a steep function by P3 and P4 elements
      Exercise 17: Investigate the approximation error in finite elements
      Problem 18: Approximate a step function by finite elements
      Exercise 19: 2D approximation with orthogonal functions
      Exercise 20: Use the Trapezoidal rule and P1 elements
      Exercise 21: Compare P1 elements and interpolation
      Exercise 22: Implement 3D computations with global basis functions
      Exercise 23: Use Simpson's rule and P2 elements
Bibliography

The finite element method is a powerful tool for solving partial differential equations. The method can easily deal with complex geometries and higher-order approximations of the solution. Below is a two-dimensional domain with a non-trivial geometry.

The idea of the finite element method is to divide the domain into triangles (elements) and seek a polynomial approximations to the unknown functions on each triangle. The method glues these piecewise approximations together to find a global solution. Linear and quadratic polynomials over the triangles are particularly popular, because of their mathematical simplicity, but higher-degree polynomials are advantageous to create very computationally efficient methods. The reason for using triangles is that they can easily approximate geometrically complicated domains, but quadrilateral elements and boxes in 3D are also widely used.