Approximation of functions

Contents:

• Approximation of functions
• 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

Index

• Index