Basic finite element methods

Contents:

• Basic finite element methods
• Approximation of vectors
  • Approximation of planar vectors
  • Approximation of general vectors
• Approximation of functions
  • The least squares method (3)
  • The Galerkin or projection method (3)
  • Example: linear approximation
  • Implementation of the least squares method
  • Perfect approximation
  • Ill-conditioning
  • Fourier series
  • Orthogonal basis functions
  • The collocation (interpolation) method
  • Lagrange polynomials
• Finite element basis functions
  • Elements and nodes
  • The basis functions
  • Calculating the linear system
  • Assembly of elementwise computations
  • Mapping to a reference element
  • Integration over a reference element
• Implementation (1)
  • Integration
  • Linear system assembly and solution
  • Example on computing approximations
  • The structure of the coefficient matrix
  • Applications
  • Sparse matrix storage and solution
• Comparison of finite element and finite difference approximation
  • Collocation or interpolation
  • 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 (2)
  • Cubic Hermite polynomials
• Numerical integration (1)
  • Newton-Cotes rules
  • Gauss-Legendre rules with optimized points
• Approximation of functions in 2D
  • Global basis functions (1)
  • Implementation (3)
• 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 (1)
  • Exercise 1: Linear algebra refresher I
  • Exercise 2: Linear algebra refresher II
  • Exercise 3: Approximate a three-dimensional vector in a plane
  • Exercise 4: Approximate the sine function by power functions
  • Exercise 5: Approximate a steep function by sines
  • Exercise 6: Fourier series as a least squares approximation
  • Exercise 7: Approximate a steep function by Lagrange polynomials
  • Exercise 8: Define finite element meshes
  • Exercise 9: Construct matrix sparsity patterns
  • Exercise 10: Perform symbolic finite element computations
  • Exercise 11: Approximate a steep function by P1 and P2 elements
  • Exercise 12: Approximate a \(\tanh\) function by P3 and P4 elements
  • Exercise 13: Investigate the approximation errors in finite elements
  • Exercise 14: Approximate a step function by finite elements
  • Exercise 15: 2D approximation with orthogonal functions
  • Exercise 16: Use the Trapezoidal rule and P1 elements
• Basic principles for approximating differential equations
  • Differential equation models
  • Residual-minimizing principles
  • Examples on using the principles
  • Integration by parts
  • Boundary function
  • Abstract notation for variational formulations
  • More examples on variational formulations
  • Example on computing with Dirichlet and Neumann conditions
  • Variational problems and optimization of functionals
• Computing with finite elements
  • Computation in the global physical domain
  • Elementwise computations (2)
• Boundary conditions: specified value
  • General construction of a boundary function
  • Modification of the linear system
  • Symmetric modification of the linear system
  • Modification of the element matrix and vector
• Boundary conditions: specified derivative
  • The variational formulation
  • Direct computation of the global linear system
  • Elementwise computations (3)
• Implementation (4)
  • Global basis functions (2)
  • Example: constant right-hand side
  • Finite elements
• Variational formulations in 2D and 3D
  • Transformation to a reference cell in 2D and 3D
  • Numerical integration (2)
  • Convenient formulas for P1 elements in 2D
• Summary
• Time-dependent problems
  • Discretization in time by a Forward Euler scheme
  • Discretization in time by a Backward Euler scheme
  • Analysis of the discrete equations
• Systems of differential equations
  • Variational forms (3)
  • A worked example
  • Identical function spaces for the unknowns
  • Different function spaces for the unknowns
  • Computations in 1D
  • Another example in 1D
• Exercises (2)
  • Exercise 17: Compute the deflection of a cable with sine functions
  • Exercise 18: Check integration by parts
  • Exercise 19: Compute the deflection of a cable with 2 P1 elements
  • Exercise 20: Compute the deflection of a cable with 1 P2 element
  • Exercise 21: Compute the deflection of a cable with a step load
  • Exercise 22: Show equivalence between linear systems
  • Exercise 23: Compute with a non-uniform mesh
  • Exercise 24: Solve a 1D finite element problem by hand
  • Exercise 25: Compare finite elements and differences for a radially symmetric Poisson equation
  • Exercise 26: Compute with variable coefficients and P1 elements by hand
  • Exercise 27: Solve a 2D Poisson equation using polynomials and sines

Indices and tables

• Index
• Module Index
• Search Page