Introduction to finite element methods

Contents:

• Introduction to finite element methods
• Approximation of vectors
  • Approximation of planar vectors
    • The least squares method (1)
    • The projection method
  • Approximation of general vectors
    • The least squares method (2)
    • The Galerkin or projection method
• Approximation of functions
  • The least squares method (3)
  • The projection (or Galerkin) method
  • Example: linear approximation
  • Implementation of the least squares method
    • Symbolic integration
    • Fallback on numerical integration
    • Plotting the approximation
  • Perfect approximation
  • Ill-conditioning
  • Fourier series
  • Orthogonal basis functions
  • Numerical computations
  • The interpolation (or collocation) method
    • Example (1)
  • Lagrange polynomials
    • Approximation of a polynomial
    • Successful example
    • Less successful example
    • Remedy for strong oscillations
• Finite element basis functions (1)
  • Elements and nodes
    • Example (2)
  • The basis functions
    • Construction principles
    • Properties of \({\varphi}_i\)
  • 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
    • Calculating a specific matrix entry
    • Calculating a general row in the matrix
  • Assembly of elementwise computations
  • Mapping to a reference element
  • Example: Integration over a reference element
• Implementation (1)
  • Integration
  • Linear system assembly and solution
  • Example on computing symbolic approximations
  • Comparison with finite elements and interpolation/collocation
  • Example on computing numerical approximations
  • The structure of the coefficient matrix
  • Applications
  • Sparse matrix storage and solution
• Comparison of finite element and finite difference approximation
  • Finite difference approximation of given functions
  • Finite difference interpretation of a finite element approximation
  • Making finite elements behave as finite differences
    • Computations in physical space
    • Elementwise computations
    • Terminology
• A generalized element concept
  • Cells, vertices, and degrees of freedom
  • Extended finite element concept
  • Implementation (2)
  • Computing the error of the approximation
  • Example: Cubic Hermite polynomials
• Numerical integration (1)
  • 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 (3)
  • Extension to 3D
• Finite elements in 2D and 3D
  • Basis functions over triangles in the physical domain
    • Element matrices and vectors
  • 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 exponential function by power functions
  • Exercise 5: Approximate the sine function by power functions
  • Exercise 6: Approximate a steep function by sines
    • Remarks
  • Exercise 7: Animate the approximation of a steep function by sines
  • Exercise 8: Fourier series as a least squares approximation
  • Exercise 9: Approximate a steep function by Lagrange polynomials
  • Exercise 10: Define nodes and elements
  • Exercise 11: Define vertices, cells, and dof maps
  • Exercise 12: Construct matrix sparsity patterns
  • Exercise 13: Perform symbolic finite element computations
  • Exercise 14: Approximate a steep function by P1 and P2 elements
  • Exercise 15: Approximate a steep function by P3 and P4 elements
  • Exercise 16: Investigate the approximation error in finite elements
  • Exercise 17: Approximate a step function by finite elements
  • Exercise 18: 2D approximation with orthogonal functions
  • Exercise 19: Use the Trapezoidal rule and P1 elements
  • Problem 20: Compare P1 elements and interpolation
  • Exercise 21: Implement 3D computations with global basis functions
  • Exercise 22: Use Simpson’s rule and P2 elements
• Basic principles for approximating differential equations
  • Differential equation models
  • Simple model problems
  • Forming the residual
  • The least squares method (4)
  • The Galerkin method (1)
  • The Method of Weighted Residuals
  • Test and Trial Functions
  • The collocation method (1)
    • The subdomain collocation method
  • Examples on using the principles
    • The model problem
    • Basis functions
    • The residual
    • The least squares method (5)
    • The Galerkin method (2)
    • The collocation method (2)
    • Comparison
  • Integration by parts
    • Weak form
  • Boundary function (1)
  • Abstract notation for variational formulations
  • Variational problems and optimization of functionals
• Examples on variational formulations
  • Variable coefficient
  • First-order derivative in the equation and boundary condition
  • Nonlinear coefficient
  • Computing with Dirichlet and Neumann conditions
  • When the numerical method is exact
• Computing with finite elements
  • Finite element mesh and basis functions
  • Computation in the global physical domain
  • Comparison with a finite difference discretization
  • Cellwise computations (1)
    • The integral for the element matrix
    • The integral for the element vector
    • Detailed calculations of the element matrix and vector
    • Contributions from the first and last cell
    • Assembly
• Boundary conditions: specified nonzero value
  • General construction of a boundary function
  • Example on computing with finite element-based a boundary function
    • Computations in physical coordinates
    • Cellwise computations on the reference element
  • Modification of the linear system (1)
    • Computations in the physical system
  • Symmetric modification of the linear system
  • Modification of the element matrix and vector
• Boundary conditions: specified derivative
  • The variational formulation
  • Boundary term vanishes because of the test functions
  • Boundary term vanishes because of linear system modifications
  • Direct computation of the global linear system
  • Cellwise computations (2)
• Implementation (4)
  • Global basis functions
  • 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
    • Time discretization (1)
    • Space discretization
  • Variational forms (1)
  • Simplified notation for the solution at recent time levels
  • Deriving the linear systems
  • Computational algorithm
  • Comparing P1 elements with the finite difference method
    • Lumping the mass matrix
  • Discretization in time by a Backward Euler scheme
    • Time discretization (2)
    • Variational forms (2)
    • Linear systems
  • Dirichlet boundary conditions
    • Boundary function (2)
    • Finite element basis functions (2)
    • Modification of the linear system (2)
  • Example: Oscillating Dirichlet boundary condition
  • Analysis of the discrete equations
    • Forward Euler discretization
    • Backward Euler discretization
    • Comparing amplification factors
• Systems of differential equations
  • Variational forms (3)
  • A worked example
  • Identical function spaces for the unknowns
    • Variational form of each individual PDE
    • Compound scalar variational form
    • Decoupled linear systems
    • Coupled linear systems
  • Different function spaces for the unknowns
  • Computations in 1D
• Exercises (2)
  • Exercise 23: Refactor functions into a more general class
  • Exercise 24: Compute the deflection of a cable with sine functions
  • Exercise 25: Check integration by parts
  • Exercise 26: Compute the deflection of a cable with 2 P1 elements
  • Exercise 27: Compute the deflection of a cable with 1 P2 element
  • Exercise 28: Compute the deflection of a cable with a step load
  • Exercise 29: Show equivalence between linear systems
  • Exercise 30: Compute with a non-uniform mesh
  • Problem 31: Solve a 1D finite element problem by hand
  • Exercise 32: Compare finite elements and differences for a radially symmetric Poisson equation
  • Exercise 33: Compute with variable coefficients and P1 elements by hand
  • Exercise 34: Solve a 2D Poisson equation using polynomials and sines
  • Exercise 35: Analyze a Crank-Nicolson scheme for the diffusion equation
• Bibliography

Indices and tables

• Index
• Module Index
• Search Page