Stationary variational forms

Contents:

• Stationary variational forms
• Basic principles for approximating differential equations
  • Differential equation models
  • Simple model problems and their solutions
  • Forming the residual
  • The least squares method
  • The Galerkin method
  • The Method of Weighted Residuals
  • Test and Trial Functions
  • The collocation method
  • Examples on using the principles
  • Integration by parts
  • Boundary function
  • Abstract notation for variational formulations
  • Variational problems and minimization 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
• Boundary conditions: specified nonzero value
  • General construction of a boundary function
  • Example on computing with a finite element-based 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
  • 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
• Implementation
  • Global basis functions
  • Example: constant right-hand side
  • Finite elements
  • Utilizing a sparse matrix
• Variational formulations in 2D and 3D
  • Integration by parts
  • Example on a multi-dimensional variational problem
  • Transformation to a reference cell in 2D and 3D
  • Numerical integration
  • Convenient formulas for P1 elements in 2D
  • A glimpse of the mathematical theory of the finite element method
• Summary
• Exercises
  • Exercise 1: Refactor functions into a more general class
  • Exercise 2: Compute the deflection of a cable with sine functions
  • Exercise 3: Compute the deflection of a cable with power functions
  • Exercise 4: Check integration by parts
  • Exercise 5: Compute the deflection of a cable with 2 P1 elements
  • Exercise 6: Compute the deflection of a cable with 1 P2 element
  • Exercise 7: Compute the deflection of a cable with a step load
  • Exercise 8: Compute with a non-uniform mesh
  • Problem 9: Solve a 1D finite element problem by hand
  • Exercise 10: Investigate exact finite element solutions
  • Exercise 11: Compare finite elements and differences for a radially symmetric Poisson equation
  • Exercise 12: Compute with variable coefficients and P1 elements by hand
  • Exercise 13: Solve a 2D Poisson equation using polynomials and sines
• Bibliography

Index

• Index