Time-dependent variational forms

Contents:

• Time-dependent variational forms
• Discretization in time by a Forward Euler scheme
  • Time discretization
  • Space discretization
  • Variational forms
  • Simplified notation for the solution at recent time levels
  • Deriving the linear systems
  • Computational algorithm
  • Example using sinusoidal basis functions
  • Comparing P1 elements with the finite difference method
• Discretization in time by a Backward Euler scheme
  • Time discretization
  • Variational forms
  • Linear systems
• Dirichlet boundary conditions
  • Boundary function
  • Finite element basis functions
  • Modification of the linear system
  • Example: Oscillating Dirichlet boundary condition
• Analysis of the discrete equations
  • Fourier components
  • Forward Euler discretization
  • Backward Euler discretization
  • Comparing amplification factors
• Exercises
  • Exercise 1: Analyze a Crank-Nicolson scheme for the diffusion equation
• Bibliography

Index

• Index