Tutorial Part B (Nonlinear Problems)

In this chapter we illustrate how to use the Picard’s and Newton’s methods to solve nonlinear problems.

• Picard’s Method with Acceleration (01-picard)
  • Picard’s method
  • Anderson acceleration
  • Model problem
  • Defining custom nonlinearity
  • Defining initial condition
  • Defining custom weak forms
  • Initializing the weak formulation
  • Picard’s iteration loop
  • Convergence of the original method
  • Convergence of the original method
  • Sample results
• Introduction to Newton’s Method
  • Residual vector and Jacobian matrix
  • How to differentiate weak forms
  • Practical formula
  • Stopping criteria
  • Linear problems
• Newton’s Method with Analytic Nonlinearity (02-newton-analytic)
  • Model problem
  • Initializing the weak formulation
  • Obtaining a good initial coefficient vector
  • The Newton’s iteration loop
  • Translating the resulting coefficient vector into a Solution
  • Cleaning up
  • Convergence
  • Sample results
• Newton’s Method with Spline Nonlinearity (02-newton-spline)
  • Model problem
  • Defining a cubic spline
  • Plotting the spline for visual control
  • Initializing the weak formulation
  • Convergence
  • Sample results