Tutorial Part II (Nonlinear Problems)

In this chapter we illustrate how to use the Picard’s and Newton’s methods to solve nonlinear problems, using a series of examples with an increasing level of complexity.

• Picard’s Method (01-picard)
  • Picard’s Method
  • Model problem
  • Defining initial condition
  • Weak forms
  • Registering weak forms
  • Picard’s iteration loop
  • Sample results
• Newton’s Method
  • Residual vector and Jacobian matrix
  • How to differentiate weak forms
  • Practical formula
  • Stopping criteria
  • Linear problems
• Constant Initial Condition (02-newton-1)
  • Model problem
  • Defining Jacobian matrix and residual vector
  • Using the ExtData and Geom structures
  • Defining nonlinearities and their derivatives
  • Setting a constant initial condition for the Newton’s method
  • Registering weak forms
  • Initializing a nonlinear DiscreteProblem
  • The Newton’s iteration loop
  • Translating the resulting vector into a Solution
  • Sample results
• General Initial Condition (03-newton-2)
  • Model problem
  • Defining nonhomogeneous Dirichlet boundary conditions
  • Setting a nonconstant initial condition for the Newton’s method
  • Sample results