newton4rheolef

Overview

Nonlinear Newton algorithm for the resolution of the following problem:

       F(u) = 0

A simple call to the algorithm writes:

    my_problem P;
    field uh (Vh);
    newton (P, uh, tol, max_iter);

The my_problem class may contains methods for the evaluation of F (aka residue) and its derivative:

    class my_problem {
    public:
      typedef value_type;
      value_type residue          (const value_type& uh)  const;
      Float dual_space_norm  (const value_type& mrh) const;
      void update_derivative      (const value_type& uh)  const;
      value_type derivative_solve (const value_type& mrh) const;
    };

The dual_space_norm returns a scalar from the weighted residual field term mrh returned by the residue function: this scalar is used as stopping criteria for the algorithm. The update_derivative and derivative_solver members are called at each step of the Newton algorithm. See the example p_laplacian.h in the user's documentation for more.

Implementation

template <class Problem, class Field>
int newton (const Problem& P, Field& uh, Float& tol, size_t& max_iter, odiststream *p_derr = 0) {
    if (p_derr) *p_derr << "# Newton:" << std::endl << "# n r" << std::endl << std::flush;
    for (size_t n = 0; true; n++) {
      Field rh = P.residue(uh);
      Float r = P.dual_space_norm(rh);
      if (p_derr) *p_derr << n << " " << r << std::endl << std::flush;
      if (r <= tol) { tol = r; max_iter = n; return 0; }
      if (n == max_iter) { tol = r; return 1; }
      P.update_derivative (uh);
      Field delta_uh = P.derivative_solve (-rh);
      uh += delta_uh;
    }
}