A comparison of the method of frozen coefficients with Newton's method for quasilinear two-point boundary-value problems (Q1099316)

The method of frozen coefficients (Kačanov method) and Newton's method are used to solve the vector two-point boundary value problem \[ (-1)^ k\frac{d^ k}{dx^ k}[p_ 1(\frac{d^ ku}{dx^ k},u,x)]+p_ 2(\frac{d^ ku}{dx^ k},u,x)=f(x), \] with fixed (Dirichlet) and free (Neumann) boundary conditions. The method of frozen coefficients is shown to be a secant procedure in which one of the reference ``points'' is kept fixed. Sufficient conditions for global linear convergence of the method of frozen coefficients and Newton's method are compared. These conditions explain why the method of frozen coefficients can perform better than Newton's method when the approximate solution is far from the exact solution.