The residual equation of a nonlinear PDE is as follows:
To obtain a discretized residual equation, apply the finite element method (FEM)
to a partial differential equation as described in Finite Element Method Basics:
The nonlinear solver uses a Gauss-Newton iteration scheme applied to the finite
element matrices. Use a Taylor series expansion to obtain the linearized system for
the residual:
Neglecting the higher-order terms, write the linearized system of equations
as
The descent direction for the residual is
The Gauss-Newton iteration minimizes the residual, that is, the solution of , using the equation
Here, ɑ ≤ 1 is a positive number, that must be set as large as possible so that
the step has a reasonable descent. For a sufficiently small ɑ,
For the Gauss-Newton algorithm to converge, must be close enough to the solution. The first guess is often
outside the region of convergence. The Armijo-Goldstein line search (a damping
strategy for choosing ɑ) helps to improve convergence from bad initial guesses. This
method chooses the largest damping coefficient ɑ out of the sequence 1, 1/2, 1/4, .
. . such that the following inequality holds:
Using the Armijo-Goldstein line search guarantees a reduction of the residual norm
by at least . Each step of the line-search algorithm must evaluate the residual .
With this strategy, when
Un approaches
the solution, →1, thus, the convergence rate increases.