| dc.description.abstract | In this dissertation, we consider modified inexact Newton methods applied to
second order nonlinear problems. In the implementation of Newton's method applied
to problems with a large number of degrees of freedom, it is often necessary to solve
the linear Jacobian system iteratively. Although a general theory for the convergence
of modified inexact Newton's methods has been developed, its application to nonlinear
problems from nonlinear PDE's is far from complete. The case where the nonlinear
operator is a zeroth order perturbation of a fixed linear operator was considered in
the paper written by Brown et al..
The goal of this dissertation is to show that one can develop modified inexact
Newton's methods which converge at a rate independent of the number of unknowns
for problems with higher order nonlinearities. To do this, we are required to first, set
up the problem on a scale of Hilbert spaces, and second, to devise a special iterative
technique which converges in a higher order Sobolev norm, i.e., H1+alpha(omega) \ H1
0(omega)
with 0 < alpha < 1/2. We show that the linear system solved in Newton's method can
be replaced with one iterative step provided that the initial iterate is close enough.
The closeness criteria can be taken independent of the mesh size.
In addition, we have the same convergence rates of the method in the norm of
H1 0(omega) using the discrete Sobolev inequalities. | en |
