On Discretization and Differentiation of Operators with Application to Newton’s Method

In the numerical solution of operator equations $Fx = 0$, discretization of the equation and then application of Newton’s method results in the same linear algebraic system of equations as application of Newton's method followed by discretization. This leads to the general problem of determining when the two frequently used operations of discretization and (Frechet) differentiation applied to a nonlinear operator are commutative. A theory of discretization processes is developed here which proves that for a wide class of operators of interest in applications, discretization and differentiation indeed “commute”. The fundamental concept of the theory is a distinction between the discretization of the linear spaces involved and the replacement of the infinitesimal parts of the operator F, i.e., those parts involving, e.g., differentiation and integration, by a discrete analogue. Using this distinction in an abstract way, a “complete” discretization process is defined precisely and the cited commutativity res...