Newton's method for gradient equations based upon the fixed point map: Convergence and complexity study

An approximate Newton method, based upon the fixed point map T, is introduced for scalar gradient equations. Although the exact Newton method coincides for such scalar equations with the standard iteration, the structure of the fixed point map provides a way of defining an R-quadratically convergent finite element iteration in the spirit of the Kantorovich theory. The loss of derivatives phenomenon, typically experienced in approximate Newton methods, is thereby avoided. It is found that two grid parameters are sssential, h and {Mathematical expression}. The latter is used to calculate the approximate residual, and is isolated as a fractional step; it is equivalent to the approximation of T. The former is used to calculate the Newton increment, and this is equivalent to the approximation of T′. The complexity of the finite element computation for the Newton increment is shown to be of optimal order, via the Vituškin inequality relating metric entropy and n-widths.