On Mesh-Independent Convergence of an Inexact Newton--Multigrid Algorithm

Abstract

In this paper we revisit and prove optimal order and mesh-independent convergence of an inexact Newton method where the linear Jacobian systems are solved with multigrid techniques. This convergence is shown using Banach spaces and the norm, max{ • 1 , • 0,∞ }, a stronger norm than is used in previous work. These results are valid for a class of second order, semi-linear, finite element, elliptic problems posed on quasi-uniform grids. Numerical results are given which validate the theory.

Mesh-independent convergence of the modified inexact Newton method for a second order non-linear problem

“…The above conditions imply the following theorem (compare to Theorem 2.1 in [4] or Theorem 2.3 in [6]). We give its proof for completeness.…”

“…Minor variations of this algorithm have been proposed and studied, for example, in [4,6,7,8]. Our analysis is also a slight modification of theirs.…”

“…Estimates which give rise to a uniform rate of iterative convergence for inexact Newton's method applied to (1.1) in the case when the nonlinearities are restricted to the zeroth order term were provided in [4] (specifically, k(u, x) ≡ k(x) and c ≡ 0). This restriction enabled them to use the L 2 (Ω)-norm for the residuals.…”

“…This restriction enabled them to use the L 2 (Ω)-norm for the residuals. Iterative methods with residual convergence in L 2 (Ω) can be constructed in the case of full elliptic regularity by multilevel methods as discussed in [4]. Unfortunately, when the coefficients of the higher derivatives involve the discrete solution, full elliptic regularity no longer holds.…”

“…We provide a general theorem for the analysis of inexact Newton methods which is a variant of those given in, e.g., [4,6,7]. The application of this theorem requires an iterative scheme which reduces the error in a norm which is related to the stability properties of the partial differential equation.…”