A general mesh independence principle for Newton's method applied to second order boundary value problems

Abstract

--ZusammenfassungA General Mesh Independence Principle for Newton's Method Applied to Second Order Boundary Value Problems. Recent work has established that for certain classes of nonlinear boundary value problems, the number of Newton iterations applied to the related standard discrete problem for a given tolerance is independent of the mesh size when the mesh is sufficiently fine. This paper develops an extension of the mesh independence principle by relaxing the assumption on the differential equation, its boundary conditions, and the related difference approximation. Ein allgemeines Gitterunabh~ingigkeitsprinzip fiir das Newton-Verfahren bei Randwertproblemen zweiterOrdnung. Wie ktirzlich gezeigt wurde, ist bei gewissen Klassen von nichtlinearen Randwertproblemen die Anzahl der Newton-Iterationen bei der L6sung der zugeh6rigen Standard-Diskretisierung ftir eine gegebene Toleranz unabh/ingig vonder Gitterweite, wenn das Gitter hinreichend rein ist. In der Arbeit wird dieses Gitterunabhfingigkeitsprinzip durch Abschw~ichung der Voraussetzungen fiber die Differentialgleichung, die Randbedingungen und die zugeh6rige Differenzenapproximation verallgemeinert.

Multilevel First-Order System Least Squares for Nonlinear Elliptic Partial Differential Equations

“…The "mesh-independence" theory developed for Newton's method in [21,4,3,2] addresses the same property of NI that we exploit here. Unfortunately, this theory cannot easily be applied to our setting because it requires more smoothness of the infinite-dimensional iterates than ours appear to possess.…”

On a mesh-independence principle for operator equations and the secant method

“…we deduce that We now complete the claims made in the introduction concerning the mesh-independence principle as follows: Both conditions (75) and (76) are almost standard in most discretization studies [1], [2], [8], [10].…”

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

“…In this paper we revisit the problem of mesh independence and optimal order convergence of inexact Newton MG (multigrid) methods for solving finite element, second order, nonlinear, elliptic equations. Previous work has shown mesh-independent convergence of exact Newton methods for such equations discretized by finite differences [1,2]. In this work, we relax the need for an exact Newton method and show mesh-independent convergence for an inexact Newton method in which multigrid techniques are used to solve the linear Jacobian systems.…”