On the convergence of adaptive iterative linearized Galerkin methods

Pascal Heid,Thomas P Wihler

doi:10.1007/s10092-020-00368-4

Abstract

A wide variety of different (fixed-point) iterative methods for the solution of nonlinear equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from our previous work [16] that covers some prominent procedures (including the Zarantonello, Kačanov and Newton iteration methods). In combination with appropriate discretization methods so-called (adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main purpose of this paper is the derivation of an abstract convergence theory for the unified ILG approach (based on general adaptive Galerkin discretization methods) proposed in [16]. The theoretical results will be tested and compared for the aforementioned three iterative linearization schemes in the context of adaptive finite element discretizations of strongly monotone stationary conservation laws.

Highlights

In this paper we analyze the convergence of adaptive iterative linearized Galerkin (ILG) methods for nonlinear problems with strongly monotone operators
We consider a real Hilbert space X with inner product (⋅, ⋅)X and induced norm denoted by ‖ ⋅ ‖X
Given a nonlinear operator F ∶ X → X⋆, we focus on the equation u∈X∶

Summary

Introduction

In this paper we analyze the convergence of adaptive iterative linearized Galerkin (ILG) methods for nonlinear problems with strongly monotone operators. Given a nonlinear operator F ∶ X → X⋆ , we focus on the equation u∈X∶ (u) = 0 in X⋆, (1). Where X⋆ denotes the dual space of X. where X⋆ denotes the dual space of X In weak form, this problem reads u ∈ X ∶ ⟨ (u), v⟩X⋆×X = 0 for all v ∈ X, (2). (F2) The operator is strongly monotone, i.e. there is a constant ν > 0 such that ν‖u − v‖2X ≤ ⟨ (u) − (v), u − v⟩X⋆×X , for all u, v ∈ X. Given the properties (F1) and (F2), the main theorem of strongly monotone operators states that (1) has a unique solution u⋆ ∈ X ; see, e.g., [20, §3.3] or [23, Theorem 25.B]

Objectives

Methods

Conclusion