On the convergence of adaptive iterative linearized Galerkin methods
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On the convergence of adaptive iterative linearized Galerkin methods Pascal Heid1 · Thomas P. Wihler1 Received: 22 November 2019 / Revised: 10 June 2020 / Accepted: 4 July 2020 / Published online: 5 August 2020 © The Author(s) 2020
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. Keywords Numerical solution methods for quasilinear elliptic PDE · Monotone problems · Fixed point iterations · Linearization schemes · Kačanov method · Newton method · Galerkin discretizations · Adaptive mesh refinement · Convergence of adaptive finite element methods Mathematics Subject Classification 35J62 · 47J25 · 47H05 · 47H10 · 49M15 · 65J15 · 65N12 · 65N30 · 65N50
The authors acknowledge the financial support of the Swiss National Science Foundation (SNF), Grant No. 200021_182524. * Thomas P. Wihler [email protected] Pascal Heid [email protected] 1
Mathematics Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
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P. Heid, T. P. Wihler
1 Introduction In this paper we analyze the convergence of adaptive iterative linearized Galerkin (ILG) methods for nonlinear problems with strongly monotone operators. To set the stage, we consider a real Hilbert space X with inner product (⋅, ⋅)X and induced norm denoted by ‖ ⋅ ‖X . Then, given a nonlinear operator 𝖥 ∶ X → X ⋆ , we focus on the equation
u∈X∶
𝖥(u) = 0 in X ⋆ ,
(1)
where X ⋆ denotes the dual space of X. In weak form, this problem reads
u∈X∶
⟨𝖥(u), v⟩X ⋆ ×X = 0 for all v ∈ X,
(2)
with ⟨⋅, ⋅⟩X ⋆ ×X signifying the duality pairing in X ⋆ × X . For the purpose of this work, we suppose that 𝖥 satisfies the following conditions: (F1) The operator 𝖥 is Lipschitz continuous, i.e. there exists a constant L𝖥 > 0 such that
�⟨𝖥(u) − 𝖥(v), w⟩ ⋆ � ≤ L ‖u − v‖ ‖w‖ , X ×X � 𝖥 X X �
for all u, v, w ∈ X . (F2) The operator 𝖥 is strongly monotone, i.e. there is a constant 𝜈 > 0 such that
𝜈‖u − v‖2X ≤ ⟨𝖥(u) − 𝖥(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]. 1.1 Iterative linearization The existence of a solution to the nonlinear equati
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