期刊
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
卷 21, 期 2, 页码 407-422出版社
WALTER DE GRUYTER GMBH
DOI: 10.1515/cmam-2021-0025
关键词
Iterative Linearized Galerkin Methods; Fixed Point Iterations; Lipschitz Continuous and Strongly Monotone Operators; Second-Order Elliptic Problems; Energy Contraction; Adaptive Mesh Refinement; Convergence of Adaptive FEM; Optimal Computational Cost
资金
- Swiss National Science Foundation (SNF) [200021_182524]
- Austrian Science Fund (FWF) [SFB F65, P33216]
- Swiss National Science Foundation (SNF) [200021_182524] Funding Source: Swiss National Science Foundation (SNF)
- Austrian Science Fund (FWF) [P33216] Funding Source: Austrian Science Fund (FWF)
Through previous research, it has been found that the methodology proposed by Heid and Wihler exhibits an energy contraction property in the context of strongly monotone problems, which is crucial for achieving linear convergence. Specifically, adaptive iterative linearized finite element methods lead to full linear convergence with optimal algebraic rates in terms of degrees of freedom and total computational time.
We revisit a unified methodology for the iterative solution of nonlinear equations in Hilbert spaces. Our key observation is that the general approach from [P. Heid and T. P. Wihler, Adaptive iterative linearization Galerkin methods for nonlinear problems, Math. Comp. 89 (2020), no. 326, 2707-2734; P. Heid and T. P. Wihler, On the convergence of adaptive iterative linearized Galerkin methods, Calcolo 57 (2020), Paper No. 24] satisfies an energy contraction property in the context of (abstract) strongly monotone problems. This property, in turn, is the crucial ingredient in the recent convergence analysis in [G. Gantner, A. Haberl, D. Praetorius and S. Schimanko, Rate optimality of adaptive finite element methods with respect to the over-all computational costs, preprint (2020)]. In particular, we deduce that adaptive iterative linearized finite element methods (AILFEMs) lead to full linear convergence with optimal algebraic rates with respect to the degrees of freedom as well as the total computational time.
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