Energy contraction and optimal convergence of adaptive iterative linearized finite element methods
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 TP Wihler …
spaces. Our key observation is that the general approach from [P. Heid and TP Wihler …
Rate optimality of adaptive finite element methods with respect to overall computational costs
We consider adaptive finite element methods for second-order elliptic PDEs, where the
arising discrete systems are not solved exactly. For contractive iterative solvers, we …
arising discrete systems are not solved exactly. For contractive iterative solvers, we …
Goal-oriented mesh adaptation method for nonlinear problems including algebraic errors
V Dolejší, O Bartoš, F Roskovec - Computers & Mathematics with …, 2021 - Elsevier
We deal with the goal-oriented error estimates and mesh adaptation for nonlinear partial
differential equations. The setting of the adjoint problem and the resulting estimates are not …
differential equations. The setting of the adjoint problem and the resulting estimates are not …
[HTML][HTML] Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation
We present an effective adaptive procedure for the numerical approximation of the steady-
state Gross–Pitaevskii equation. Our approach is solely based on energy minimization, and …
state Gross–Pitaevskii equation. Our approach is solely based on energy minimization, and …
[HTML][HTML] Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver
A Haberl, D Praetorius, S Schimanko… - Numerische Mathematik, 2021 - Springer
We consider a second-order elliptic boundary value problem with strongly monotone and
Lipschitz-continuous nonlinearity. We design and study its adaptive numerical …
Lipschitz-continuous nonlinearity. We design and study its adaptive numerical …
Cost-optimal adaptive iterative linearized FEM for semilinear elliptic PDEs
R Becker, M Brunner, M Innerberger… - ESAIM: Mathematical …, 2023 - esaim-m2an.org
We consider scalar semilinear elliptic PDEs where the nonlinearity is strongly monotone, but
only locally Lipschitz continuous. We formulate an adaptive iterative linearized finite element …
only locally Lipschitz continuous. We formulate an adaptive iterative linearized finite element …
Adaptive iterative linearization Galerkin methods for nonlinear problems
A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in
Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local …
Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local …
Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs
M Brunner, M Innerberger, A Miraçi… - IMA Journal of …, 2024 - academic.oup.com
We consider a general nonsymmetric second-order linear elliptic partial differential equation
in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite …
in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite …
On full linear convergence and optimal complexity of adaptive FEM with inexact solver
The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to
compute an approximation of user-prescribed accuracy at quasi-minimal computational …
compute an approximation of user-prescribed accuracy at quasi-minimal computational …
A numerical energy reduction approach for semilinear diffusion-reaction boundary value problems based on steady-state iterations
We present a novel energy-based numerical analysis of semilinear diffusion-reaction
boundary value problems, where the nonlinear reaction terms need to be neither monotone …
boundary value problems, where the nonlinear reaction terms need to be neither monotone …