[HTML][HTML] Rate-optimal goal-oriented adaptive FEM for semilinear elliptic PDEs
R Becker, M Brunner, M Innerberger, JM Melenk… - … & Mathematics with …, 2022 - Elsevier
We formulate and analyze a goal-oriented adaptive finite element method for a semilinear
elliptic PDE and a linear goal functional. The discretization is based on finite elements of …
elliptic PDE and a linear goal functional. The discretization is based on finite elements of …
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 …
[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 …
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 …
[HTML][HTML] On the convergence of adaptive iterative linearized Galerkin methods
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 …
equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from …
[HTML][HTML] Iterative Galerkin discretizations for strongly monotone problems
S Congreve, TP Wihler - Journal of Computational and Applied …, 2017 - Elsevier
In this article we investigate the use of fixed point iterations to solve the Galerkin
approximation of strictly monotone problems. As opposed to Newton's method, which …
approximation of strictly monotone problems. As opposed to Newton's method, which …
Virtual element method for semilinear elliptic problems on polygonal meshes
In this paper, the virtual element method is employed to approximate semilinear elliptic
problems over arbitrary polygonal meshes. The nonlinear load term is approximated by …
problems over arbitrary polygonal meshes. The nonlinear load term is approximated by …
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 …