Solving partial differential equations using large-data models: a literature review
AM Hafiz, I Faiq, M Hassaballah - Artificial Intelligence Review, 2024 - Springer
Abstract Mathematics lies at the heart of engineering science and is very important for
capturing and modeling of diverse processes. These processes may be naturally-occurring …
capturing and modeling of diverse processes. These processes may be naturally-occurring …
hp-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs
M Innerberger, A Miraçi, D Praetorius… - ESAIM: Mathematical …, 2024 - esaim-m2an.org
In this work, we formulate and analyze a geometric multigrid method for the iterative solution
of the discrete systems arising from the finite element discretization of symmetric second …
of the discrete systems arising from the finite element discretization of symmetric second …
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 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 review on some discrete variational techniques for the approximation of essential boundary conditions
F Chouly - 2023 - hal.science
We review different techniques to enforce essential boundary conditions, such as the
(nonhomogeneous) Dirichlet boundary condition, within a discrete variational framework …
(nonhomogeneous) Dirichlet boundary condition, within a discrete variational framework …
[PDF][PDF] Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs
We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity
of interest G (u⋆) with a linear goal functional G and the solution u⋆ to a general second …
of interest G (u⋆) with a linear goal functional G and the solution u⋆ to a general second …
-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs
M Innerberger, A Miraçi, D Praetorius… - arXiv preprint arXiv …, 2022 - arxiv.org
In this work, we formulate and analyze a geometric multigrid method for the iterative solution
of the discrete systems arising from the finite element discretization of symmetric second …
of the discrete systems arising from the finite element discretization of symmetric second …
MATLAB implementation of HP finite elements on rectangles using hierarchical basis functions
A Moskovka, J Valdman - … Conference on Parallel Processing and Applied …, 2022 - Springer
A MATLAB implementation of hierarchical shape functions on 2D rectangles is explained
and available for download. Global shape functions are ordered for a given polynomial …
and available for download. Global shape functions are ordered for a given polynomial …
Iterative solvers in adaptive FEM
P Bringmann, A Miraçi, D Praetorius - arXiv preprint arXiv:2404.07126, 2024 - arxiv.org
This chapter provides an overview of state-of-the-art adaptive finite element methods
(AFEMs) for the numerical solution of second-order elliptic partial differential equations …
(AFEMs) for the numerical solution of second-order elliptic partial differential equations …