In moving mesh methods, the underlying mesh is dynamically adapted without changing the connectivity of the mesh. We specifically consider the generation of meshes which are adapted to a scalar monitor function through equidistribution. Together with an optimal transport condition, this leads to a Monge-Ampère equation for a scalar mesh potential. We adapt an existing finite element scheme for the standard Monge-Ampère equation to this mesh generation problem; this is a mixed finite element scheme, in which an extra discrete variable is introduced to represent the Hessian matrix of second derivatives. The problem we consider has additional nonlinearities over the basic Monge-Ampère equation due to the implicit dependence of the monitor function on the resulting mesh. We also derive the equivalent Monge-Ampère-like equation for generating meshes on the sphere. The finite element scheme is extended to the sphere, and we provide numerical examples. All numerical experiments are performed using the open-source finite element framework Firedrake.
G-Adaptivity: optimised graph-based mesh relocation for finite element methods
Department of Applied Mathematics；Theoretical Physics, University of Cambridge, UK；Department of Mathematical Sciences, University of Bath, UK；Department of Computer Science, University of Bath, UK；School of Computing；Mathematical Sciences, University of Leicester, UK；Laboratoire Jacques-Louis Lions, Sorbonne Universit\'e, France；Department of Computer Science；Technology, University of Cambridge, UK
Goal-oriented space-time adaptivity for the Navier--Stokes equations based on the dual weighted residual method
Helmut Schmidt University Hamburg, Faculty of Mechanical and Civil Engineering（汉堡赫尔穆特·施密特大学机械与土木工程学院）；University of Magdeburg, Institute for Analysis and Numerics（马格德堡大学分析与数值研究所）
A posteriori error analysis and adaptivity of a space-time finite element method for the wave equation in second order formulation
Inria（法国国家信息与自动化研究所）；CERMICS, Ecole nationale des ponts et chaussées, IP Paris, CNRS（国立桥路学校CERMICS实验室，巴黎理工学院，法国国家科学研究中心）；The Maxwell Institute for Mathematical Sciences and Department of Mathematics, Heriot-Watt University（赫瑞-瓦特大学麦克斯韦数学科学研究所与数学系）；Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens（雅典国立工业大学应用数理科学学院数学系）；IACM-FORTH（希腊计算与应用数学研究所）；Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca（米兰比可卡大学数学与应用科学系）；IMATI-CNR（意大利国家研究委员会应用数学研究所）；Fakultät für Mathematik, Universität Wien（维也纳大学数学系）
A mesh adaptivity scheme on the Landau-de Gennes functional minimization case in 3D, and its driving efficiency
Fakulteta za matematiko in fiziko, Univerza v Ljubljani（卢布尔雅那大学数学与物理学院）；Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions（索邦大学，巴黎第六大学，UMR 7598，雅克-路易·利翁斯实验室）；CNRS, UMR 7598, Laboratoire Jacques-Louis Lions（法国国家科学研究中心，UMR 7598，雅克-路易·利翁斯实验室）；INRIA-Paris-Rocquencourt（法国国家信息与自动化研究所巴黎罗昆库尔中心）；Inštitut Jožef Stefan（约瑟夫斯蒂芬研究所）
Solving the (Navier-)Stokes equations with space and time adaptivity using deal.II
Technical University of Berlin（柏林工业大学）；Charles University（查理大学）；Ruhr University Bochum（波鸿鲁尔大学）；Otto von Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）
Parametric Neural r-Adaptivity for Isogeometric Analysis via Residual Minimization
Universidad del Paı́s Vasco/Euskal Herriko Unibertsitatea (UPV/EHU)（巴斯克大学）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；Basque Foundation for Science (Ikerbasque)（巴斯克科学基金会）
An automatic-adaptivity stabilized finite element method via residual minimization for heterogeneous, anisotropic advection-diffusion-reaction problems
Gradient flow finite element discretisations with energy-based adaptivity for excited states of Schrödingers equation
我们提出了一种有效的数值方法，该方法基于[Heid et al.，arXiv：1906.06954]，用于薛定谔方程激发态的数值逼近。特别是，该过程采用梯度流迭代和局部网格细化的自适应相互作用，从而在算法的每个步骤中保证能量衰减。计算测试突出表明，该策略能够提供高度准确的结果，相对于自由度的数量具有最佳的收敛速度。
A posteriori error estimates and adaptivity for locally conservative methods. Inexpensive implementation and evaluation, polytopal meshes, iterative linearization and algebraic solvers, and applications to complex porous media flows
A posteriori error analysis and adaptivity for a VEM discretization of the Navier-Stokes equations
我们考虑由Beirão da Veiga，Lovadina和Vacca在2016年引入的用于数值解定常不可压缩Navier-斯托克斯方程的虚拟单元法（VEM）;该方法具有任意阶$k \geq 2$并保证无发散速度。对于这样的离散化，我们开发了一个残差为基础的后验误差估计，这是一个标准的条款在VEM分析（残差项，数据振荡，VEM稳定）的组合，加上一些其他条款起源于非线性对流项的VEM离散化。我们表明，速度和压力误差的线性组合是由估计（可靠性）的多个上限。我们还建立了一些效率的结果，涉及误差的下界。一些数值测试说明了估计器及其组件的性能，同时均匀地细化网格，产生预期的衰减率。最后，我们将一种自适应网格加密策略应用于低雷诺数下管道内方柱绕流的计算。
Mathematical modeling and numerical multigoal-oriented a posteriori error control and adaptivity for a stationary, nonlinear, coupled flow temperature model with temperature dependent density
Polynomial spline spaces of non-uniform bi-degree on T-meshes: Combinatorial bounds on the dimension
Institute for Computational Engineering and Sciences, University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Université Côte d’Azur, Inria Sophia Antipolis Méditerranée（蔚蓝海岸大学，法国国家信息与自动化研究所索菲亚安提贝斯分部）
Error estimation and adaptivity for stochastic collocation finite elements Part I: single-level approximation
提出并分析了一种求解线性椭圆型偏微分方程的自适应加密策略。该自适应策略扩展了Guignard和Nyndham在2018年引入的后验误差估计框架（SIAM J. Numer.肛门，56，3121- 3143）来覆盖具有非仿射参数系数依赖性的问题。一个次优的，但仍然可靠和方便的实施策略涉及近似的解耦PDE问题与一个共同的有限元近似空间。本文（第一部分）介绍了使用这种单级策略获得的计算结果。使用一个潜在的更有效的多级近似策略，其中网格是单独定制的，将在本工作的第二部分中讨论的结果。用于生成数值结果的代码可在GitHub上获得
An \textsf{AT1} phase-field framework for quasi-static anti-plane shear fracture: Unifying $ξ$-based adaptivity and nonlinear strain energy density function
Error indicators and adaptivity for a least-squares method applied to the Monge-Ampère equation
Geneva School of Business Administration (HEG-GENÈVE), University of Applied Sciences and Arts Western Switzerland (HES-SO)（日内瓦商业管理学院（HEG-GENÈVE），瑞士西部应用科学艺术大学（HES-SO））；Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院数学研究所）
Adaptive finite element methods with optimally preconditioned GMRES guarantee optimal complexity
Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）；Sorbonne Université, Université Paris Cité, CNRS, Laboratoire Jacques-Louis Lions, LJLL（索邦大学，巴黎西岱大学，法国国家科学研究中心，雅克-路易·利翁斯实验室）；TU Wien, Institute of Analysis and Scientific Computing（维也纳工业大学，分析与科学计算研究所）
