The paper develops a finite element method for partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method uses traces of bulk finite element functions on a surface embedded in a volumetric domain. The bulk finite element space is defined on an octree grid which is locally refined or coarsened depending on error indicators and estimated values of the surface curvatures. The cartesian structure of the bulk mesh leads to easy and efficient adaptation process, while the trace finite element method makes fitting the mesh to the surface unnecessary. The number of degrees of freedom involved in computations is consistent with the two-dimension nature of surface PDEs. No parametrization of the surface is required; it can be given implicitly by a level set function. In practice, a variant of the marching cubes method is used to recover the surface with the second order accuracy. We prove the optimal order of accuracy for the trace finite element method in $H^1$ and $L^2$ surface norms for a problem with smooth solution and quasi-uniform mesh refinement. Experiments with less regular problems demonstrate optimal convergence with respect to the number of degrees of freedom, if grid adaptation is based on an appropriate error indicator. The paper shows results of numerical experiments for a variety of geometries and problems, including advection-diffusion equations on surfaces. Analysis and numerical results of the paper suggest that combination of cartesian adaptive meshes and the unfitted (trace) finite elements provide simple, efficient, and reliable tool for numerical treatment of PDEs posed on surfaces.
A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes
我们提出了一个完全耦合的，隐式的时间框架，用于解决一个完全一致的Cahn-Hilliard Navier-Stokes系统，模拟两相流。在这项工作中，我们扩展了Khanwale等人[ \textit{Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes}，J. Comput.（2020）]，一个完全耦合的，可证明的二阶精确的计划在时间上，同时保持能量稳定性。新方法在每次牛顿迭代中需要更少的矩阵集合，从而导致更快的求解时间。该方法是基于一个全隐式的Crank-Nicolson计划的时间和压力稳定的等阶Galerkin制定。也就是说，我们使用一个符合连续Galerkin（CG）有限元法在空间配备了一个基于残差的变分多尺度（RBVMS）程序来稳定的压力。我们部署这种方法的大规模并行数值实现使用并行八叉树为基础的自适应网格。我们提出了全面的数值实验，详细的比较与文献中的典型情况下，包括单气泡上升，瑞利-泰勒不稳定性，和盖子驱动的空腔流动问题的结果。我们详细分析了我们的数值实现的缩放。
Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes
Field conserving adaptive mesh refinement (AMR) scheme on massively parallel adaptive octree meshes
Iowa State University（爱荷华州立大学）；Stanford University（斯坦福大学）；University of Utah（犹他大学）；Tufts University（塔夫茨大学）；RIKEN Center for Computational Science（理化学研究所计算科学中心）
Analysis of adaptive two-grid finite element algorithms for linear and nonlinear problems
提出了求解线性和非线性偏微分方程的自适应二重网格有限元算法。这些算法的主要思想是利用$k$级自适应网格上的解来求解$(k+1)$级自适应网格上的解，而$(k+1)$级自适应网格是通过对$k$级自适应网格进行自适应单元剖分而构造的。这些算法将非对称正定（非SPD）PDE（分别为，非线性偏微分方程）转化为对称正定（SPD）偏微分方程（分别，线性PDE）。所提出的算法具有以下优点：不需要求解非对称或非线性系统;很小;很容易实现;插值误差很小。其次，本文构造了残差型{ \em a posteriori}误差估计量，并证明了其可靠性和有效性。证明有效性的关键因素是建立振荡项的上界，其可能不是高阶项（h. o.t.）由于数值解的低规则性。此外，还证明了当网格剖分时算法的收敛性。最后，提供数值实验来验证ATG有限元算法的精度和效率，与常规自适应有限元算法和两网格有限元算法[27]进行比较。
Adaptive-Growth Randomized Neural Networks for PDEs: Algorithms and Numerical Analysis
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University（西安交通大学数学与统计学院及动力工程多相流国家重点实验室）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
An adaptive numerical method for free surface flows passing rigidly mounted obstacles
Institute of Numerical Mathematics of the Russian Academy of Sciences（俄罗斯科学院计算数学研究所）；Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences（俄罗斯科学院凯尔迪什应用数学研究所）；Department of Mathematics, University of Houston（休斯顿大学数学系）；School of Earth, Energy & Environmental Sciences, Stanford University（斯坦福大学地球、能源与环境科学学院）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
