This work presents spectral, mesh-free, Green's theorem-based numerical quadrature schemes for integrating functions over planar regions bounded by rational parametric curves. Our algorithm proceeds in two steps: (1) We first find intermediate quadrature rules for line integrals along the region's boundary curves corresponding to Green's theorem. (2) We then use a high-order quadrature rule to compute the numerical antiderivative of the integrand along a coordinate axis, which is used to evaluate the Green's theorem line integral. We present two methods to compute the intermediate quadrature rule. The first is spectrally accurate (it converges faster than any algebraic order with respect to number of quadrature points) and is relatively easy to implement, but has no guarantee of polynomial exactness. The second guarantees exactness for polynomial integrands up to a pre-specified degree k with an a priori-known number of quadrature points and retains the convergence properties of the first, but is slightly more complicated. The quadrature schemes have applications to computation of geometric moments, immersogeometric analysis, conservative field transfer between high-order meshes, and initialization of simulations with rational geometry. We compare the quadrature schemes produced using our method to other methods in the literature and show that they are much more efficient both in terms of number of quadrature points and computational time.
The effect of quadrature rules on finite element solutions of Maxwell variational problems. Consistency estimates on meshes with straight and curved elements
The Modified Direct Method: an Approach for Smoothing Planar and Surface Meshes
修正的直接法（Modified Direct Method，MDM）是一种用于光滑平面和曲面网格的迭代网格光滑方法，它是在Balendran [1]提出的非迭代光滑方法的基础上发展起来的。当平滑平面网格时，对于三角形和四边形网格，MDM的性能实际上与拉普拉斯平滑的性能相同;然而，MDM优于三四边形网格的拉普拉斯平滑。在光滑曲面网格时，对于三角形、四边形和四边形为主的混合网格，在光滑过程中，所有网格单元的平均质量（MQ）总是增加，均方误差（MSE）总是减小;对于三角形为主的混合网格，三角形的质量总是下降，四边形的质量总是上升。算例表明，该方法对平面和曲面三角形、四边形和四边形网格都是收敛的。
High-Order Quadrature for Implicitly Defined Surfaces and Regions
Tsinghua University（清华大学）；Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
The Influence of Quadrature Errors on Isogeometric Mortar Methods
Istituto Universitario di Studi Superiori Pavia（帕维亚高等大学研究所）；Istituto di Matematica Applicata e Tecnologie Informatiche del CNR（意大利国家研究委员会应用数学与信息技术研究所）；M2 – Zentrum Mathematik, Technische Universität München（慕尼黑工业大学数学中心）
Construction of near-boundary Voronoi mesh layers for planar domains
Dorodnicyn Computing Center FRC CSC RAS（多罗德尼琴科计算中心俄罗斯科学院联邦研究中心）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Keldysh Institute of Applied Mathematics RAS（凯尔迪什应用数学研究所俄罗斯科学院）
Simple quadrature rules for a nonparametric nonconforming quadrilateral element
本文对四自由度非参数四边形单元族引入了简单的求积规则。我们的求积规则是由孟{ \it et al.} \cite{meng2018new}。首先，我们引入了由Meng { \it et al.}提出的中间参考域上保留四个自由度的MVP（Mean Value Property）族。然后设计了中间参考域上的两点和三点求积规则。在等正交权的假设下，两点法和三点法的高斯点距四边形中心的偏差均为修正了常数项的二次多项式。这样，两点规则和三点规则就被一次构造出来了。当网格步长足够小时，求积规则是渐近最优的。数值实验表明了新的求积规则的有效性和收敛性。
Gauss--Chebyshev Quadrature, $\mathbb S^1$-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry
Multilevel quadrature for elliptic parametric partial differential equations in case of polygonal approximations of curved domains
Universität Bonn（波恩大学）；Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)（弗劳恩霍夫算法与科学计算研究所）；Universität Basel（巴塞尔大学）；Università della Svizzera italiana（瑞士意大利语大学）
Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with Summation-by-Parts Property for Hyperbolic Conservation Laws
University of Cologne（科隆大学）；Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart（斯图加特大学空气动力学与气体动力学研究所）；Center for Data and Simulation Science (CDS), University of Cologne（科隆大学数据与模拟科学中心）
A novel Hermite RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation with Neumann boundary condition
Jiangsu Normal University（江苏师范大学）；De Montfort University（德蒙福特大学）
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）
