We study numerical integration of smooth functions defined over the $s$-dimensional unit cube. A recent work by Dick et al. (2019) has introduced so-called extrapolated polynomial lattice rules, which achieve the almost optimal rate of convergence for numerical integration and can be constructed by the fast component-by-component search algorithm with smaller computational costs as compared to interlaced polynomial lattice rules. In this paper we prove that, instead of polynomial lattice point sets, truncated higher order digital nets and sequences can be used within the same algorithmic framework to explicitly construct good quadrature rules achieving the almost optimal rate of convergence. The major advantage of our new approach compared to original higher order digital nets is that we can significantly reduce the precision of points, i.e., the number of digits necessary to describe each quadrature node. This finding has a practically useful implication when either the number of points or the smoothness parameter is so large that original higher order digital nets require more than the available finite-precision floating point representations.
Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number
Beijing Computational Science Research Center（北京计算科学研究中心）；Department of Mathematics, Nanjing University（南京大学数学系）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
An extrapolation cascadic multigrid method combined with a fourth order compact scheme for 3D poisson equation
School of Mathematics and Statistics, Central South University（中南大学数学与统计学院）；School of Aerospace Engineering and Applied Mechanics, Tongji University（同济大学航空航天与力学学院）
