标题： 赫尔姆霍兹-霍奇分解的拟插值 显示英文标题

标题： Quasi-interpolation for the Helmholtz-Hodge decomposition

摘要： 本文旨在提出一种高效且稳定的基于拟插值的方法，用于数值计算向量场的Helmholtz-Hodge分解。 为此，我们首先从多项调和样条中显式构造一个一般形式的矩阵核，使其包括无散/无旋/调和矩阵核作为特例。 然后我们将该矩阵核与卷积技术结合，应用于向量分解以及Helmholtz-Hodge分解。 更准确地说，我们证明了如果将一个向量场与缩放的无散（无旋）矩阵核进行卷积，则得到的无散（无旋）卷积序列将收敛到该场的Helmholtz-Hodge分解的相应无散（无旋）部分。 最后，通过使用某种求积规则对卷积序列进行离散化，我们构建了一族（无散/无旋）拟插值算子，用于Helmholtz-Hodge分解（在全空间和有界域上均有定义）。 论文中推导出的相应误差估计表明，我们的基于拟插值的方法能够给出向量场及其Helmholtz-Hodge分解的收敛逼近。 显示英文摘要

摘要： The paper aims at proposing an efficient and stable quasi-interpolation based method for numerically computing the Helmholtz-Hodge decomposition of a vector field. To this end, we first explicitly construct a matrix kernel in a general form from polyharmonic splines such that it includes divergence-free/curl-free/harmonic matrix kernels as special cases. Then we apply the matrix kernel to vector decomposition via the convolution technique together with the Helmholtz-Hodge decomposition. More precisely, we show that if we convolve a vector field with a scaled divergence-free (curl-free) matrix kernel, then the resulting divergence-free (curl-free) convolution sequence converges to the corresponding divergence-free (curl-free) part of the Helmholtz-Hodge decomposition of the field. Finally, by discretizing the convolution sequence via certain quadrature rule, we construct a family of (divergence-free/curl-free) quasi-interpolants for the Helmholtz-Hodge decomposition (defined both in the whole space and over a bounded domain). Corresponding error estimates derived in the paper show that our quasi-interpolation based method yields convergent approximants to both the vector field and its Helmholtz-Hodge decomposition