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

翻译：本文旨在提出一种高效且稳定的基于拟插值的数值方法，用于计算向量场的 Helmholtz-Hodge 分解。为此，我们首先从多调和样条出发，显式地构造了一个一般形式的矩阵核，使其将无散/无旋/调和矩阵核作为特例包含在内。然后，我们通过卷积技术结合 Helmholtz-Hodge 分解，将该矩阵核应用于向量分解。更精确地说，我们证明，若将一个向量场与一个缩放后的无散（或无旋）矩阵核进行卷积，则所得的无散（或无旋）卷积序列将收敛于该向量场 Helmholtz-Hodge 分解中对应的无散（或无旋）部分。最后，通过使用特定的求积法则对卷积序列进行离散化，我们为 Helmholtz-Hodge 分解（定义在全空间或有界域上）构造了一族（无散/无旋）拟插值算子。文中推导的相应误差估计表明，我们基于拟插值的方法能够生成对向量场及其 Helmholtz-Hodge 分解均收敛的近似解。