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Quasi-interpolation for the Helmholtz-Hodge decomposition

Nicholas Fisher, Gregory Fasshauer, Wenwu Gao

TL;DR

This work addresses the stable numerical computation of the Helmholtz–Hodge decomposition for vector fields by developing a quasi-interpolation framework based on divergence-free, curl-free, and harmonic matrix kernels derived from polyharmonic splines. A generalized matrix kernel ${}_{\eta,\beta,\gamma}\bm{\Psi}_{\ell,k}$ is constructed so that, when convolved with a vector field, it reproduces the corresponding component (divergence-free or curl-free) and leads to a convergent quasi-interpolant $Q_h$ via a Schoenberg-style discretization. The authors provide simultaneous error estimates for the reconstructed field and its decomposition components on both $\mathbb{R}^d$ and bounded domains, and validate the theory through numerical simulations showing anticipated rates. The method offers a stable, structure-preserving, computationally efficient alternative to standard projection methods, with potential impact for CFD and vector-field processing.

Abstract

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

Quasi-interpolation for the Helmholtz-Hodge decomposition

TL;DR

This work addresses the stable numerical computation of the Helmholtz–Hodge decomposition for vector fields by developing a quasi-interpolation framework based on divergence-free, curl-free, and harmonic matrix kernels derived from polyharmonic splines. A generalized matrix kernel is constructed so that, when convolved with a vector field, it reproduces the corresponding component (divergence-free or curl-free) and leads to a convergent quasi-interpolant via a Schoenberg-style discretization. The authors provide simultaneous error estimates for the reconstructed field and its decomposition components on both and bounded domains, and validate the theory through numerical simulations showing anticipated rates. The method offers a stable, structure-preserving, computationally efficient alternative to standard projection methods, with potential impact for CFD and vector-field processing.

Abstract

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

Paper Structure

This paper contains 15 sections, 10 theorems, 106 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Polyharmonic splines satisfying higher-order Strang--Fix conditions
  5. The Helmholtz--Hodge decomposition
  6. The Main Results
  7. Divergence-free, curl-free, and harmonic matrix kernels
  8. Kernel projections
  9. Construction of quasi-interpolation schemes
  10. Quasi-interpolation defined in the whole space $\mathbb{R}^d$
  11. Quasi-interpolation defined over a bounded domain $\Omega$
  12. Numerical Simulations
  13. Approximations on $\mathbb{R}^d$
  14. Approximation on bounded domains
  15. Conclusions and Discussions

Key Result

Lemma 1

Let $\psi_{\ell,k}$ be a level-$k\ell$ polyharmonic spline defined in Equation polyharmonicspline. Then the decay condition holds true for any $0\leq |\alpha|< 2k$ as $\|\bold x\|$ tends to infinity. Moreover, the (generalized) Fourier transform $\widehat{\psi_{\ell,k}}$ of $\psi_{\ell,k}$ satisfies $\widehat{\psi_{\ell,k}}\in \mathcal{M}^{2k}(\mathbb{R}^d)$ together with the Strang--Fix conditio

Figures (3)

  • Figure 1: Errors and convergence rates on $\mathbb{R}^d$ with $\alpha = 0$ (left) and $\alpha = 1$ (right).
  • Figure 2: Errors and convergence rates on $[0,1]^2$ with $\alpha = 0$ using $\boldsymbol{\Psi}^{div}_{2,2}(\bold x)$ (left) and $\boldsymbol{\Psi}^{div}_{3,3}(\bold x)$ (right).
  • Figure 3: Quasi-interpolation for the Helmholtz--Hodge decomposition on a bounded domain.

Theorems & Definitions (19)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 1
  • proof
  • ...and 9 more