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On Particles and Splines in Bounded Domains

Matthias Kirchhart

TL;DR

This work develops a robust particle-based advection scheme for general bounded domains by coupling particle fields with Cartesian tensor-product splines in a fictitious-domain setting. It replaces previous $V_\sigma^n$ spaces with spline spaces on an unfitted grid and proves stability and consistency in $W^{s,p}$ norms, including a main error bound that mirrors whole-space blob analyses while offering $\mathcal{O}(1)$ evaluation cost. The method relies on a ghost-penalty stabilization and a boundary-m mesh projection to extend particle fields smoothly to the fictitious domain, enabling accurate initialization and regularization near boundaries. Key contributions include a rigorous $L^p$-based stability framework, a convergent approximate extension operator $A_\varepsilon^{-1}$ with super-convergence on $\Omega$, and practical guidance for balancing regularization and quadrature errors in bounded domains. The approach enhances accuracy, stability, and efficiency for advection-dominated problems in complex geometries, with potential for adaptive and multi-resolution extensions.

Abstract

We propose numerical schemes that enable the application of particle methods for advection problems in general bounded domains. These schemes combine particle fields with Cartesian tensor product splines and a fictitious domain approach. Their implementation only requires a fitted mesh of the domain's boundary, and not the domain itself, where an unfitted Cartesian grid is used. We establish the stability and consistency of these schemes in $W^{s,p}$-norms, $s\in\mathbb{R}$, $1<p\leq\infty$.

On Particles and Splines in Bounded Domains

TL;DR

This work develops a robust particle-based advection scheme for general bounded domains by coupling particle fields with Cartesian tensor-product splines in a fictitious-domain setting. It replaces previous spaces with spline spaces on an unfitted grid and proves stability and consistency in norms, including a main error bound that mirrors whole-space blob analyses while offering evaluation cost. The method relies on a ghost-penalty stabilization and a boundary-m mesh projection to extend particle fields smoothly to the fictitious domain, enabling accurate initialization and regularization near boundaries. Key contributions include a rigorous -based stability framework, a convergent approximate extension operator with super-convergence on , and practical guidance for balancing regularization and quadrature errors in bounded domains. The approach enhances accuracy, stability, and efficiency for advection-dominated problems in complex geometries, with potential for adaptive and multi-resolution extensions.

Abstract

We propose numerical schemes that enable the application of particle methods for advection problems in general bounded domains. These schemes combine particle fields with Cartesian tensor product splines and a fictitious domain approach. Their implementation only requires a fitted mesh of the domain's boundary, and not the domain itself, where an unfitted Cartesian grid is used. We establish the stability and consistency of these schemes in -norms, , .

Paper Structure

This paper contains 21 sections, 13 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Grid-based Schemes
  3. Particle Methods
  4. Particle Methods in Bounded Domains
  5. Novelty and Main Result
  6. Spaces of Functions and Functionals
  7. Sobolev Spaces of Integer Order
  8. Sobolev and Besov Spaces of Fractional Order
  9. Sobolev Embeddings and Stein Extension
  10. Spline Spaces
  11. Some Properties of Splines
  12. Particle Initialization
  13. Particle Approximations of Splines
  14. Error Bounds
  15. Particle Regularization
  16. ...and 6 more sections

Key Result

Lemma 2.4

Let $\square\subset\mathbb{R}^\mathrm{D}$ be a finite collection of entire, uncut cubes $Q_\mathbf{i}$ from the Cartesian grid of size $\sigma>0$. Then every function $v_\sigma\in V_\sigma^{n}(\square)$, $n\in\mathbb{N}$, can be written as with a uniquely determined coefficient vector $\mathsf{u}_{\sigma} = \bigl(\mathsf{u}_{\sigma,\boldsymbol{\lambda}}\bigr)_{\boldsymbol{\lambda}\in\Lambda_{\sig

Figures (2)

  • Figure 1: \newlabelfig:regularisation0 Approximation of the exponential function (blue) on the interval $[0,2]$. On the left: a highly accurate particle approximation. The particle weights, depicted by the arrows' heights, usually do not correlate well with the local function values. On the right: conventional smoothing of the particle field yields a globally smooth approximation (red) of the target function's non-smooth zero-extension. This results in poor approximations near its discontinuities at the boundaries. The stabilized $L^2$-projection (brown) kirchhart2017b yields an approximation of a smooth extension. It is not only accurate on the entire interval but also extra-polates well after its ends.
  • Figure 1: \newlabelfig:fictitious_domains0 An illustration of the fictitious domain approach. The domain $\Omega$ (blue), in this case a circle, may intersect the infinite Cartesian grid in an arbitrary manner. The fictitious domain $\Omega_\sigma$ (red) is defined as the union of all intersected cells. The domain $\Omega_\sigma^\circ$ in this case consists of the four elements entirely lying in $\Omega$. The set of faces $\mathcal{F}_\sigma$ is highlighted using bold lines. It can be thought of forming a bridge between $\Omega_\sigma^\circ$ and the remaining elements in $\Omega_\sigma^\Gamma$.

Theorems & Definitions (24)

  • Definition 2.1: Cartesian Grid and Fictitious Domains
  • Definition 2.2: Spline Spaces
  • Definition 2.3: B-Splines
  • Lemma 2.4: Stability of the B-Spline Basis
  • Lemma 2.5: Inverse Estimates
  • Lemma 2.6: Quasi-interpolator
  • Remark 2.7
  • Lemma 2.8
  • Theorem 3.1
  • Proof 1
  • ...and 14 more