Meshfree cell quadrature and fill radius

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be open. Write $B(x,r)$ for the open Euclidean ball from Open ball, closed ball and sphere in a metric space with distance induced by The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. A family of cells is locally finite here when every compact subset of $\mathrm{\Omega }$ meets only finitely many cells. A bounded set is Jordan measurable when its Jordan inner and outer contents agree (Jordan inner and outer content and Jordan measurable bounded sets in ${\mathbb{R}}^m$). A cell particle rule is a nonempty locally finite family of bounded Jordan-measurable cells $C_j\subseteq \mathrm{\Omega }$ whose interiors are disjoint and whose union covers $\mathrm{\Omega }$ up to their shared boundaries, together with nodes $x_j\in C_j$ and positive volumes $V_j=\int 1_{C_j}$. The cell volume integral is Riemann-defined and equals the Jordan content (The Riemann integral of a bounded function over a bounded Jordan measurable set). Its fill radius is $h={\sup }_j{\sup }_{y\in C_j}\mathrm{\mid }y-x_j\mathrm{\mid }$, using the Euclidean norm from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$; require $0<h<\mathrm{\infty }$. Assume finite constants $C_V,C_L>0$ satisfy $V_j\le C_V{h}^d$ and the local volume bound

${\sum }_{j:C_j\cap B(x,r)\mathrm{\ne }\mathrm{\varnothing }}{V}_j\le C_L(r+h{)}^d$

for every $x\in {\mathbb{R}}^d$ and $r>0$. For continuous $g:{\mathbb{R}}^d\to \mathbb{R}$ with compact support contained in $\mathrm{\Omega }$ (support as in The support of a function on ${\mathbb{R}}^n$ and its compactly supported Riemann integral), define $Q_{h}g={\sum }_j{V}_{j}g(x_j)$; local finiteness makes this sum finite. For specified parameters $R>0$ and $\epsilon >0$, an evaluation point $x\in {\mathbb{R}}^d$ is interior relative to $(R,\epsilon )$ when $B(x,R\epsilon +2h)\subset \mathrm{\Omega }$; in a kernel application, $R$ is the chosen support-radius bound for the unscaled kernel. For periodic calculations, use the periodic extensions of the field and particle rule on ${\mathbb{R}}^d$, so no kernel support is truncated.

The cell rule is integration-compatible: for every continuous $g:{\mathbb{R}}^d\to \mathbb{R}$ with compact support contained in $\mathrm{\Omega }$, only finitely many cells meet $\mathrm{supp}(g)$, every cell integral appearing below is Riemann-defined, and ${\int }_{\mathrm{\Omega }}g(y)dy={\sum }_{j:C_j\cap \mathrm{supp}(g)\mathrm{\ne }\mathrm{\varnothing }}{\int }_{C_j}g(y)dy\mathrm{.}$ This finite partition identity is part of the quadrature data.