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Explicit cellwise error for kernel value and gradient quadrature
Statement
Assume the cell rule of Meshfree cell quadrature and fill radius, h≤ε≤1, x is interior, and f∈C¹(R^d) is bounded with bounded first derivative. Define
Let S_εf(x)=∫f(y)W_ε(x−y)dy and D_εf(x)=∫f(y)∇W_ε(x−y)dy. There is C=C(d,R,C_L,W), independent of ε,h,x, such that
Facts & Assumptions
Given: An interior point, a cell rule with the fill-radius and local-volume bounds of Meshfree cell quadrature and fill radius, a fixed admissible kernel, ε,h with h≤ε≤1, and f∈C¹ with bounded f and ∇f.
Scaled symmetric compactly supported kernel family: W is supported in the closed ball \overline B(0,R), and M₀,M₁,M₂ bound W, ∇W and D²W.
Meshfree cell quadrature and fill radius: 0<h<∞, cells meeting B(x,r) have total volume at most C_L(r+h)^d, and an interior point satisfies B(x,Rε+2h)⊂Ω. Hence supp(g₀),supp(g₁)⊂\overline B(x,Rε)⊂B(x,Rε+2h)⊂Ω, so both integrands meet the cellwise lemma's support hypothesis and the full-space integrals equal the corresponding Ω integrals.
Scaling identities for kernel moments and derivatives: ||W_ε||∞=ε^(−d)M₀, ||∇W_ε||∞=ε^(−d−1)M₁, and ||D²W_ε||∞=ε^(−d−2)M₂.
Cellwise oscillation bound for particle quadrature: cell quadrature error is bounded by 2h times the Lipschitz constant of the integrand times the total active cell volume.
The mean value inequality: if is continuous and differentiable on with , then : a continuously differentiable function on a line segment with derivative norm at most L changes by at most L times the segment length.
The chain rule for total derivatives: and Sums, scalar multiples, products and quotients: , , , and when : after restriction to a line segment, the total chain rule differentiates each factor and the one-variable product rule differentiates each component product.
Cauchy-Schwarz with its equality case, the triangle inequality for , the parallelogram law and polarisation: Euclidean inner products satisfy Cauchy-Schwarz.
Proof
Put g₀(y)=f(y)W_ε(x−y). On any segment y(t)=y₀+t(y₁−y₀), [F6] gives the chain and product rules, and Cauchy-Schwarz [F7] gives |(g₀∘y)'(t)|≤(ε^(−d)M₀||∇f||∞+ε^(−d−1)M₁||f||∞)|y₁−y₀| by [F3]. Applying [F5] on that segment gives the same Lipschitz constant for g₀. Its support is contained in the closed ball \overline B(x,Rε), hence in B(x,(R+1)ε). By [F2], the total volume of cells meeting that support is at most C_L((R+1)ε+h)^d≤C_L(R+2)^dε^d. Apply [F4] and cancel ε^d; the factor 2 is absorbed into C. This gives the first inequality.
Put g₁(y)=f(y)∇W_ε(x−y). On any segment y(t)=y₀+t(y₁−y₀), [F6] gives componentwise chain and product rules, and Cauchy-Schwarz [F7] gives ||(g₁∘y)'(t)||₂≤(ε^(−d−1)M₁||∇f||∞+ε^(−d−2)M₂||f||∞)|y₁−y₀| by [F3]. Applying the vector mean value inequality [F5] bounds its Lipschitz constant by this expression. Its support is contained in the same closed ball, so [F2] gives the active-volume bound from step 1.1. Apply [F4] to each vector component; taking the Euclidean norm costs at most a factor √d, and [F4] contributes a factor 2; both are absorbed into C. Cancelling ε^d proves the second inequality.
Depends on
- Scaled symmetric compactly supported kernel family
- Meshfree cell quadrature and fill radius
- Scaling identities for kernel moments and derivatives
- Cellwise oscillation bound for particle quadrature
- The mean value inequality: if $f : [a,b] \to \mathbb{R}^m$ is continuous and differentiable on $(a,b)$ with $\lVert f'\rVert_2 \le M$, then $\lVert f(b)-f(a)\rVert_2 \le M(b-a)$
- The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
- Cauchy-Schwarz $\lvert\langle x,y\rangle\rvert \le \lVert x\rVert_2\lVert y\rVert_2$ with its equality case, the triangle inequality for $\lVert\cdot\rVert_2$, the parallelogram law and polarisation
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Sources
- Volker Springel, Smoothed Particle Hydrodynamics in Astrophysics, Annual Review of Astronomy and Astrophysics 48 (2010), author full text (standard reference, not scraped)
- Benjamin Seibold, MIT OpenCourseWare 18.336, Numerical Methods for Partial Differential Equations, Lecture 25: Particle Methods (Spring 2009) (standard reference, not scraped)