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Cellwise oscillation bound for particle quadrature
Statement
Let g be compactly supported and Lipschitz with constant L_g, meaning |g(y)−g(x)|≤L_g|y−x|, with supp(g)⊂Ω, and use the cells and nodes of Meshfree cell quadrature and fill radius. Define osc_C(g)=sup_{y,z∈C}|g(y)−g(z)| and let J_g={j:C_j∩supp(g)≠∅}. Then
Facts & Assumptions
Given: A compactly supported Lipschitz g with supp(g)⊂Ω and a cell particle rule.
Meshfree cell quadrature and fill radius: only finitely many cells meet supp(g), their integrals sum to ∫Ωg, V_j=∫1{C_j}, and |y−x_j|≤h on C_j.
Cauchy-Schwarz with its equality case, the triangle inequality for , the parallelogram law and polarisation: the Euclidean norm satisfies the triangle inequality.
Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in : on a bounding rectangle, the Riemann integral is linear, monotone, and satisfies |∫u|≤∫|u|; the Jordan-cell integrals use their zero extensions to such rectangles.
Proof
By the integration-compatible partition in [F1] and linearity in [F3], subtracting Q_hg cell by cell gives ∫Ωg−Q_hg=Σ{j∈J_g}∫{C_j}(g(y)−g(x_j))dy, using ∫{C_j}g(x_j)=V_jg(x_j). On a bounding rectangle for each cell, [F3] gives |∫{C_j}(g−g(x_j))|≤∫{C_j}|g−g(x_j)|≤V_j osc_{C_j}(g). Summing proves the first inequality.
For y,z∈C_j, [F1] gives |y−x_j|≤h and |z−x_j|≤h; the triangle inequality in [F2] gives |y−z|≤2h. Hence |g(y)−g(z)|≤2hL_g and osc_{C_j}(g)≤2hL_g. Substituting this bound in step 1.1 proves the second inequality.
Depends on
- Meshfree cell quadrature and fill radius
- 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
- Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $\mathbb{R}^m$
Used by
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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)