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Elementwise P1 product-cutoff energy estimate

Statement

Assume Countable Choice. Let $d \geq 2$ and let $K$ be a nondegenerate affine $d$ -simplex in a shape-regular mesh with $h_{K} / ρ_{K} \leq σ$ . Let $θ, w \in P_{1} (K)$ . Let $I_{K} (θ w) \in P_{1} (K)$ be the affine function whose values at the vertices of $K$ are the nodal values of $θ w$ . There is a constant $C$ depending only on $d$ and the shape regularity bound $σ$ such that $∥ \nabla I_{K} (θ w) ∥_{L^{2} (K)} \leq C (∥ θ \nabla w ∥_{L^{2} (K)} + ∥ w \nabla θ ∥_{L^{2} (K)}) .$

Facts & Assumptions

Given: Countable Choice, a nondegenerate affine simplex $K$ from a shape-regular family, and two affine scalar functions $θ, w$ on $K$ .

[F1]

Matching affine simplicial meshes and shape regularity gives each cell as a nondegenerate affine simplex with affinely independent vertices $x_{0}, \dots, x_{d}$ . Thus there are unique barycentric affine functions $λ_{0}, \dots, λ_{d}$ satisfying $\sum_{i} λ_{i} = 1$ and $x = \sum_{i} λ_{i} (x) x_{i}$ ; these are the P1 nodal basis functions used below.

[F2]

Under Countable Choice, Shape regularity controls affine simplex geometry: for simplex diameter $h_{K}$ , each barycentric basis function satisfies $∥ \nabla λ_{i} ∥_{L^{\infty} (K)} \leq C h_{K}^{- 1}$ , and affine maps have uniform volume scaling under the shape-regularity bound.

[F3]

Finite dimensional norm equivalence over a complete valued field: on the fixed reference simplex the coordinate maximum of a P1 polynomial's vertex values is bounded by a constant times its $L^{2}$ norm; affine scaling transfers this as $\max_{i} ∣ v (x_{i}) ∣ \leq C h_{K}^{- d / 2} ∥ v ∥_{L^{2} (K)}$ .

Proof

1.1F1F2algebra

Write the vertices as $x_{0}, \dots, x_{d}$ , their barycentric coordinates as $λ_{0}, \dots, λ_{d}$ , and set $θ_{i} = θ (x_{i})$ and $w_{i} = w (x_{i})$ . Since $\sum_{i} \nabla λ_{i} = 0$ , $\nabla I_{K} (θ w) = \sum_{i = 1}^{d} (θ_{i} w_{i} - θ_{0} w_{0}) \nabla λ_{i} .$ For each $i \geq 1$ , $θ_{i} w_{i} - θ_{0} w_{0} = θ_{i} (w_{i} - w_{0}) + w_{0} (θ_{i} - θ_{0})$ . Affineness gives $∣ w_{i} - w_{0} ∣ \leq h_{K} ∣ \nabla w ∣$ and $∣ θ_{i} - θ_{0} ∣ \leq h_{K} ∣ \nabla θ ∣$ . Applying [F2] to the barycentric gradients therefore gives pointwise $∣ \nabla I_{K} (θ w) ∣ \leq C (\max_{i} ∣ θ_{i} ∣ ∣ \nabla w ∣ + \max_{i} ∣ w_{i} ∣ ∣ \nabla θ ∣) .$

2.1F2F3step 1.1algebra∎

By [F3], the two vertex maxima in step 1.1 are at most $C h_{K}^{- d / 2} ∥ θ ∥_{L^{2} (K)}$ and $C h_{K}^{- d / 2} ∥ w ∥_{L^{2} (K)}$ , respectively. The gradients of affine functions are constant on $K$ , so multiplying the pointwise estimate by $∣ K ∣^{1 / 2} ≍ h_{K}^{d / 2}$ yields $∥ \nabla I_{K} (θ w) ∥_{L^{2} (K)} \leq C (∣ \nabla w ∣ ∥ θ ∥_{L^{2} (K)} + ∣ \nabla θ ∣ ∥ w ∥_{L^{2} (K)}),$ which is the asserted estimate.

Depends on

Used by

Stable two-level P1 decomposition for structured Poisson meshes Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jinchao Xu, Iterative Methods by Space Decomposition and Subspace Correction, SIAM Review 34(4) (1992) (standard reference, not scraped)
Peter Bastian, Lecture Notes on Parallel Solution of Large Sparse Linear Systems, University of Heidelberg (2015) (standard reference, not scraped)