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P1 interpolation scaling on uniformly curved cells

Statement

Assume AC and Countable Choice. In dimension two let $K = Ψ (\tilde{K})$ , where $\tilde{K}$ is a shape-regular affine triangle of diameter comparable to $h_{K}$ , and $Ψ$ extends to a $C^{2}$ diffeomorphism on a neighborhood of $\overline{\tilde{K}}$ . Assume $∥ D Ψ ∥_{\infty} + ∥ D Ψ^{- 1} ∥_{\infty} + ∥ D^{2} Ψ ∥_{\infty} + ∥ J_{Ψ}^{- 1} ∥_{\infty}$ are uniformly bounded. Let $v$ denote its continuous representative and set $I_{K} v = (I_{\tilde{K}} (v \circ Ψ)) \circ Ψ^{- 1}$ , the mapped nodal P1 interpolant. For $v \in H^{2} (K)$ , $∥ v - I_{K} v ∥_{L^{2} (K)} \leq C h_{K}^{2} ∥ v ∥_{H^{2} (K)}$ and $∣ v - I_{K} v ∣_{H^{1} (K)} \leq C h_{K} ∥ v ∥_{H^{2} (K)}$ , with $C$ uniform over the family.

Facts & Assumptions

Given: AC and Countable Choice; a uniformly shape-regular affine triangle family; uniformly bi-Lipschitz $C^{2}$ cell maps with bounded second derivatives and Jacobian inverse; and $v \in H^{2} (K)$ .

[F1]

Local interpolation estimates on shape-regular affine meshes: under Countable Choice and, for automatic nodal boundedness, AC, on a shape-regular affine mesh, the nodal P1 interpolant of an $H^{2}$ function has $L^{2}$ error at most $C h_{K}^{2} ∣ v ∣_{H^{2}}$ and $H^{1}$ seminorm error at most $C h_{K} ∣ v ∣_{H^{2}}$ .

[F2]

Admissible isoparametric curved cells: admissible maps are diffeomorphisms with positive Jacobian and uniform derivative and inverse bounds.

[F3]

AC is used for continuous-representative nodal evaluation in [F5] and its automatic nodal-boundedness application in [F1]; Countable Choice is used through [F1] and [F4] (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F4]

C^k boundary flattening preserves local W^{k,p}: under Countable Choice, composition with a $C^{2}$ diffeomorphism and its inverse having bounded derivatives through order two is bounded on local $W^{2, 2}$ spaces, with the weak derivatives given by the chain-rule polynomials.

[F5]

Sobolev threshold for Lagrange point evaluations on a reference simplex: under AC, in dimension $d = 2$ , point evaluation is bounded on $W^{2, 2}$ because $2 > d / p = 1$ , so the nodal values are defined for the continuous representative.

Proof

1.1F2F4F5givenalgebra

First apply [F4] to $w = v \circ Ψ$ on the interiors of the reference and curved cells. The maps extend to a $C^{2}$ diffeomorphism on neighborhoods of the compact cell closures, and the stated derivative, inverse-derivative, and Jacobian bounds give $w \in H^{2} (\tilde{K})$ with $∥ w ∥_{H^{2} (\tilde{K})} \leq C ∥ v ∥_{H^{2} (K)}$ . Now [F5] supplies the unique continuous representative $\tilde{v}$ of $w$ on $\overline{\tilde{K}}$ , so its three reference-vertex values are defined. The function $\tilde{v} \circ Ψ^{- 1}$ is continuous on $\overline{K}$ and represents $v$ , by the inverse-composition part of [F4]. If $g$ is any other continuous representative of $v$ on $\overline{K}$ , then $g \circ Ψ$ is a continuous representative of $w$ and equals $\tilde{v}$ by [F5]; hence the representative on $K$ is unique. Its values at the curved-cell vertices are exactly the pulled-back nodal values. Applying [F4] to the inverse map also gives the uniform $L^{2}$ and $H^{1}$ norm bounds needed when the interpolation error is pushed back to $K$ .

2.1F1F5step 1.1given

Apply [F1] to $w$ on the affine triangle $\tilde{K}$ . The mapped interpolant is exactly $(I_{\tilde{K}} w) \circ Ψ^{- 1}$ because its three nodal values are those of $v$ at the three image vertices. The affine estimates give $∥ w - I_{\tilde{K}} w ∥_{L^{2} (\tilde{K})} \leq C h_{K}^{2} ∥ w ∥_{H^{2} (\tilde{K})}$ and $∣ w - I_{\tilde{K}} w ∣_{H^{1} (\tilde{K})} \leq C h_{K} ∥ w ∥_{H^{2} (\tilde{K})}$ , using the assumed diameter comparability.

3.1F1F2F3F4F5step 1.1step 2.1∎

Push the error through $Ψ^{- 1}$ using step 1.1. The resulting uniform norm bounds are the two claimed estimates. AC is used in [F5] for the continuous representative and nodal evaluations, including the automatic nodal-boundedness application in [F1]; Countable Choice is used through [F1] and [F4].

Depends on

Used by

Strang error bound for a parabolic isoparametric P1 family Theorem

Dependency tree · two levels

29 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

Jean-Luc Guermond, MATH 661 Fall 2025, Part III, Chapter 11 (standard reference, not scraped)
Joachim Schöberl, Numerical Methods for Partial Differential Equations, §4.2 (standard reference, not scraped)