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Affine Piola transport preserves commuting interpolation diagrams

Statement

Let ${\hat{V}}_{j}$ and $V_{j}$ be reference and physical spaces related by invertible affine Piola maps $P_{j} : {\hat{V}}_{j} \to V_{j}$ . Let ${\hat{D}}_{j}$ and $D_{j}$ be the corresponding differential operators, and assume the proved affine identities $D_{j} P_{j} = P_{j + 1} {\hat{D}}_{j}$ from Affine Piola maps commute with differential operators and preserve moments. Suppose reference interpolation operators ${\hat{I}}_{j}$ are defined on spaces containing the relevant inputs and satisfy ${\hat{D}}_{j} {\hat{I}}_{j} = {\hat{I}}_{j + 1} {\hat{D}}_{j}$ there. Define the physical interpolants by Piola conjugation $I_{j} = P_{j} {\hat{I}}_{j} P_{j}^{- 1}$ . Then $D_{j} I_{j} = I_{j + 1} D_{j}$ on every input for which these compositions are defined. This is conditional on the reference commuting identity; it does not assert that arbitrary unisolvent degrees of freedom commute.

Facts & Assumptions

Given: Unisolvent reference interpolants as in Reference finite elements and unisolvent degrees of freedom and Unisolvent degrees of freedom define a unique reference interpolant, invertible affine Piola maps, and the reference commuting identities in the statement.

[F1]

The physical and reference differential operators satisfy $D_{j} P_{j} = P_{j + 1} {\hat{D}}_{j}$ on their domains. (Affine Piola maps commute with differential operators and preserve moments)

[F2]

The interpolation operator defined by the dual shape functions is a linear map on its declared domain. (Unisolvent degrees of freedom define a unique reference interpolant)

Proof

1.1F1givenalgebra

By definition $I_{j} = P_{j} {\hat{I}}_{j} P_{j}^{- 1}$ . Using [F1] and the assumed reference commutation, $D_{j} I_{j} = D_{j} P_{j} {\hat{I}}_{j} P_{j}^{- 1} = P_{j + 1} {\hat{D}}_{j} {\hat{I}}_{j} P_{j}^{- 1} = P_{j + 1} {\hat{I}}_{j + 1} {\hat{D}}_{j} P_{j}^{- 1} .$

2.1F1F2step 1.1algebra∎

Apply [F1] to $P_{j}^{- 1}$ , equivalently ${\hat{D}}_{j} P_{j}^{- 1} = P_{j + 1}^{- 1} D_{j}$ . Substitution into step 1.1 gives $D_{j} I_{j} = P_{j + 1} {\hat{I}}_{j + 1} P_{j + 1}^{- 1} D_{j} = I_{j + 1} D_{j},$ which is the claimed physical commuting identity. The only interpolation property used was the explicitly assumed reference identity, proving the conditional scope.

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Chapter 16 (standard reference, not scraped)
Joachim Schöberl, Numerical Methods for Partial Differential Equations, Chapter 8 (standard reference, not scraped)