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Unisolvent degrees of freedom define a unique reference interpolant

Statement

Let $(\hat{K}, \hat{P}, \hat{Σ})$ be a reference finite element with $N = \dim \hat{P}$ , dual shape functions ${\hat{ϕ}}_{i}$ , and bounded extensions of its degrees of freedom to a normed function space $X$ containing $\hat{P}$ . Then the operator $\hat{I} v : = \sum_{i = 1}^{N} {\hat{σ}}_{i} (v) {\hat{ϕ}}_{i}$ is the unique linear map $X \to \hat{P}$ satisfying ${\hat{σ}}_{i} (\hat{I} v) = {\hat{σ}}_{i} (v)$ for every $i$ , and $\hat{I} p = p$ for every $p \in \hat{P}$ .

Facts & Assumptions

Given: The reference finite-element convention of Reference finite elements and unisolvent degrees of freedom and an input $v \in X$ on which every degree of freedom is defined.

[F1]

The degree-of-freedom map $J_{\hat{Σ}} : \hat{P} \to K^{N}$ is bijective, and the shape functions satisfy ${\hat{σ}}_{i} ({\hat{ϕ}}_{j}) = δ_{i j}$ . (Reference finite elements and unisolvent degrees of freedom)

Proof

1.1F1given

If $N = 0$ , then $\hat{P} = {0}$ and the empty sum defines the unique linear map $X \to \hat{P}$ ; all degree-of-freedom identities are vacuous and reproduction holds for the sole element $p = 0$ . For $N \geq 1$ , define $\hat{I} v = \sum_{i = 1}^{N} {\hat{σ}}_{i} (v) {\hat{ϕ}}_{i}$ . The sum is finite and each coefficient is linear in $v$ , so $\hat{I}$ is a well-defined linear map into $\hat{P}$ . Applying the $j$ th degree of freedom gives ${\hat{σ}}_{j} (\hat{I} v) = \sum_{i} {\hat{σ}}_{i} (v) {\hat{σ}}_{j} ({\hat{ϕ}}_{i}) = {\hat{σ}}_{j} (v)$ by [F1].

2.1F1step 1.1∎

If $p \in \hat{P}$ , then both $p$ and $\hat{I} p$ have the same degree-of-freedom vector, by step 1.1; injectivity of $J_{\hat{Σ}}$ therefore gives $\hat{I} p = p$ . If another $J : X \to \hat{P}$ has the same degree-of-freedom values, then $J_{\hat{Σ}} (J v - \hat{I} v) = 0$ , and injectivity again gives $J v = \hat{I} v$ . This proves uniqueness.

Depends on

Reference finite elements and unisolvent degrees of freedom

Used by

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Sources

Joachim Schöberl, Numerical Methods for Partial Differential Equations, Chapter 4 (standard reference, not scraped)
Jean-Luc Guermond, MATH 661 Fall 2025, Chapter 11 (standard reference, not scraped)