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A stable local commuting projector and optimal hp approximation estimates in \(\varvec{H}(\textrm{curl})\)

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Abstract

We design an operator from the infinite-dimensional Sobolev space \(\varvec{H}(\textrm{curl})\) to its finite-dimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh that has the following properties: (1) it is defined over the entire \(\varvec{H}(\textrm{curl})\), including boundary conditions imposed on a part of the boundary; (2) it is defined locally in a neighborhood of each mesh element; (3) it is based on simple piecewise polynomial projections; (4) it is stable in the \(\varvec{L}^2\)-norm, up to data oscillation; (5) it has optimal (local-best) approximation properties; (6) it satisfies the commuting property with its sibling operator on \(\varvec{H}(\textrm{div})\); (7) it is a projector, i.e., it leaves intact objects that are already in the Nédélec piecewise polynomial space. This operator can be used in various parts of numerical analysis related to the \(\varvec{H}(\textrm{curl})\) space. We in particular employ it here to establish the two following results: (i) equivalence of global-best, tangential-trace- and curl-constrained, and local-best, unconstrained approximations in \(\varvec{H}(\textrm{curl})\) including data oscillation terms; and (ii) fully h- and p- (mesh-size- and polynomial-degree-) optimal approximation bounds valid under the minimal Sobolev regularity only requested elementwise. As a result of independent interest, we also prove a p-robust equivalence of curl-constrained and unconstrained best-approximations on a single tetrahedron in the \(\varvec{H}(\textrm{curl})\)-setting, including hp data oscillation terms.

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Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No 647134 GATIPOR).

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  1. Théophile Chaumont-Frelet

    Present address: Inria Univ. Lille and Laboratoire Paul Painlevé, 59655, Villeneuve-d’Ascq, France

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    Théophile Chaumont-Frelet

  2. Inria, 48 rue Barrault, 75647, Paris, France

    Martin Vohralík

  3. CERMICS, Ecole des Ponts, 6 et 8 Avenue Blaise Pascal, 77455, Marne-la-Vallée, France

    Martin Vohralík

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Appendices

p-robust equivalence of constrained and unconstrained best-approximation in \(\varvec{H}(\textrm{curl})\) on a tetrahedron

In this Appendix, we extend [16, Lemma 1] to functions \(\varvec{v}\) with nonpolynomial curl, in the spirit of [26, Lemma A1]. This is a consequence of the breakthrough result of Costabel and McIntosh in [20, Proposition 4.2]. Interestingly enough, the constant hidden in the inequality is here independent of the polynomial degree p.

Lemma 10

(Equivalence of constrained and unconstrained best-approximation on a tetrahedron) Let a polynomial degree \(p \ge 0\), a tetrahedron \({K}\), and an arbitrary \(\varvec{v}\in \varvec{H}(\textrm{curl}, {K})\) be fixed. Let

$$\begin{aligned} \varvec{\tau }_h :=\arg \min _{\begin{array}{c} \varvec{w}_h \in \varvec{\mathcal {R\hspace{-0.1em}T}}\hspace{-0.25em}_p({K})\\ \nabla {\cdot }\varvec{w}_h = 0 \end{array}} \Vert \nabla {\times }\varvec{v}- \varvec{w}_h\Vert _{K}. \end{aligned}$$
(A.1)

Then

$$\begin{aligned} \begin{aligned} \min _{\varvec{v}_h \in \varvec{\mathcal {N}}\hspace{-0.2em}_p({K})}\Vert \varvec{v}- \varvec{v}_h\Vert _{K}&\le \min _{\begin{array}{c} \varvec{v}_h \in \varvec{\mathcal {N}}\hspace{-0.2em}_p({K})\\ \nabla {\times }\varvec{v}_h = \varvec{\tau }_h \end{array}} \Vert \varvec{v}- \varvec{v}_h\Vert _{K}\\&\lesssim \Big (\min _{\varvec{v}_h \in \varvec{\mathcal {N}}\hspace{-0.2em}_p({K})}\Vert \varvec{v}- \varvec{v}_h\Vert _{K}+ \frac{h_{{K}}}{p+1}\big \Vert \nabla {\times }\varvec{v}- \varvec{\Pi }_{\varvec{\mathcal {R\hspace{-0.1em}T}}}^{p}(\nabla {\times }\varvec{v})\big \Vert _{K}\Big ), \end{aligned}\nonumber \\ \end{aligned}$$
(A.2)

where the hidden constant only depends on the shape-regularity \(\kappa _{K}:=h_{K}/ \rho _{K}\) of \({K}\).

Proof

The first inequality is obvious, since the second minimization set has an additional curl constraint. In order to show the second one, denote respectively by

$$\begin{aligned} \varvec{\iota }_h :=\arg \min _{\begin{array}{c} \varvec{v}_h \in \varvec{\mathcal {N}}\hspace{-0.2em}_p({K})\\ \nabla {\times }\varvec{v}_h = \varvec{\tau }_h \end{array}} \Vert \varvec{v}- \varvec{v}_h\Vert _{K}\end{aligned}$$
(A.3)

and

$$\begin{aligned} {\tilde{\varvec{\iota }}}_h :=\arg \min _{\varvec{v}_h \in \varvec{\mathcal {N}}\hspace{-0.2em}_p({K})} \Vert \varvec{v}- \varvec{v}_h\Vert _{K}\end{aligned}$$
(A.4)

the constrained and unconstrained minimizers. We then need to show

$$\begin{aligned} \Vert \varvec{v}- \varvec{\iota }_h\Vert _{K}\lesssim \Vert \varvec{v}- {\widetilde{\varvec{\iota }}}_h\Vert _{K}+ \frac{h_{{K}}}{p+1}\big \Vert \nabla {\times }\varvec{v}- \varvec{\Pi }_{\varvec{\mathcal {R\hspace{-0.1em}T}}}^{p}(\nabla {\times }\varvec{v})\big \Vert _{K}, \end{aligned}$$
(A.5)

where \(\lesssim \) means inequality up to a constant only depending on the shape-regularity \(\kappa _{K}\).

Using (A.1) and (4.2), \(\varvec{\tau }_h - \nabla {\times }{\tilde{\varvec{\iota }}}_h \in \{\varvec{w}_h \in \varvec{\mathcal {R\hspace{-0.1em}T}}\hspace{-0.25em}_p({K}); \, \nabla {\cdot }\varvec{w}_h = 0\}\). Thus, we can use [20, Proposition 4.2], cf. also the reformulation in [12, Theorem 2], stipulating the existence of \(\varvec{\varphi }_h \in \varvec{\mathcal {N}}\hspace{-0.2em}_p({K})\) with \(\nabla {\times }\varvec{\varphi }_h = \varvec{\tau }_h - \nabla {\times }{\tilde{\varvec{\iota }}}_h\) such that

$$\begin{aligned} \Vert \varvec{\varphi }_h\Vert _{K}\lesssim \min _{\begin{array}{c} \varvec{\varphi }\in \varvec{H}(\textrm{curl}, {K})\\ \nabla {\times }\varvec{\varphi }= \varvec{\tau }_h - \nabla {\times }{\tilde{\varvec{\iota }}}_h \end{array}} \Vert \varvec{\varphi }\Vert _{K}. \end{aligned}$$
(A.6)

Shifting now the right-hand side of (A.6) by \({\tilde{\varvec{\iota }}}_h\), we arrive at

$$\begin{aligned} \Vert \varvec{\varphi }_h\Vert _{K}\lesssim \min _{\begin{array}{c} \varvec{\varphi }\in \varvec{H}(\textrm{curl}, {K})\\ \nabla {\times }\varvec{\varphi }= \varvec{\tau }_h \end{array}} \Vert \varvec{\varphi }- {\tilde{\varvec{\iota }}}_h\Vert _{K}. \end{aligned}$$

A primal–dual equivalence as in, e.g., [13, Lemma 5.5] implies (as in Sect. 2.3, \(\varvec{H}_0(\textrm{curl}, {K})\) is composed of those \(\varvec{\varphi }\in \varvec{H}(\textrm{curl}, {K})\) that verify \(\varvec{\varphi }{\times }\varvec{n}_{K}=0\) on \(\partial {K}\) in the weak sense (2.3))

$$\begin{aligned} \min _{\begin{array}{c} \varvec{\varphi }\in \varvec{H}(\textrm{curl}, {K})\\ \nabla {\times }\varvec{\varphi }= \varvec{\tau }_h \end{array}} \Vert \varvec{\varphi }- {\tilde{\varvec{\iota }}}_h\Vert _{K}= &\sup _{\begin{array}{c} \varvec{\varphi }\in \varvec{H}_0(\textrm{curl}, {K})\\ \Vert \nabla {\times }\varvec{\varphi }\Vert _{K}= 1 \end{array}} \big \{(\varvec{\tau }_h, \varvec{\varphi })_{K}- ({\tilde{\varvec{\iota }}}_h, \nabla {\times }\varvec{\varphi })_{K}\big \} \\ \le &\sup _{\begin{array}{c} \varvec{\varphi }\in \varvec{H}_0(\textrm{curl}, {K})\\ \Vert \nabla {\times }\varvec{\varphi }\Vert _{K}= 1 \end{array}} \big \{(\nabla {\times }\varvec{v}, \varvec{\varphi })_{K}- ({\tilde{\varvec{\iota }}}_h, \nabla {\times }\varvec{\varphi })_{K}\big \} \\ &+ \sup _{\begin{array}{c} \varvec{\varphi }\in \varvec{H}_0(\textrm{curl}, {K})\\ \Vert \nabla {\times }\varvec{\varphi }\Vert _{K}= 1 \end{array}} (\varvec{\tau }_h - \nabla {\times }\varvec{v}, \varvec{\varphi })_{K}\\ \lesssim &\min _{\begin{array}{c} \varvec{\varphi }\in \varvec{H}(\textrm{curl}, {K})\\ \nabla {\times }\varvec{\varphi }= \nabla {\times }\varvec{v} \end{array}} \Vert \varvec{\varphi }- {\tilde{\varvec{\iota }}}_h\Vert _{K}+ \frac{h_{{K}}}{p+1}\big \Vert \nabla {\times }\varvec{v}- \varvec{\Pi }_{\varvec{\mathcal {R\hspace{-0.1em}T}}}^{p}(\nabla {\times }\varvec{v})\big \Vert _{K}, \end{aligned}$$

where, to estimate the second term on the middle line, we have used the technical result of Lemma 11 below.

Consequently, since \(\varvec{v}\in \varvec{H}(\textrm{curl}, {K})\) satisfies the curl constraint above,

$$\begin{aligned} \Vert \varvec{\varphi }_h\Vert _{K}\lesssim \Vert \varvec{v}- {\tilde{\varvec{\iota }}}_h\Vert _{K}+ \frac{h_{{K}}}{p+1}\big \Vert \nabla {\times }\varvec{v}- \varvec{\Pi }_{\varvec{\mathcal {R\hspace{-0.1em}T}}}^{p}(\nabla {\times }\varvec{v})\big \Vert _{K}. \end{aligned}$$
(A.7)

Now note that \((\varvec{\varphi }_h + {\tilde{\varvec{\iota }}}_h) \in \varvec{\mathcal {N}}\hspace{-0.2em}_p({K})\) with \(\nabla {\times }(\varvec{\varphi }_h + {\tilde{\varvec{\iota }}}_h) = \varvec{\tau }_h\). Thus, \(\varvec{\varphi }_h + {\tilde{\varvec{\iota }}}_h\) belongs to the minimization set in (A.3), and the minimization property (A.3) of \(\varvec{\iota }_h\) implies \(\Vert \varvec{v}- \varvec{\iota }_h\Vert _{K}\le \Vert \varvec{v}- (\varvec{\varphi }_h + {\tilde{\varvec{\iota }}}_h)\Vert _{K}\). Thus, by virtue of the triangle inequality and using (A.7), we altogether infer

$$\begin{aligned} \Vert \varvec{v}- \varvec{\iota }_h\Vert _{K}&\le \Vert \varvec{v}- (\varvec{\varphi }_h + {\tilde{\varvec{\iota }}}_h)\Vert _{K}\le \Vert \varvec{v}- {\tilde{\varvec{\iota }}}_h\Vert _{K}+ \Vert \varvec{\varphi }_h\Vert _{K}\\&\lesssim \Vert \varvec{v}- {\tilde{\varvec{\iota }}}_h\Vert _{K}+ \frac{h_{{K}}}{p+1}\big \Vert \nabla {\times }\varvec{v}- \varvec{\Pi }_{\varvec{\mathcal {R\hspace{-0.1em}T}}}^{p}(\nabla {\times }\varvec{v})\big \Vert _{K}, \end{aligned}$$

i.e., (A.5), and the proof is finished. \(\square \)

Lemma 11

(hp data oscillation) Let the assumptions of Lemma 10 be verified. Then

$$\begin{aligned} \sup _{\begin{array}{c} \varvec{\varphi }\in \varvec{H}_0(\textrm{curl}, {K})\\ \Vert \nabla {\times }\varvec{\varphi }\Vert _{K}= 1 \end{array}} (\varvec{\tau }_h - \nabla {\times }\varvec{v}, \varvec{\varphi })_{K}\lesssim \frac{h_{{K}}}{p+1}\big \Vert \nabla {\times }\varvec{v}- \varvec{\Pi }_{\varvec{\mathcal {R\hspace{-0.1em}T}}}^{p}(\nabla {\times }\varvec{v})\big \Vert _{K}, \end{aligned}$$

where the hidden constant only depends on the shape-regularity \(\kappa _{K}:=h_{K}/ \rho _{K}\) of \({K}\).

Proof

Fix \(\varvec{\varphi }\in \varvec{H}_0(\textrm{curl}, {K})\) with \(\Vert \nabla {\times }\varvec{\varphi }\Vert _{K}= 1\). From (2.15), there exists \(\varvec{\psi }\in {\varvec{H}}^1({K}) \cap \varvec{H}_0(\textrm{curl}, {K})\) such that \(\nabla {\times }\varvec{\psi }= \nabla {\times }\varvec{\varphi }\) and

$$\begin{aligned} \Vert \nabla \varvec{\psi }\Vert _{K}\le \Vert \nabla {\times }\varvec{\varphi }\Vert _{K}= 1. \end{aligned}$$

Since \(\varvec{\tau }_h \in \varvec{\mathcal {R\hspace{-0.1em}T}}\hspace{-0.25em}_p({K})\) with \(\nabla {\cdot }\varvec{\tau }_h = 0\), there exists \(\varvec{w}_h \in \varvec{\mathcal {N}}\hspace{-0.2em}_p({K})\) such that \(\nabla {\times }\varvec{w}_h = \varvec{\tau }_h\). Thus, by the Green theorem,

$$\begin{aligned} (\varvec{\tau }_h - \nabla {\times }\varvec{v}, \varvec{\varphi })_{K}&= (\nabla {\times }(\varvec{w}_h - \varvec{v}), \varvec{\varphi })_{K}= (\varvec{w}_h - \varvec{v}, \nabla {\times }\varvec{\varphi })_{K}\\&= (\varvec{w}_h - \varvec{v}, \nabla {\times }\varvec{\psi })_{K}= (\varvec{\tau }_h - \nabla {\times }\varvec{v}, \varvec{\psi })_{K}, \end{aligned}$$

and we conclude by the following Lemma 12. \(\square \)

Lemma 12

(hp data estimate) Let the assumptions of Lemma 10 be verified. Let \(\varvec{\psi }\in {\varvec{H}}^1({K})\). Then

$$\begin{aligned} \vert (\varvec{\tau }_h - \nabla {\times }\varvec{v}, \varvec{\psi })_{K}\vert \lesssim \frac{h_{{K}}}{p+1}\big \Vert \nabla {\times }\varvec{v}- \varvec{\Pi }_{\varvec{\mathcal {R\hspace{-0.1em}T}}}^{p}(\nabla {\times }\varvec{v})\big \Vert _{K}\Vert \nabla \varvec{\psi }\Vert _{K}, \end{aligned}$$

where the hidden constant only depends on the shape-regularity \(\kappa _{K}:=h_{K}/ \rho _{K}\) of \({K}\).

Proof

Let \(q \in H^1_0({K})\) be such that

$$\begin{aligned} (\nabla q, \nabla v)_{K}= (\varvec{\psi }, \nabla v)_{K}\qquad \forall v \in H^1_0({K}). \end{aligned}$$
(A.8)

Defining \(\varvec{\chi }:=\varvec{\psi }- \nabla q\), we have

$$\begin{aligned} \varvec{\chi }\in \varvec{H}(\textrm{div},{K}) \text { with } \nabla {\cdot }\varvec{\chi }= 0. \end{aligned}$$
(A.9)

Moreover, by the Green theorem, the right-hand side in (A.8) can be equivalently written as \((\varvec{\psi }, \nabla v)_{K}= - (\nabla {\cdot }\varvec{\psi }, v)_{K}\). Thus, since \({K}\) is convex, the elliptic regularity shift, see, e.g., [38, Chapter 3], gives \(q \in H^2({K})\) with

$$\begin{aligned} \Vert \nabla (\nabla q)\Vert _{K}\lesssim \Vert \nabla {\cdot }\varvec{\psi }\Vert _{K}\le \Vert \nabla \varvec{\psi }\Vert _{K}, \end{aligned}$$

where \(\lesssim \) means inequality up to a constant only depending on the shape-regularity \(\kappa _{K}\). Thus, in addition to (A.9), there also holds \(\varvec{\chi }\in {\varvec{H}}^1({K})\) with

$$\begin{aligned} \Vert \nabla \varvec{\chi }\Vert _{K}\le \Vert \nabla \varvec{\psi }\Vert _{K}+ \Vert \nabla (\nabla q)\Vert _{K}\lesssim \Vert \nabla \varvec{\psi }\Vert _{K}. \end{aligned}$$
(A.10)

Now, since from (A.1) \(\varvec{\tau }_h - \nabla {\times }\varvec{v}\in \varvec{H}(\textrm{div},{K})\) with \(\nabla {\cdot }(\varvec{\tau }_h - \nabla {\times }\varvec{v}) = 0\), there follows by the Green theorem

$$\begin{aligned} (\varvec{\tau }_h - \nabla {\times }\varvec{v}, \nabla q)_{K}= 0. \end{aligned}$$

Consequently,

$$\begin{aligned} (\varvec{\tau }_h - \nabla {\times }\varvec{v}, \varvec{\psi })_{K}= (\varvec{\tau }_h - \nabla {\times }\varvec{v}, \varvec{\chi })_{K}. \end{aligned}$$
(A.11)

Let respectively

$$\begin{aligned} \varvec{\chi }_h :=\arg \min _{\begin{array}{c} \varvec{w}_h \in \varvec{\mathcal {R\hspace{-0.1em}T}}\hspace{-0.25em}_p({K})\\ \nabla {\cdot }\varvec{w}_h = 0 \end{array}} \Vert \varvec{\chi }- \varvec{w}_h\Vert _{K}\end{aligned}$$

and

$$\begin{aligned} {\tilde{\varvec{\chi }}}_h :=\arg \min _{\varvec{w}_h \in \varvec{\mathcal {R\hspace{-0.1em}T}}\hspace{-0.25em}_p({K})} \Vert \varvec{\chi }- \varvec{w}_h\Vert _{K}\end{aligned}$$

be the constrained and unconstrained Raviart–Thomas approximations of \(\varvec{\chi }\). The Euler–Lagrange conditions of (A.1) allow us to subtract \(\varvec{\chi }_h\) (but not \({\tilde{\varvec{\chi }}}_h\)) in (A.11), so that the Cauchy–Schwarz inequality and, crucially, the p-robust constrained–unconstrained \(\varvec{H}(\textrm{div})\) equivalence of [26, Lemma A1] (note that \(\nabla {\cdot }\varvec{\chi }= 0\)) lead to

$$\begin{aligned} \begin{aligned} (\varvec{\tau }_h - \nabla {\times }\varvec{v}, \varvec{\chi })_{K}&= (\varvec{\tau }_h - \nabla {\times }\varvec{v}, \varvec{\chi }- \varvec{\chi }_h)_{K}\le \Vert \varvec{\tau }_h - \nabla {\times }\varvec{v}\Vert _{K}\Vert \varvec{\chi }- \varvec{\chi }_h\Vert _{K}\\&\lesssim \Vert \varvec{\tau }_h - \nabla {\times }\varvec{v}\Vert _{K}\Vert \varvec{\chi }- {\tilde{\varvec{\chi }}}_h\Vert _{K}. \end{aligned} \end{aligned}$$
(A.12)

Now, since \(\varvec{\mathcal {R\hspace{-0.1em}T}}\hspace{-0.25em}_p({K})\) contains by (2.9) polynomials up to degree p in each component and since the minimization for \({\tilde{\varvec{\chi }}}_h\) is unconstrained, the hp approximation bound (2.17) gives

$$\begin{aligned} \Vert \varvec{\chi }- {\tilde{\varvec{\chi }}}_h\Vert _{K}\lesssim \frac{h_{{K}}}{p+1} \Vert \nabla \varvec{\chi }\Vert _{K}. \end{aligned}$$
(A.13)

Finally, in addition to (A.1), let

$$\begin{aligned} {\tilde{\varvec{\tau }}}_h :=\arg \min _{\varvec{w}_h \in \varvec{\mathcal {R\hspace{-0.1em}T}}\hspace{-0.25em}_p({K})} \Vert \nabla {\times }\varvec{v}- \varvec{w}_h\Vert _{K}, \end{aligned}$$

i.e., from (2.11b), \({\tilde{\varvec{\tau }}}_h = \varvec{\Pi }_{\varvec{\mathcal {R\hspace{-0.1em}T}}}^{p}(\nabla {\times }\varvec{v})\). It follows once again from the p-robust constrained–unconstrained \(\varvec{H}(\textrm{div})\) equivalence of [26, Lemma A1] (note that \(\nabla {\cdot }(\nabla {\times }\varvec{v}) = 0\)) that

$$\begin{aligned} \Vert \varvec{\tau }_h - \nabla {\times }\varvec{v}\Vert _{K}\lesssim \Vert {\tilde{\varvec{\tau }}}_h - \nabla {\times }\varvec{v}\Vert _{K}= \big \Vert \nabla {\times }\varvec{v}- \varvec{\Pi }_{\varvec{\mathcal {R\hspace{-0.1em}T}}}^{p}(\nabla {\times }\varvec{v})\big \Vert _{K}. \end{aligned}$$
(A.14)

Thus, the desired result is a combination of (A.11), (A.12), (A.13), and (A.14) together with (A.10). \(\square \)

Broken regular decomposition in a patch

Let \({\mathcal {T}_{{\varvec{a}}}}\) be a vertex patch for a mesh vertex \({\varvec{a}}\in \mathcal {V}_h\) as in Sect. 2.2. Recall the local spaces (2.6) and (2.7), including the notation \({\gamma _{\mathrm D}}\) standing for the faces sharing the vertex \({\varvec{a}}\) and lying in \({\overline{{\Gamma _{\mathrm D}}}}\). Then there holds:

Lemma 13

(Broken regular decomposition in a patch) Let \(\varvec{\varphi }\in \varvec{H}^\dagger (\textrm{curl}, {\omega _{\varvec{a}}})\). Then there exists \(\varvec{\psi }\in \varvec{H}^\dagger (\textrm{curl}, {\omega _{\varvec{a}}})\) with

$$\begin{aligned} \varvec{\psi }\vert _{K}\in {\varvec{H}}^1({K}) \qquad \forall {K}\in {\mathcal {T}_{{\varvec{a}}}}\end{aligned}$$
(B.1a)

such that

$$\begin{aligned} \nabla {\times }\varvec{\psi }= \nabla {\times }\varvec{\varphi }\end{aligned}$$
(B.1b)

and

$$\begin{aligned} \Vert \varvec{\psi }\Vert _{\omega _{\varvec{a}}}&\lesssim h_{\omega _{\varvec{a}}}\Vert \nabla {\times }\varvec{\varphi }\Vert _{\omega _{\varvec{a}}}, \end{aligned}$$
(B.1c)
$$\begin{aligned} \Bigg \{\sum _{{K}\in {\mathcal {T}_{{\varvec{a}}}}} \Vert \nabla \varvec{\psi }\Vert _{K}^2 \Bigg \}^{1/2}&\lesssim \Vert \nabla {\times }\varvec{\varphi }\Vert _{\omega _{\varvec{a}}}, \end{aligned}$$
(B.1d)

where the hidden constants only depend on the shape-regularity parameter \(\kappa _{{\mathcal {T}_h}}\) of the mesh \({\mathcal {T}_h}\).

Proof

Our proof involves a reference patch. For a fixed shape-regularity parameter \(\kappa _{{\mathcal {T}_h}}\), there is a maximal number of tetrahedra \(\vert {\mathcal {T}_{{\varvec{a}}}}\vert \) in each vertex patch \({\mathcal {T}_{{\varvec{a}}}}\). Therefore, there exists a finite number of ways these tetrahedra can be connected together in the patch, and we may, for each value of \(\kappa _{{\mathcal {T}_h}}\), define a finite set \(R \) of reference patches of unit diameter in such a way that each vertex patch \({\mathcal {T}_{{\varvec{a}}}}\) in the mesh is the image of an element \(\widetilde{\mathcal {T}} \in R \) through a piecewise affine map that preserves connectivity.

Let \(\widetilde{\mathcal {T}} \in R \) be the reference patch associated with the given \({\mathcal {T}_{{\varvec{a}}}}\), and denote by \({\mathcal {A}}: {\widetilde{\omega }} \rightarrow {\omega _{\varvec{a}}}\) the corresponding piecewise affine map, where \({\widetilde{\omega }}\) is the open subdomain corresponding to \(\widetilde{\mathcal {T}}\). Then, for each \({K}\in {\mathcal {T}_{{\varvec{a}}}}\), we denote by

$$\begin{aligned} \big ({\mathcal {A}}^{\textrm{c}} \varvec{v}\big )\vert _{K}:={\mathbb {J}}^{\mathrm{-T}}_{{K}} \varvec{v}\circ {\mathcal {A}}^{-1}, \qquad \big ({\mathcal {A}}^{\textrm{d}} \varvec{v}\big )\vert _{K}:=(\det {\mathbb {J}}_{{K}})^{-1} {\mathbb {J}}_{{K}} \varvec{v}\circ {\mathcal {A}}^{-1}, \end{aligned}$$
(B.2)

the curl- and divergence-preserving Piola mappings associated with \({\mathcal {A}}\), where \({\mathbb {J}}_{{K}}\) is the (constant) Jacobian matrix of \({\mathcal {A}}\) on \({K}\), see e.g. [28, Section 7.2]. It is standard that \({\mathcal {A}}^{\textrm{c}}\) maps \(\varvec{H}^\dagger (\textrm{curl}, {\widetilde{\omega }})\) into \(\varvec{H}^\dagger (\textrm{curl}, {\omega _{\varvec{a}}})\), where \(\varvec{H}^\dagger (\textrm{curl}, {\widetilde{\omega }})\) embeds essential boundary conditions on \({\mathcal {A}}^{-1}({\gamma _{\mathrm D}})\). \({\mathcal {A}}^{\textrm{c}}\) and \({\mathcal {A}}^{\textrm{d}}\) are bijective, and we denote by \(({\mathcal {A}}^{\textrm{c}})^{-1}\) and \(({\mathcal {A}}^{\textrm{d}})^{-1}\) their inverses. Finally, we have the commuting property \(\nabla {\times }({\mathcal {A}}^{\textrm{c}} \cdot ) = {\mathcal {A}}^{\textrm{d}}(\nabla {\times }\cdot )\).

Consider \(\varvec{\varphi }\in \varvec{H}^\dagger (\textrm{curl}, {\omega _{\varvec{a}}})\) and let \({\widetilde{\varvec{\varphi }}} :=({\mathcal {A}}^{\textrm{c}})^{-1}(\varvec{\varphi }) \in \varvec{H}^\dagger (\textrm{curl}, {\widetilde{\omega }})\). We can employ (2.15) (where we take \(\varvec{w}\) in \({\varvec{H}}^1_{0,\textrm{D}}(\omega )\)) on the reference patch \({\widetilde{\omega }}\), where \(C_{\textrm{L},{\widetilde{\omega }}}\) is now a generic constant only depending on \(\kappa _{{\mathcal {T}_h}}\) due to the above discussion. Therefore, there exists \({\widetilde{\varvec{\psi }}} \in {\varvec{H}}^1({\widetilde{\omega }})\), with \({\widetilde{\varvec{\psi }}} = 0\) on \({\mathcal {A}}^{-1}({\gamma _{\mathrm D}})\) if \({\gamma _{\mathrm D}}\) includes at least one face and of componentwise mean value zero on \({\widetilde{\omega }}\) otherwise such that

$$\begin{aligned} \nabla {\times }{\widetilde{\varvec{\psi }}} = \nabla {\times }{\widetilde{\varvec{\varphi }}} \quad \text { and } \quad \quad \Vert \nabla {\widetilde{\varvec{\psi }}}\Vert _{{\widetilde{\omega }}} \le C_{\textrm{L},{\widetilde{\omega }}} \Vert \nabla {\times }{\widetilde{\varvec{\varphi }}}\Vert _{{\widetilde{\omega }}}. \end{aligned}$$

We also note that we have

$$\begin{aligned} \Vert {\widetilde{\varvec{\psi }}}\Vert _{{\widetilde{\omega }}} \lesssim \Vert \nabla {\widetilde{\varvec{\psi }}}\Vert _{{\widetilde{\omega }}} \lesssim \Vert \nabla {\times }{\widetilde{\varvec{\psi }}}\Vert _{{\widetilde{\omega }}} = \Vert \nabla {\times }{\widetilde{\varvec{\varphi }}}\Vert _{{\widetilde{\omega }}} \end{aligned}$$
(B.3)

by the Poincaré–Friedrichs inequality (2.16), since \({\widetilde{\omega }}\) is of unit diameter.

We then introduce \(\varvec{\psi }:={\mathcal {A}}^{\textrm{c}}({\widetilde{\varvec{\psi }}}) \in \varvec{H}^\dagger (\textrm{curl}, {\omega _{\varvec{a}}})\) and claim that it satisfies all the requirements in (B.1). Indeed, elementwise, the Piola mapping \({\mathcal {A}}^{\textrm{c}}\) amounts to an affine change of coordinates and a multiplication by a constant matrix. Therefore, it is clear that it preserves smoothness elementwise and (B.1a) follows from \({\widetilde{\varvec{\psi }}} \in {\varvec{H}}^1({\widetilde{\omega }})\). Moreover, (B.1b) holds true since

$$\begin{aligned} \nabla {\times }\varvec{\psi }&= \nabla {\times }{\mathcal {A}}^{\textrm{c}}({\widetilde{\varvec{\psi }}}) = {\mathcal {A}}^{\textrm{d}}(\nabla {\times }{\widetilde{\varvec{\psi }}}) = {\mathcal {A}}^{\textrm{d}}(\nabla {\times }{\widetilde{\varvec{\varphi }}}) = {\mathcal {A}}^{\textrm{d}}(\nabla {\times }(({\mathcal {A}}^{\textrm{c}})^{-1}(\varvec{\varphi })))\\&={\mathcal {A}}^{\textrm{d}}(({\mathcal {A}}^{\textrm{d}})^{-1}(\nabla {\times }\varvec{\varphi })) =\nabla {\times }\varvec{\varphi }. \end{aligned}$$

The estimates (B.1c) and (B.1d) then follow from (B.3) and the usual scaling arguments due to the structure (B.2) of the Piola mappings. Indeed, for each \({K}\in {\mathcal {T}_{{\varvec{a}}}}\), we have, on the one hand,

$$\begin{aligned} \Vert \varvec{\psi }\Vert _{{K}}^2&= \Vert {\mathcal {A}}^{\textrm{c}}({\widetilde{\varvec{\psi }}})\Vert _{{K}}^2 = \Vert {\mathbb {J}}_{K}^{-T}{\widetilde{\varvec{\psi }}} \circ {\mathcal {A}}^{-1}\Vert _{{K}}^2\\&\lesssim h_{K}^{-2} \Vert {\widetilde{\varvec{\psi }}} \circ {\mathcal {A}}^{-1}\Vert _{K}^2 \lesssim h_{K}^{-2} \vert {K}\vert \Vert {\widetilde{\varvec{\psi }}}\Vert _{{\widetilde{{K}}}}^2 \lesssim h_{K}\Vert {\widetilde{\varvec{\psi }}}\Vert _{{\widetilde{{K}}}}^2, \end{aligned}$$

where \({\widetilde{{K}}} = {\mathcal {A}}^{-1}({K})\). On the other hand, it is true that

$$\begin{aligned} \begin{aligned} \Vert \nabla {\times }{\widetilde{\varvec{\psi }}}\Vert _{{\widetilde{{K}}}}^2&= \Vert ({\mathcal {A}}^{\textrm{d}})^{-1}(\nabla {\times }\varvec{\psi })\Vert _{{\widetilde{{K}}}}^2 = \Vert (\det {\mathbb {J}}_{{K}}) {\mathbb {J}}_{K}^{-1} (\nabla {\times }\varvec{\psi }) \circ {\mathcal {A}}\Vert _{{\widetilde{{K}}}}^2\\&\lesssim (\det {\mathbb {J}}_{K})^2 h_{K}^{-2} \Vert (\nabla {\times }\varvec{\psi }) \circ {\mathcal {A}}\Vert _{{\widetilde{{K}}}}^2 \lesssim \frac{(\det {\mathbb {J}}_{K})^2 h_{K}^{-2}}{\vert {K}\vert } \Vert \nabla {\times }\varvec{\psi }\Vert _{{K}}^2\\&\lesssim h_{K}\Vert \nabla {\times }\varvec{\psi }\Vert _{K}^2, \end{aligned} \end{aligned}$$
(B.4)

so that (B.1c) follows from (B.3).

Finally, we establish (B.1d) by directly differentiating in (B.2). Specifically, we have

$$\begin{aligned} \partial _\ell \varvec{\psi }={\mathbb {J}}_{K}^{-T} \partial _\ell ({\widetilde{\varvec{\psi }}} \circ {\mathcal {A}}^{-1}) ={\mathbb {J}}_{K}^{-T} \partial _\ell ({\mathcal {A}}^{-1}_j) (\partial _j{\widetilde{\varvec{\psi }}} \circ {\mathcal {A}}^{-1}), \end{aligned}$$

so that

$$\begin{aligned} \vert \partial _\ell \varvec{\psi }\vert \le h_{K}^{-2} \vert \nabla {\widetilde{\varvec{\psi }}} \circ {\mathcal {A}}^{-1}\vert \end{aligned}$$

and

$$\begin{aligned} \Vert \partial _\ell \varvec{\psi }\Vert _{K}^2 \lesssim h^{-4}_{K}\vert {K}\vert \Vert \nabla {\widetilde{\varvec{\psi }}}\Vert _{{\widetilde{{K}}}}^2. \end{aligned}$$

We therefore arrive at

$$\begin{aligned} \sum _{{K}\in {\mathcal {T}_{{\varvec{a}}}}} \Vert \nabla \varvec{\psi }\Vert _{{K}}^2 \lesssim h_{{\omega _{\varvec{a}}}}^{-1}\Vert \nabla {\widetilde{\varvec{\psi }}}\Vert _{{\widetilde{\omega }}}^2, \end{aligned}$$

and the conclusion follows from (B.3) and (B.4). \(\square \)

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Chaumont-Frelet, T., Vohralík, M. A stable local commuting projector and optimal hp approximation estimates in \(\varvec{H}(\textrm{curl})\). Numer. Math. 156, 2293–2342 (2024). https://doi.org/10.1007/s00211-024-01431-w

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