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1 Introduction↩︎

We consider conforming finite element discretizations of the Stokes complex with mixed boundary conditions. Let \(\Omega \subset \mathbb{R}^3\) be a polyhedral domain whose boundary is partitioned into suitably regular subsets \(\Gamma_0\) and \(\Gamma_1\), and define the following spaces: \[\begin{alignat}{2} \label{eq:hkd-first-def} H^k_{\Gamma_0}(\Omega) &:= \{ \phi \in H^k(\Omega) : D^{\alpha} \phi|_{\Gamma_0} = 0, \;\forall |\alpha| \leq k-1 \}, \qquad & &k \in \mathbb{N}, \\ H^1_{\Gamma_0}(\Omega; \mathop{\mathrm{curl}}) &:= \{ v \in H^1_{\Gamma_0}(\Omega)^3 : \mathop{\mathrm{curl}}v \in H^1_{\Gamma_0}(\Omega)^3 \}, \qquad & & \end{alignat}\tag{1}\] where we may drop the subscript to denote the case \(|\Gamma_0| = 0\) and use the subscript “0” to denote \(|\Gamma_1| = 0\). These spaces fit into the so-called Stokes complex: \[\label{eq:stokes-complex-bcs} \begin{tikzcd} 0 \arrow[r] & H_{\Gamma_0}^2(\Omega) \arrow[r, "\mathop{\mathrm{grad}}"] & H_{\Gamma_0}^1(\Omega; \mathop{\mathrm{curl}}) \arrow[r, "\mathop{\mathrm{curl}}"] & H_{\Gamma_0}^1(\Omega)^3 \arrow[r, "\undefined"] & L^2(\Omega) \arrow[r] & 0. \end{tikzcd}\tag{2}\] In particular, that 2 is a complex means that the composition of any two operators is zero (e.g. \(\mathop{\mathrm{curl}}\mathop{\mathrm{grad}}= 0\)) and that the image of each operator lies in the succeeding space (e.g. \(\mathop{\mathrm{curl}}H_{\Gamma_0}^1(\mathop{\mathrm{curl}};\Omega) \subset H_{\Gamma_0}^1(\Omega)^3\)). Problems involving the spaces in the complex 2 arise in a variety of applications. Many fourth-order problems involve the space \(H^2(\Omega)\), such as the separation of binary alloys [1] or displacement formulations of strain gradient theory [2] to name a few. The space \(H^1(\mathop{\mathrm{curl}}; \Omega)\) appears in displacement formulations of couple stress theory [3]. The final two spaces most famously appear in incompressible flow.

A conforming finite element discretization or subcomplex of 2 is another complex \[\label{eq:stokes-complex-bcs-intro-discrete} \begin{tikzcd} 0 \arrow[r] & V^{0, h}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{grad}}"] & V^{1, h}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{curl}}"] & V^{2, h}_{\Gamma_0} \arrow[r, "\undefined"] & V^{3, h}_{\Gamma_0} \arrow[r] & 0, \end{tikzcd}\tag{3}\] where each space \(V^{k, h}_{\Gamma_0}\) is a conforming finite element subspace of the corresponding space in 2 . Choosing finite element discretizations from an underlying complex 3 can offer many benefits. In velocity-pressure formulations of incompressible flow, taking \(V^{2, h}_{\Gamma_0}\) and \(V^{3, h}_{\Gamma_0}\) from 3 lead to mass conserving and “pressure-robust" discretizations [4] and schemes with uniform stability properties for time-dependent or singularly perturbed flows [5], [6]. All four spaces from 3 can also be used to construct an energy and enstrophy stable scheme for the incompressible Navier-Stokes equations [7]. The complex 3 also encodes information that is useful for preconditioning parameter-dependent problems. For example, consider the weighted bilinear forms for \(\alpha, \beta \in \mathbb{R}_+\): \[\label{eq:riesz-maps} \begin{alignat}{2} &(\mathop{\mathrm{grad}}u, \alpha \mathop{\mathrm{grad}}v)_{L^2(\Omega)} + (\mathop{\mathrm{grad}}\mathop{\mathrm{curl}}u, \beta \mathop{\mathrm{grad}}\mathop{\mathrm{curl}}v)_{L^2(\Omega)} \qquad & &\forall u, v \in V_{\Gamma_0}^{1, h}, \\ &(\mathop{\mathrm{grad}}u, \alpha \mathop{\mathrm{grad}}v)_{L^2(\Omega)} + (\undefined u, \beta \undefined v)_{L^2(\Omega)} \qquad & &\forall u, v \in V_{\Gamma_0}^{2, h}, \end{alignat}\tag{4}\] which arise in couple stress theory [3] and augmented Lagrangian preconditioning for incompressible flow [8], [9]. Constructing preconditioners that are robust in the parameters \(\alpha\) and \(\beta\) typically requires knowledge of the kernel of \(\mathop{\mathrm{curl}}: V_{\Gamma_0}^{1, h} \to V_{\Gamma_0}^{2, h}\) and \(\undefined: V_{\Gamma_0}^{2, h} \to V_{\Gamma_0}^{3, h}\) [10], [11], which is precisely encoded in the algebraic structure of the complex 3 .

More specifically, the cohomology of a complex consists of the kernel of an operator modulo the range of the previous operator. For example, the first and second cohomology of the discrete complex 3 are \[\begin{align} \mathfrak{H}^{1, h}_{\Gamma_0} := \frac{ \ker(\mathop{\mathrm{curl}}: V_{\Gamma_0}^{1, h} \to V_{\Gamma_0}^{2, h}) }{ \mathop{\mathrm{im}}(\mathop{\mathrm{grad}}: V_{\Gamma_0}^{0, h} \to V_{\Gamma_0}^{1, h}) } \quad \text{and} \quad \mathfrak{H}^{2, h}_{\Gamma_0} := \frac{ \ker(\undefined: V_{\Gamma_0}^{2, h} \to V_{\Gamma_0}^{3, h}) }{ \mathop{\mathrm{im}}(\mathop{\mathrm{curl}}: V_{\Gamma_0}^{1, h} \to V_{\Gamma_0}^{2, h}) }. \end{align}\] Characterizing the cohomology is thus crucial for constructing robust preconditioners for 4 . The cohomology also plays a key role in finite element exterior calculus (FEEC), including the well-posedness of the Hodge-Laplace problems associated with 3 (see e.g. [12]) which includes incompressible flow as well as other mixed problems involving the spaces and operators in 3 . More generally, the properties of “simpler" complexes like 3 2 , including cohomology, are critical in understanding properties of more complicated sequences via the Bernstein–Gelfand–Gelfand construction [13]–[15]. For a recent review on the importance of cohomology in a variety of applications, we refer to [16]. Nevertheless, the cohomology of any conforming finite element subcomplex of 2 on nontrivial domains and/or with mixed boundary conditions seems to not have been addressed in the literature.

Our first main result shows that if the finite element spaces in 3 are chosen to be the Alfeld-split macroelements in [17], then the cohomology of the discrete complex 3 is isomorphic to the cohomology of the continuous complex 2 . This result extends [17] to the case of nontrivial domains and to the case of mixed boundary conditions. The main technique is applying the recent framework of [18] and modifying the final steps to account for boundary conditions. The second set of results show that, in a certain sense, the Alfeld-split macroelement Stokes complex analog of Whitney forms [19]–[21] and lowest-order complete polynomial complexes can be constructed to form “minimal” subcomplexes of 3 with the same cohomology structure. The final two spaces in these subcomplexes consist of low-order \(H^1(\Omega)^3\)-conforming element from [22] and piecewise constants. For the first two spaces in the subcomplexes, we obtain novel \(H^2(\Omega)\)-conforming and \(H^1(\mathop{\mathrm{curl}}; \Omega)\)-conforming finite elements whose dimension is, in a certain sense, “minimal”.

2 Summary of main results and outline↩︎

We assume that the domain \(\Omega \subset \mathbb{R}^3\) and boundary partition \(\partial \Omega = \bar{\Gamma}_0 \cup \bar{\Gamma}_N\) satisfy the assumptions in [23]: \(\Omega\) is a bounded strongly Lipschitz domain and \(\Gamma_0\) and \(\Gamma_1\) are strongly Lipschitz subsets, so that \(\Omega\) and \(\Gamma_0\) form a strong Lipschitz pair in the sense of [24]. We further assume that \(\partial \Omega\) and \(\Gamma_0\) are both the union of a finite number of polygons.

We rewrite the Stokes complex 2 in standard FEEC notation as follows: \[\label{eq:stokes-complex-bcs-feec} \begin{tikzcd} 0 \arrow[r] & V_{\Gamma_0}^0 \arrow[r, "\mathop{\mathrm{d}}^0"] & V_{\Gamma_0}^1 \arrow[r, "\mathop{\mathrm{d}}^1"] & V_{\Gamma_0}^2 \arrow[r, "\mathop{\mathrm{d}}^2"] & V_{\Gamma_0}^3 \arrow[r] & 0, \end{tikzcd}\tag{5}\] where \[\begin{align} {4} V_{\Gamma_0}^0 &:= H^2_{\Gamma_0}(\Omega), \qquad & V_{\Gamma_0}^1 &:= H^1_{\Gamma_0}(\mathop{\mathrm{curl}}; \Omega), \qquad & V_{\Gamma_0}^2 &:= H^1_{\Gamma_0}(\Omega)^3, & \qquad V_{\Gamma_0}^3 &:= L^2(\Omega), \\ \mathop{\mathrm{d}}^0 &:= \mathop{\mathrm{grad}}, \qquad & \mathop{\mathrm{d}}^1 &:= \mathop{\mathrm{curl}}, \qquad & \mathop{\mathrm{d}}^2 &:= \undefined, \qquad & \mathop{\mathrm{d}}^3 &:= 0. \end{align}\] As with the Sobolev spaces, we may drop the subscript “\(D\)” if \(|\Gamma_0| = 0\) or use the subscript “0” if \(|\Gamma_1| = 0\). The Stokes complex 5 may be viewed as a smoother subcomplex of the standard de Rham complex: \[\label{eq:de-rham-complex-bcs-feec} \begin{tikzcd} 0 \arrow[r] & W_{\Gamma_0}^0 \arrow[r, "\mathop{\mathrm{d}}^0"] & W_{\Gamma_0}^1 \arrow[r, "\mathop{\mathrm{d}}^1"] & W_{\Gamma_0}^2 \arrow[r, "\mathop{\mathrm{d}}^2"] & W_{\Gamma_0}^3 \arrow[r] & 0, \end{tikzcd}\tag{6}\] where \[\begin{align} {4} W_{\Gamma_0}^0 &:= H^1_{\Gamma_0}(\Omega), \quad & W_{\Gamma_0}^1 &:= H_{\Gamma_0}(\mathop{\mathrm{curl}}; \Omega), \quad & W_{\Gamma_0}^2 &:= H_{\Gamma_0}(\undefined; \Omega), \quad & W_{\Gamma_0}^3 &:= L^2(\Omega). \end{align}\] Here, \(H_{\Gamma_0}(\mathop{\mathrm{curl}}; \Omega)\), respectively \(H_{\Gamma_0}(\undefined; \Omega)\), is the space of \(L^2(\Omega)^3\) vector fields whose \(\mathop{\mathrm{curl}}\), respectively \(\undefined\), is also square integrable and whose tangential, respectively normal, trace vanishes on \(\Gamma_0\).

The \(k\)th-cohomology, or harmonic form, of the Stokes complex is defined by \[\begin{align} \label{eq:harmonic-forms-bcs} \mathfrak{H}_{\Gamma_0}^k := \frac{ \ker(\mathop{\mathrm{d}}^k : V_{\Gamma_0}^k \to V_{\Gamma_0}^{k+1}) }{ \mathop{\mathrm{im}}(\mathop{\mathrm{d}}^{k-1} : V_{\Gamma_0}^{k-1} \to V_{\Gamma_0}^k) }, \qquad k \in 0:3, \end{align}\tag{7}\] where we adopt the convention that any space or operator with form index \(k < 0\) or \(k > 3\) is the trivial space or operator, and \(m:n := \{ m, m+1,\dots, n-1, n\}\) for nonnegative integers \(m, n\). Theorem 5.12⁴ of [23] shows that the harmonic forms of the Stokes complex and the de Rham complex are finite and isomorphic: \[\begin{align} \label{eq:harmonic-forms-same-dim-diff-smoothness} \dim \mathfrak{H}_{\Gamma_0}^k = \dim \frac{ \ker(\mathop{\mathrm{d}}^k : W_{\Gamma_0}^k \to W_{\Gamma_0}^{k+1}) }{ \mathop{\mathrm{im}}(\mathop{\mathrm{d}}^{k-1} : W_{\Gamma_0}^{k-1} \to W_{\Gamma_0}^k) } \qquad \forall k \in 0:3. \end{align}\tag{8}\]

The dimensions of the harmonic forms correspond to relative Betti numbers. For \(k \in 0:3\), let \(b_k(\Omega, \Gamma_0)\) be the \(k\)-th relative Betti number defined as the dimension of the \(k\)-th singular homology group of \(\Omega\) relative to \(\Gamma_0\) (see e.g. [26]). Applying [27] to 8 then shows that \[\begin{align} \label{eq:harmonic-forms-dim} \dim \mathfrak{H}_{\Gamma_0}^k = b_k(\Omega, \Gamma_0) \qquad \forall k \in 0:3. \end{align}\tag{9}\] Note that \(b_k(\Omega, \emptyset)\) corresponds to the usual Betti number of \(\Omega\); e.g. \(b_0(\Omega)\) is the number of connected components of \(\Omega\). We also have the duality relation \(b_{k}(\Omega, \Gamma_0) = b_{3-k}(\Omega, \Gamma_1)\) for \(k \in 0:3\) thanks to [27]. Combining this with [27], we have \[\begin{align} b_0(\Omega, \Gamma_0) = \begin{cases} 1 & \text{if } |\Gamma_0| = 0, \\ 0 & \text{otherwise}, \end{cases} \quad \text{and} \quad b_3(\Omega, \Gamma_0) = \begin{cases} 1 & \text{if } |\Gamma_1| = 0, \\ 0 & \text{otherwise}. \end{cases} \end{align}\] Unfortunately, \(b_{1}(\Omega, \Gamma_0)\) and \(b_2(\Omega, \Gamma_0)\) do not have such simple expressions.

2.1 Cohomology of full polynomials spaces on Alfeld splits↩︎

We first recall a finite element Stokes complex on Alfeld-split meshes from [17]. For a collection of tetrahedra \(\mathcal{S}\) forming a conforming mesh of an open domain \(\mathcal{O}\), we define the following spaces of polynomials for \(p \in \mathbb{N}_0\): \[\begin{align} \mathop{\mathrm{DG}}^p(\mathcal{S}) &:= \{ v \in L^2(\mathcal{O}) : v|_{K} \in \mathcal{P}_p(K) \;\forall K \in \mathcal{S} \} \;\text{and} \; \mathop{\mathrm{CG}}^p(\mathcal{S}) := \mathop{\mathrm{DG}}^p(\mathcal{S}) \cap C(\bar{\mathcal{O}}). \end{align}\] Given a tetrahedron \(K\), let \(K_A\) denote the collection of four tetrahedra in the Alfeld split of \(K\) formed by connecting the four vertices of \(K\) to the barycenter of \(K\), and for \(p \geq 3\), define the following local Alfeld-split macroelement spaces: \[\label{eq:local-full-alfeld} \begin{alignat}{2} V^{0, h}(K_A) &:= \{v \in \mathop{\mathrm{CG}}^{p+2}(K_A) : \mathop{\mathrm{grad}}v \in \mathop{\mathrm{CG}}^{p+1}(K_A)^3\},\\ V^{1, h}(K_A) &:= \{v \in \mathop{\mathrm{CG}}^{p+1}(K_A)^3 : \mathop{\mathrm{curl}}v \in \mathop{\mathrm{CG}}^{p}(K_A)^3\},\\ V^{2, h}(K_A) &:= \mathop{\mathrm{CG}}^p(K_A)^3, \\ V^{3, h}(K_A) &:= \mathop{\mathrm{DG}}^{p-1}(K_A). \end{alignat}\tag{10}\] These local spaces form a complex \[\label{eq:stokes-complex-local-full-alfeld} \begin{tikzcd} \mathbb{R} \arrow[r, "\subset"] & V^{0, h}(K_A) \arrow[r, "\mathop{\mathrm{d}}^0"] & V^{1, h}(K_A) \arrow[r, "\mathop{\mathrm{d}}^1"] & V^{2, h}(K_A) \arrow[r, "\mathop{\mathrm{d}}^2"] & V^{3, h}(K_A) \arrow[r] & 0, \end{tikzcd}\tag{11}\] and the complex is exact for all \(p \geq 3\) [17], which means that the range of each operator is the kernel of the succeeding operator; i.e., all harmonic forms are trivial.

The local spaces 10 may be extended to a multi-element mesh in a typical finite element fashion provided that extra vertex smoothness is imposed on first two spaces. In particular, let \(\mathcal{T}\) be a conforming, shape regular simplicial mesh of \(\Omega\) such that there exists a submesh of triangles \(\mathcal{U}\) that forms a conforming mesh of \(\Gamma_0\). Additionally, we define the global ambient spaces by \[\begin{align} \label{eq:global-ambient-spaces} \mathbb{X}^0 := \mathbb{X}^3 := \mathbb{R} \quad \text{and} \quad \mathbb{X}^1 := \mathbb{X}^2 := \mathbb{R}^3. \end{align}\tag{12}\] \[\label{eq:global-full-alfeld} Then, for k \in 0:1, the global spaces are given by \begin{multline} V^{k, h} := \{ v \in C(\Omega) \otimes \mathbb{X}^k : \mathop{\mathrm{d}}^k v \in C(\Omega) \otimes \mathbb{X}^{k+1}, \; \text{v is C^{2-k} at \Delta_0(\mathcal{T})} \\ \text{ and } v|_{K} \in V^{k, h}(K_A) \;\forall K \in \mathcal{T}\}, \end{multline} where \Delta_{\ell}(\mathcal{S}) denotes the set of all \ell-dimensional subsimplices of a collection of d-dimensional simplices \mathcal{S} with \ell \in 0:d (e.g. \mathcal{T}= \Delta_3(\mathcal{T})). For k \in 2:3, the global spaces are simply \begin{align} V^{k, h} := \{ v \in C^{2-k}(\Omega) \otimes \mathbb{X}^{k} : v|_{K} \in V^{k, h}(K_A) \;\forall K \in \mathcal{T}\}, \end{align} where C^{-1}(\Omega) := L^2(\Omega).\tag{13}\] The corresponding spaces incorporating the boundary conditions are \[\begin{align} \label{eq:global-full-alfeld-bcs} V^{k, h}_{\Gamma_0} := V^{k, h} \cap V_{\Gamma_0}^k, \end{align}\tag{14}\] which form a conforming subcomplex of 5 : \[\label{eq:stokes-complex-full-alfeld-bcs} \begin{tikzcd} 0 \arrow[r] & V^{0, h}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{d}}^0"] & V^{1, h}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{d}}^1"] & V^{2, h}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{d}}^2"] & V^{3, h}_{\Gamma_0} \arrow[r] & 0. \end{tikzcd}\tag{15}\]

Our first main result shows that the discrete harmonic forms \[\begin{align} \label{eq:harmonic-forms-full-alfeld-bcs} \mathfrak{H}^{k, h}_{\Gamma_0} := \frac{\ker(\mathop{\mathrm{d}}^k : V_{\Gamma_0}^{k, h} \to V_{\Gamma_0}^{k+1, h})}{ \mathop{\mathrm{im}}(\mathop{\mathrm{d}}^{k-1} : V_{\Gamma_0}^{k-1, h} \to V_{\Gamma_0}^{k, h})}, \qquad k \in 0:3, \end{align}\tag{16}\] have the same dimension as the continuous harmonic forms in 7 .

The proof of 1, appearing below in 3.7, applies the general framework of [18] and extends it to the case of mixed boundary conditions for the particular discrete complex 15 . We note that 1 is the extension of [17] to nontrivial domains with mixed boundary conditions.

2.2 “Minimal” conforming complexes↩︎

We now seek a “minimal” conforming finite element subcomplex of the Stokes complex 5 \[\label{eq:stokes-complex-reduced-alfeld-bcs} \begin{tikzcd} 0 \arrow[r] & \tilde{V}^{0, h}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{d}}^0"] & \tilde{V}^{1, h}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{d}}^1"] & \tilde{V}^{2, h}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{d}}^2"] & \tilde{V}^{3, h}_{\Gamma_0} \arrow[r] & 0 \end{tikzcd}\tag{17}\] whose cohomology is isomorphic to that of 5 . We restrict ourselves to subcomplexes of 17 to stay within the setting of Alfeld-split meshes; see [28] for spaces on other types of splits and [28] for minimal two-dimensional elements. One of the key properties of the full polynomial spaces 13 that is crucial in the proof of 1 is that one may choose degrees of freedom for the spaces \(V^{k, h}\) to include the following linear functionals: \[\label{eq:whitney-dofs-currents} \begin{alignat}{2} C(\bar{\Omega}) \otimes \mathbb{X}^0 \ni v &\mapsto \mathcal{I}_{z}^0(v) := v(z) \qquad & &\forall z \in \Delta_0(\mathcal{T}), \\ C(\bar{\Omega}) \otimes \mathbb{X}^1 \ni v &\mapsto \mathcal{I}_{e}^1(v) := \int_{e} v \cdot \hat{t}_e \,\mathrm{d}{s} \qquad & &\forall e \in \Delta_1(\mathcal{T}), \\ C(\bar{\Omega}) \otimes \mathbb{X}^2 \ni v &\mapsto \mathcal{I}_{f}^2(v) := \int_{f} v \cdot \hat{n}_f \,\mathrm{d}{s} \qquad & &\forall f \in \Delta_2(\mathcal{T}), \\ C(\bar{\Omega}) \otimes \mathbb{X}^3 \ni v &\mapsto \mathcal{I}_{K}^3(v) := \int_{K} v \,\mathrm{d}{x} \qquad & &\forall K \in \Delta_3(\mathcal{T}), \end{alignat}\tag{18}\] where \(\hat{n}_f\) and \(\hat{t}_e\) are unit normal and tangent vectors with a fixed global orientation. Note that 18 are simply the canonical set of degrees of freedom for the Whitney forms [19]–[21], the lowest-order conforming discretization of the de Rham complex 6 .

Our starting point is then to choose \[\begin{align} \label{eq:min-l2-space} \tilde{V}^{3, h}(K_A) := \mathop{\mathrm{DG}}^0(K), \quad \tilde{V}^{3, h} := \mathop{\mathrm{DG}}^0(\mathcal{T}), \quad \text{and} \quad \tilde{V}^{3, h}_{\Gamma_0} := \tilde{V}^{3, h} \cap V^3_{\Gamma_0}, \end{align}\tag{19}\] the Whitney forms of index 3. Moving one space to the left in the complex 17 , we seek an \(H^1(\Omega)^3\)-conforming finite element space whose divergence lies in \(\mathop{\mathrm{DG}}^0(\mathcal{T})\) and is large enough to be equipped with the degrees of freedom in 18 . The Guzmán-Neilan element [22] exactly meets these requirements and is defined as follows. For \(f \in \Delta_2(K)\), let \(b_f \in \mathcal{P}_3(K)\) be the face bubble function satisfying \(b_f|_{\partial K \setminus f} = 0\), normalized so that \(\int_f b_f \,\mathrm{d}{s} = |f|\). Henceforth, \(b_f\) is also used to denote the restriction \(b_f|_f\). Moreover, for \(K \in \mathcal{T}\), let \(S_{K} : \mathop{\mathrm{CG}}^{3}(K_A)^3 \to \mathop{\mathrm{CG}}^{3}(K_A)^3\) be any fixed linear operator satisfying the following for all \(v \in \mathop{\mathrm{CG}}^{3}(K_A)^3\): \[\begin{align} \label{eq:generic-gn-extension-operator} S_{K} v|_{\partial K} = v|_{\partial K}, \quad \undefined S_{K} v \in \mathcal{P}_0(K), \;\;\text{and} \;\; | S_K v |_{H^{\ell}(K)} \leq C_{S} |v |_{H^{\ell}(K)}, \;\ell \in 0:1, \end{align}\tag{20}\] where \(C_{S} > 0\) is independent of \(K\). Then, the local and global Guzmán-Neilan spaces are give by \[\begin{align} \tag{21} \mathop{\mathrm{GN}}(K_A) &:= \mathcal{P}_1(K)^3 \oplus \mathop{\mathrm{span}}\{ S_{K}( b_f \hat{n}_f) : f \in \Delta_2(K) \} \qquad \forall K \in \mathcal{T}, \\ \tag{22} \mathop{\mathrm{GN}}(\mathcal{T}) &:= \{v \in C(\Omega)^3 : v|_{K} \in \mathop{\mathrm{GN}}(K_A) \;\forall K \in \mathcal{T}\}. \end{align}\] In particular, \(\mathop{\mathrm{CG}}^1(\mathcal{T})^3 \subset \mathop{\mathrm{GN}}(\mathcal{T}) \subset V^{1, h}\), \(\undefined\mathop{\mathrm{GN}}(\mathcal{T}) \subseteq \mathop{\mathrm{DG}}^0(\mathcal{T})\), and a global set of degrees of freedom are given by [22]: \[\begin{align} \label{eq:gn-dofs} v(z) \qquad \forall z \in \Delta_0(\mathcal{T}) \quad \text{and} \quad \int_f v \cdot n_f \,\mathrm{d}{s} \qquad \forall f \in \Delta_2(\mathcal{T}). \end{align}\tag{23}\]

The choice \[\begin{align} \label{eq:min-h1-space} \tilde{V}^{2, h}(K_A) := \mathop{\mathrm{GN}}(K_A), \quad \tilde{V}^{2, h} := \mathop{\mathrm{GN}}(\mathcal{T}), \quad \text{and} \quad \tilde{V}^{2, h}_{\Gamma_0} := \tilde{V}^{2, h} \cap V^2_{\Gamma_0} \end{align}\tag{24}\] is then “minimal", as one typically requires vertex degrees of freedom as in 23 to ensure continuity. The construction of the remaining spaces \(\tilde{V}_{\Gamma_0}^{0, h}\) and \(\tilde{V}_{\Gamma_0}^{1, h}\) will be detailed in 4 below. The main result is the following.

The proof of 3 appears in 5.1 below. We will also show below in 6 [ lem:h1curl-local-unisolvence]{reference-type=“ref” reference=” lem:h1curl-local-unisolvence”} that \(\tilde{V}^{0, h}\) may be equipped with the degrees of freedom \[\begin{align} \label{eq:walkington-dofs} D^{\alpha} v(z) \qquad \forall |\alpha| \leq 2, \;\forall z \in \Delta_0(\mathcal{T}), \end{align}\tag{27}\] while \(\tilde{V}^{1, h}\) may be equipped with \[\tag{28} \begin{alignat}{2} \tag{29} &D^{\alpha} v(z) \qquad & & \forall|\alpha| \leq 1, \;\forall z \in \Delta_0(\mathcal{T}), \\ \tag{30} &\int_{e} v \cdot \hat{t}_e \,\mathrm{d}{s} \qquad & & \forall e \in \Delta_1(\mathcal{T}). \end{alignat}\] In view of the \(C^{2-k}\)-continuity imposed at the mesh vertices of elements in \(V^{k, h}\), \(k \in 0:1\), we see that \(\tilde{V}^{k, h}\) are then “minimal” subspaces of \(V^{k, h}\) whose degrees of freedom can be chosen to include 18 and satisfy \(\mathcal{P}_{3-k}(K) \otimes \mathbb{X}^k \subset \tilde{V}^{k, h}(K_A)\).

In view of this property, the local complex 25 bears resemblance to the complex of complete polynomials used to discretize the de Rham complex 6 . Locally, the complete polynomial complex reads for \(K \in \mathcal{T}\): \[\label{eq:de-rham-complex-complete-poly-element} \begin{tikzcd} \mathbb{R} \arrow[r, "\subset"] & \mathcal{P}_{3}(K) \arrow[r, "\mathop{\mathrm{d}}^0"] & \mathcal{P}_{2}(K)^3 \arrow[r, "\mathop{\mathrm{d}}^1"] & \mathcal{P}_{1}(K)^3 \arrow[r, "\mathop{\mathrm{d}}^2"] & \mathcal{P}_{0}(K) \arrow[r] & 0. \end{tikzcd}\tag{31}\] In fact, 3 shows that 31 is a subcomplex of 25 . The additional complexities of the spaces in 25 only arise due to the additional global continuity imposed by being conforming subspaces of \(V_{\Gamma_0}^{k}\) rather than \(W_{\Gamma_0}^{k}\). Thus, the global complex 17 may be seen as the Alfeld-split macroelement Stokes complex analog of the lowest-order discretization of the de Rham complex with complete polynomials.

Of course, one can further reduce the local complex 31 to the local Whitney complex or lowest-order trimmed polynomial complex: \[\label{eq:de-rham-complex-reduce-poly-element} \begin{tikzcd} \mathbb{R} \arrow[r, "\subset"] & W^{0, h}(K) \arrow[r, "\mathop{\mathrm{d}}^0"] & W^{1, h}(K) \arrow[r, "\mathop{\mathrm{d}}^1"] & W^{2, h}(K) \arrow[r, "\mathop{\mathrm{d}}^2"] & W^{3, h}(K) \arrow[r] & 0, \end{tikzcd}\tag{32}\] where \[\label{eq:whitney-forms-local} \begin{alignat}{2} W^{0, h}(K) &:= \mathcal{P}_1(K), \qquad & W^{1, h}(K) &:= \mathcal{P}_{0}(K)^3 + x \times \mathcal{P}_0(K)^3, \\ W^{2, h}(K) &:= \mathcal{P}_{0}(K)^3 + x \mathcal{P}_0(K), \qquad & W^{3, h}(K) &:= \mathcal{P}_0(K). \end{alignat}\tag{33}\] The following result shows that the two spaces \(\tilde{V}^{k, h}(K_A)\), \(k \in 0:1\), can be reduced further while ensuring that 32 is a subcomplex of the corresponding local complex.

On noting that \(\mathop{\mathrm{hess}}W^{0, h}(K) = 0\) and \(\mathop{\mathrm{sym}}\mathop{\mathrm{grad}}W^{1, h}(K) = 0\), the proof of 4 is completely analogous to the proof of 3 and is therefore omitted. One may readily see that if any additional degrees of freedom from 27 or 28 are set to zero, then we would lose the inclusion \(W^{k, h}(K) \subset \hat{V}^{k, h}(K_A)\), and so 36 may be seen as the Alfeld-split macroelement Stokes complex analog of the Whitney complex.

2.3 Bounded commuting cochain projections for “minimal” complex↩︎

With the cohomology of 17 fully characterized, the last remaining components used extensively in the FEEC literature are bounded commuting cochain projections. One possible avenue is to modify the construction of locally \(L^2\)-bounded cochain projections in [18] to take into account the boundary conditions analogously to the construction of Clément interpolants [29]. Instead, we construct Scott-Zhang [30] type interpolants that also commute.

To describe the result, we define for an open set \(\mathcal{O}\) the following norms: \(\| \cdot \|_{V^0(\mathcal{O})}\) the \(H^2(\mathcal{O})\) norm, \(\| \cdot \|_{V^1(\mathcal{O})}\) the \(H^1(\mathop{\mathrm{curl}}; \mathcal{O})\) norm, \(\| \cdot \|_{V^2(\mathcal{O})}\) the \(H^1(\mathcal{O})^3\) norm, and \(\| \cdot \|_{V^3(\mathcal{O})}\) the \(L^2(\mathcal{O})\) norm. Additionally, given a tetrahedron \(K \in \mathcal{T}\), let \(\omega_K\) denote the 1 element neighborhood of \(K\): \[\begin{align} \omega_K := \mathrm{int} \left( \bigcup \{ \bar{K}' \in \mathcal{T}: \bar{K} \cap \bar{K}' \neq \emptyset \} \right). \end{align}\] The locally bounded cochain projections are summarized in the following result.

The proof of 5 appears below in 6. Owing to the regularity of the spaces appearing in the Stokes complex 5 , the construction of these interpolants involves standard techniques similar to those in [30], [31] in contrast to the more sophisticated techniques employed for discretizations of the de Rham complex 6 ; see e.g. [32]–[34] and references therein.

2.4 Outline↩︎

The remainder of the manuscript is organized as follows. In 3, we show how the framework from [18] applies to the complex with full polynomial spaces 15 and modify the framework to account for boundary conditions. Then, in 4, we construct the reduced spaces \(\tilde{V}^{k, h}\), \(k \in 0:1\), show that the degrees of freedom in 27 28 are unisolvent, and demonstrate how boundary conditions may be incorporated into the spaces. The cohomology of the complex 17 is the focus of 5, and the bounded cochain projections are constructed in 6.

3 Cohomology of full polynomial spaces↩︎

We mostly follow the framework in [18], originally developed for determining the cohomology of discrete complexes spaces without boundary conditions, with some modification to handle the case of mixed boundary conditions in the Stokes complex. We first require some additional notation. Give a collection of \(d\)-dimensional simplices \(\mathcal{S}\), let \(\Delta(\mathcal{S}) := \bigcup_{\ell=0}^{d} \Delta_{\ell}(\mathcal{S})\) denote the collection of all subsimplices of \(\mathcal{S}\). Moreover, given a simplex \(\tau\), we say \(\eta \mathop{\mathrm{\unlhd}}\tau\) if \(\eta\) is a subsimplex of \(\tau\) and \(\eta \mathop{\mathrm{\lhd}}\tau\) if \(\eta \mathop{\mathrm{\unlhd}}\tau\) and \(\eta \neq \tau\).

3.1 Trace structure↩︎

We begin by defining ambient spaces for the various trace operators we will define. For \(K \in \mathcal{T}\), we take \(A^k(K) := V^{k, h}(K_A)\). For \(\tau \in \Delta_{1}(\mathcal{T}) \cup \Delta_{2}(\mathcal{T})\) and \(z \in \Delta_0(\mathcal{T})\), we define \[\begin{align} {2} A^0(\tau) &:= \mathcal{P}_{p+2}(\tau) \oplus \mathcal{P}_{p+1}(\tau)^3 \oplus \bigoplus_{z \in \Delta_0(\tau)} \mathbb{R}^{3 \times 3}_{\mathop{\mathrm{sym}}}, \qquad & A^0(z) &:= \mathbb{R} \oplus \mathbb{R}^3 \oplus \mathbb{R}^{3 \times 3}, \\ A^1(\tau) &:= \mathcal{P}_{p+1}(\tau)^3 \oplus \mathcal{P}_{p}(\tau)^3 \oplus \bigoplus_{z \in \Delta_0(\tau)} \mathbb{R}^{3 \times 3}, \qquad & A^1(z) &:= \mathbb{R}^3 \oplus \mathbb{R}^{3 \times 3}, \\ A^2(\tau) &:= \mathcal{P}_{p}(\tau)^3, \qquad & A^2(z) &:= \mathbb{R}^3, \end{align}\] where \(\mathbb{R}^{3 \times 3}_{\mathop{\mathrm{sym}}}\) denotes the set of \(3\times 3\) symmetric matrices with real entries.

For each \(k \in 0:3\), we show that \(A^k := \{ A^k(\tau) : \tau \in \Delta(\mathcal{T}) \}\) may be equipped with a trace structure in the sense of [18]. That is, for \(\tau \in \Delta(\mathcal{T})\) and \(\eta \mathop{\mathrm{\unlhd}}\tau\), we define linear operators \(\mathop{\mathrm{Tr}}^k_{\eta \mathop{\mathrm{\leftarrow}}\tau} : A^k(\tau) \to A^k(\eta)\) satisfying the following properties:

1. \(\mathop{\mathrm{Tr}}^k_{\tau \mathop{\mathrm{\leftarrow}}\tau}\) is the identity map for all \(\tau \in \Delta(\mathcal{T})\).

2. For all \(\sigma \in \mathcal{T}\) and \(\eta \mathop{\mathrm{\unlhd}}\tau \mathop{\mathrm{\unlhd}}\sigma\), there holds \[\begin{align} \label{eq:trace-vanishing-composition} \mathop{\mathrm{Tr}}^k_{\eta \mathop{\mathrm{\leftarrow}}\sigma} u = 0 \implies \mathop{\mathrm{Tr}}^k_{\eta \mathop{\mathrm{\leftarrow}}\tau} \circ \mathop{\mathrm{Tr}}^k_{\tau \mathop{\mathrm{\leftarrow}}\sigma} u = 0 \qquad \forall u \in A^k(\sigma). \end{align}\tag{39}\]

3. \(\mathop{\mathrm{Tr}}^k := \{ \mathop{\mathrm{Tr}}^k_{\eta \mathop{\mathrm{\leftarrow}}\tau} : \eta \mathop{\mathrm{\unlhd}}\tau, \;\tau \in \Delta(\mathcal{T}) \}\) characterizes \(V^{k, h}\): \[\begin{gather} \label{eq:trace-characterizes-space} V^{k, h} = \{ u \in L^2(\Omega) \otimes \mathbb{X}^k : u|_{K} \in A^k(K) \text{ and } \\ \mathop{\mathrm{Tr}}^k_{\tau \mathop{\mathrm{\leftarrow}}K} u|_{K} = \mathop{\mathrm{Tr}}^k_{\tau \mathop{\mathrm{\leftarrow}}K'} u|_{K'} \;\forall \tau \mathop{\mathrm{\unlhd}}K, K', \;\forall K, K' \in \mathcal{T}\}. \end{gather}\tag{40}\]

Recall that \(\mathbb{X}^k\) are the global ambient spaces defined in 12 . If \((A^k, \mathop{\mathrm{Tr}}^k)\) satisfy (i-ii), then \((A^k, \mathop{\mathrm{Tr}}^k)\) is a trace structure in the sense of [18], while (iii) ensures that \(V^{k, h}\) is the “global space” with respect to the trace structure [18]. In the following subsections, we construct the trace operators \(\mathop{\mathrm{Tr}}^k_{\eta \mathop{\mathrm{\leftarrow}}\tau}\) for \(\eta \neq \tau\), tacitly assuming that (i) always holds.

3.1.1 Trace structure on \(A^0\)↩︎

For \(K \in \mathcal{T}\) and \(\tau \mathop{\mathrm{\lhd}}K\), we define for \(v \in A^0(K)\) \[\begin{align} \mathop{\mathrm{Tr}}^0_{\tau \mathop{\mathrm{\leftarrow}}K} v &= v|_{\tau} \oplus \mathop{\mathrm{grad}}v|_{\tau} \oplus \bigoplus_{z \in \Delta_0(\tau)} \mathop{\mathrm{hess}}v(z), \end{align}\] where \(w|_{\tau} := w(\tau)\) if \(\dim \tau = 0\) and we recall that \(\mathop{\mathrm{hess}}\) denotes the Hessian operator. Given \(\tau \in \Delta_{1}(\mathcal{T}) \cup \Delta_{2}(\mathcal{T})\) and \(\eta \mathop{\mathrm{\lhd}}\tau\), we define for \(\phi \oplus \psi \oplus \bigoplus_{z \in \Delta_0(\tau)} M_z \in A^0(\tau)\) \[\begin{align} \mathop{\mathrm{Tr}}^0_{\eta \mathop{\mathrm{\leftarrow}}\tau} \left( \phi \oplus \psi \oplus \bigoplus_{z \in \Delta_0(\tau)} M_z \right) &= \phi|_{\eta} \oplus \psi|_{\eta} \oplus \bigoplus_{z \in \Delta_0(\eta)} M_z. \end{align}\] Then, one may readily verify that 40 39 hold for \(k = 0\).

3.1.2 Trace structure on \(A^1\)↩︎

For \(K \in \mathcal{T}\) and \(\tau \mathop{\mathrm{\lhd}}K\), we define for \(v \in A^1(K)\) \[\begin{align} \mathop{\mathrm{Tr}}^1_{\tau \mathop{\mathrm{\leftarrow}}K} := \begin{dcases} v|_{\tau} \oplus \mathop{\mathrm{curl}}v|_{\tau} \oplus \bigoplus_{z \in \Delta_0(\tau)} \mathop{\mathrm{grad}}v(z) & \text{if } \dim \tau > 0, \\ v(\tau) \oplus \mathop{\mathrm{grad}}v(\tau) & \text{if } \dim \tau = 0. \end{dcases} \end{align}\] For \(\tau \in \Delta_1(\mathcal{T}) \cup \Delta_2(\mathcal{T})\) and \(\eta \mathop{\mathrm{\lhd}}\tau\), we define for \(\phi \oplus \psi \oplus \bigoplus_{z \in \Delta_0(\tau)} M_z \in A^1(\tau)\) \[\begin{align} \mathop{\mathrm{Tr}}^1_{\eta \mathop{\mathrm{\leftarrow}}\tau} \left( \phi \oplus \psi \oplus \bigoplus_{z \in \Delta_0(\tau)} M_z \right) := \begin{dcases} \phi|_{\eta} \oplus \psi|_{\eta} \oplus \bigoplus_{z \in \Delta_0(\eta)} M_z & \text{if } \dim \eta > 0, \\ \phi(\eta) \oplus M_{\eta} & \text{if } \dim \eta = 0. \end{dcases} \end{align}\] Then, we may readily verify that 39 40 hold for \(k = 1\).

3.1.3 Trace structure on \(A^2\) and \(A^3\)↩︎

For \(\tau \in \Delta(\mathcal{T})\) and \(\eta \mathop{\mathrm{\lhd}}\tau\), we define for \({v \in A^2(\tau)}\) \[\begin{align} \mathop{\mathrm{Tr}}_{\eta \mathop{\mathrm{\leftarrow}}\tau}^2 v = v|_{\eta}, \end{align}\] while for \(w \in A^3(\tau)\), we set \(\mathop{\mathrm{Tr}}_{\eta \mathop{\mathrm{\leftarrow}}\tau}^3 w = 0\). Then, 39 40 hold for \(k = 2,3\).

3.2 Bubble spaces↩︎

Following [18], for \(K \in \mathcal{T}\) and \(\tau \mathop{\mathrm{\unlhd}}K\), we define \(B^k(\tau; K)\) by \[\begin{align} \label{eq:generic-bubble-space} B^k(\tau; K) := \{ v \in \mathop{\mathrm{Tr}}^k_{\tau \mathop{\mathrm{\leftarrow}}K} A^k(K) : \mathop{\mathrm{Tr}}^k_{\eta \mathop{\mathrm{\leftarrow}}\tau} v = 0 \;\forall \eta \mathop{\mathrm{\lhd}}\tau \}. \end{align}\tag{41}\] We now show that each trace structure \((A^k, \mathop{\mathrm{Tr}}^k)\) satisfies the geometric decomposition property [18]: \[\begin{align} \label{eq:geometric-decomposition} \sum_{\tau \mathop{\mathrm{\unlhd}}K} \dim B^k(\tau; K) = \dim A^k(K) \qquad \forall K \in \mathcal{T}. \end{align}\tag{42}\] In the following subsections, let \(K \in \mathcal{T}\), \(f \in \Delta_2(K)\), \(e \in \Delta_1(K)\), and \(z \in \Delta_0(K)\).

3.2.1 The case \(k=0\)↩︎

Expanding definitions, we see that \[\begin{align} B^0(K; K) &= V_0^{0, h}(K_A), \\ B^0(f; K) &= \left\{ v \oplus w \oplus 0 \in A^0(f) : v \in H^2_0(f), \;w \in H^1_0(f)^3, \right. \\ &\qquad \qquad \left. \text{and } (I - \hat{n}_f \otimes \hat{n}_f) w = \mathop{\mathrm{grad}}_f v \right\}, \\ B^0(e; K) &= \left\{ v \oplus w \oplus 0 \in A^0(e) : v \in H^3_0(e), w \in H^2_0(e)^3, \text{ and } w \cdot \hat{t}_e = \partial_{\hat{t}_e} v \right\}, \\ B^0(z; K) &= \mathbb{R} \oplus \mathbb{R}^3 \oplus \mathbb{R}_{\mathop{\mathrm{sym}}}^{3 \times 3}, \end{align}\] where \(\mathop{\mathrm{grad}}_f\) denotes the surface gradient (viewed as an element of \(\mathbb{R}^3\)). As a consequence, we obtain \[\begin{align} \dim B^0(f; K) &= \dim \mathcal{P}_{p-4}(f) + \dim \mathcal{P}_{p-2}(f), \\ \dim B^0(e; K) &= \dim \mathcal{P}_{p-4}(e) + \dim \mathcal{P}_{p-3}(e)^2, \end{align}\] and so performing a direct calculation and applying [17] shows that 42 holds for \(k=0\).

3.2.2 The case \(k=1\)↩︎

Expanding definitions gives \[\begin{align} B^1(K; K) &= V_0^{1, h}(K_A), \\ B^1(f; K) &= \left\{ v \oplus w \oplus 0 \in A^1(f) : v,w \in H^1_0(f)^3, \; \text{and } w \cdot \hat{n}_f = \mathop{\mathrm{rot}}_f v \right\}, \\ B^1(e; K) &= \left\{ v \oplus w \oplus 0 \in A^1(e) : v \in H^2_0(e) \text{ and } w \in H^1_0(e) \right\}, \\ B^1(z; K) &= \mathbb{R}^3 \oplus \mathbb{R}^{3 \times 3}, \end{align}\] where \(\mathop{\mathrm{rot}}_f\) is the surface curl defined so that \(\mathop{\mathrm{rot}}_f v|_f := \mathop{\mathrm{curl}}v \cdot \hat{n}_f|_{f}\) for \(v \in C^{\infty}(\mathbb{R}^3)^3\). Note that every \(v \in \mathcal{P}_{p+1}(f)\) satisfies \[\begin{align} v|_{\partial f} = 0 \text{ and } \mathop{\mathrm{rot}}_f v|_{\partial f} = 0 \implies \frac{\partial }{\partial \hat{t}_e \times \hat{n}_f} v \cdot \hat{t}_{e} = 0 \qquad \forall e \in \Delta_1(f), \end{align}\] and so \[\begin{align} \dim B^1(f; K) &= \dim \mathcal{P}_{p-2}(f)^3 - \sum_{e' \in \Delta_1(f)} \dim \mathcal{P}_{p-2}(e') + \dim \mathcal{P}_{p-3}(f)^2 \\ &= \dim \mathcal{P}_{p-2}(f) + \left( p(p-1) - 3(p-1) \right) + \dim \mathcal{P}_{p-3}(f)^2. \end{align}\] Moreover, \(\dim B^1(e; K) = \dim \mathcal{P}_{p-3}(e)^3 + \dim \mathcal{P}_{p-2}(e)^3\), and so performing a direct calculation and applying [17] shows that 42 holds for \(k=1\).

3.2.3 The case \(k=2,3\)↩︎

The case \(k=2\) corresponds to the usual bubble spaces: \(B^2(K; K) = V^{2, h}_0(K_A)\), \(B^2(\tau; K) = \mathcal{P}_{p}(\tau)^3 \cap H^1_0(\tau)^3\) for \(\tau \in \Delta_1(K) \cup \Delta_2(K)\), and \(B^2(z; K) = \mathbb{R}^3\). Moreover, \(B^3(K; K) = A^3(K)\), and so 42 holds for \(k=2,3\).

3.3 Trace complexes↩︎

Again let \(K \in \mathcal{T}\), \(f \in \Delta_2(K)\), \(e \in \Delta_1(K)\), and \(z \in \Delta_0(K)\), and consider the following diagram: \[\label{eq:full-spaces-trace-complex} \begin{tikzcd}[ampersand replacement = \&, column sep = 8em] A^0(K) \arrow[r, "\mathop{\mathrm{d}}_K^0"] \arrow[d, "\mathop{\mathrm{Tr}}"] \arrow[dd, bend right=60, "\mathop{\mathrm{Tr}}" description] \arrow[ddd, bend right=70, "\mathop{\mathrm{Tr}}" description] \& A^1(K) \arrow[r, "\mathop{\mathrm{d}}_K^1"] \arrow[d, "\mathop{\mathrm{Tr}}"] \& A^2(K) \arrow[d, "\mathop{\mathrm{Tr}}"] \\ A^0(f) \arrow[r, "\mathop{\mathrm{d}}^{0}_{f}"] \& A^1(f) \arrow{r}{\mathop{\mathrm{d}}^{1}_{f}} \& A^2(f) \\ A^0(e) \arrow{r}{\mathop{\mathrm{d}}^{0}_{e}} \& A^1(e) \arrow{r}{\mathop{\mathrm{d}}^{1}_{e}} \arrow[from=uu, bend right=60, "\mathop{\mathrm{Tr}}" description, pos=0.66] \& A^2(e) \arrow[from=uu, bend right=60, "\mathop{\mathrm{Tr}}" description, pos=0.66] \\ A^0(z) \arrow{r}{\mathop{\mathrm{d}}^{0}_{z}} \& A^1(z) \arrow{r}{\mathop{\mathrm{d}}^{1}_{z}} \arrow[from=uuu, bend right=70, "\mathop{\mathrm{Tr}}" description] \& A^2(z), \arrow[from=uuu, bend right=70, "\mathop{\mathrm{Tr}}" description] \end{tikzcd}\tag{43}\] where the vertical arrows are the corresponding trace operators (with sub and superscripts omitted) and the “differential” operators on the horizontal arrows are defined as follows: Let \(\mathop{\mathrm{d}}^0_K := \mathop{\mathrm{grad}}\) and \(\mathop{\mathrm{d}}^1_K = \mathop{\mathrm{curl}}\). For \(\tau \in \Delta_1(K) \cup \Delta_2(K)\), let \[\begin{align} \mathop{\mathrm{d}}_{\tau}^0 \left( v \oplus w \oplus \bigoplus_{z \in \Delta_0(\tau)} M_z \right) &:= w \oplus 0 \oplus \bigoplus_{z \in \Delta_0(\tau)} M_z, \\ \mathop{\mathrm{d}}_{\tau}^1 \left( v \oplus w \oplus \bigoplus_{z \in \Delta_0(\tau)} M_{\tau} \right) &:= w. \end{align}\] while for \(z \in \Delta_0(K)\), we define \[\begin{align} \mathop{\mathrm{d}}_z^0 \left( c \oplus v \oplus M \right) &= v \oplus M \quad \text{and} \quad \mathop{\mathrm{d}}_z^1 \left( v \oplus M \right) = \sum_{i, j, k = 1}^{3}\epsilon_{ijk} M_{j, i} \hat{e}_k, \end{align}\] where \(\hat{e}_k\) is the standard unit vector in the \(k\)-th direction, and \(\epsilon_{ijk}\) is the permutation symbol. Then, one may verify that 43 commutes and each row is a complex. Thus, \((A^k, \mathop{\mathrm{Tr}}^k, \mathop{\mathrm{d}}^k)\) is a conforming finite element subcomplex with trace structure (FECTS) ([18]) of the de Rham complex.

3.4 Compatible currents and bubble complex↩︎

For \(k \in 0:3\) and \(\tau \in \Delta_k(\mathcal{T})\), we recall the “currents" \(\mathcal{I}_{\tau}^k : C(\bar{\Omega}) \otimes \mathbb{X}^k \to \mathbb{R}\) defined in 18 . In particular, the Stokes formula holds for \(\sigma \in \Delta_{k+1}(\mathcal{T})\) and \(w \in C(\bar{\Omega}) \otimes \mathbb{X}^k\): \[\begin{align} \label{eq:currents-stokes-thm} \mathcal{I}^{k+1}_{\sigma}(\mathop{\mathrm{d}}^k w) = \sum_{\tau \in \Delta_{k}(\sigma)} \mathcal{O}(\tau, \sigma) \mathcal{I}^{k}_{\tau}(w), \end{align}\tag{44}\] where \(\mathcal{O}(\tau, \sigma)\) denotes the orientation of \(\tau\) relative to \(\sigma\). Thus, \((\mathcal{I}, \mathbb{R})\) is a family of generalized currents [18]. Note that [18] assumes that the domain of \(\mathcal{I}_{\tau}^k\) is \(C^{\infty}(\bar{\Omega}) \otimes \mathbb{X}^k\); however, one only needs that \(\mathcal{I}^k_{\tau}\) is well-defined on \(V^{k, h}\).

We also see that for each \(k \in 0:3\) and \(\tau \in \Delta_k(\mathcal{T})\), the functionals \({\tilde{\mathcal{I}}_{\tau}^k : A^k(\tau) \to \mathbb{R}}\) defined by \[\begin{align} {2} \tilde{\mathcal{I}}^0_{\tau} ( c \oplus v \oplus M ) &:= c, \qquad & \tilde{\mathcal{I}}^2_{\tau} (w) &:= \int_{\tau} w \cdot \hat{n}_{\tau} \,\mathrm{d}{s}, \\ \tilde{\mathcal{I}}^1_{\tau} \left(\phi \oplus \psi \oplus \bigoplus_{z \in \Delta_0(\tau)} M_z \right) &:= \int_{\tau} \phi \cdot \hat{t}_{\tau} \,\mathrm{d}{s}, & \qquad \tilde{\mathcal{I}}^3_{\tau} (q) &:= \int_{\tau} q \,\mathrm{d}{x}, \end{align}\] satisfy \[\begin{align} \label{eq:current-trace-compatibility} \tilde{\mathcal{I}}_{\tau}^k( \mathop{\mathrm{Tr}}^k_{\tau \mathop{\mathrm{\leftarrow}}\sigma} w) = \mathcal{I}_{\tau}^k(w) \qquad \forall w \in A^{k}(\sigma), \;\forall \sigma \in \mathcal{T}: \tau \mathop{\mathrm{\unlhd}}\sigma. \end{align}\tag{45}\] Note that above, we are viewing \(A^k(\sigma)\) as defined on all of \(\Omega\), where any \(V^{k}\)-continuous extension is taken.

The bubble spaces 41 do not depend on the parent tetrahedron, so we drop “\(K\)" from the notation. Define the modified bubble spaces \[\begin{align} \tilde{B}^k(\tau) := \begin{cases} B^k(\tau) \cap \ker \tilde{\mathcal{I}}_{\tau}^k & \text{if } k = \dim \tau, \\ B^k(\tau) & \text{otherwise}. \end{cases} \end{align}\] The next result shows that these modified bubble spaces form an exact sequence.

Thanks to 45 and 1, the FECTS \((A^k, \mathop{\mathrm{Tr}}^k, \mathop{\mathrm{d}}^k)\) is compatible with respect to the currents \((\mathcal{I}^k, \mathbb{R})\) in the sense of [18].

3.5 Lifted bubble and skeletal complexes↩︎

For \(k \in 0:3\), let \((\cdot,\cdot)_{A^k(\tau)}\) denote the natural \(L^2(\tau)\) inner product on \(A^k(\tau)\). Let \(\mathbb{Q}^k_{\tau} : A^k(\tau) \to \mathop{\mathrm{d}}_{\tau}^{k-1} \tilde{B}^{k-1}(\tau)\) denote the \((\cdot,\cdot)_{A^k(\tau)}\)-orthogonal projection onto \(\mathop{\mathrm{d}}_{\tau}^{k-1} \tilde{B}^{k-1}(\tau)\): \[\begin{align} (\mathbb{Q}^k_{\tau} u, \mathop{\mathrm{d}}_{\tau}^{k-1} v)_{A^k(\tau)} = (u, \mathop{\mathrm{d}}_{\tau}^{k-1} v)_{A^k(\tau)} \qquad \forall v \in \tilde{B}^{k-1}(\tau), \;\forall u \in A^k(\tau). \end{align}\] We define the harmonic inner product \(A^k(\tau)\) [18] as follows: \[\begin{align} \langle u, v \rangle_{A^k(\tau)} := (\mathbb{Q}^k_{\tau} u, \mathbb{Q}^k_{\tau} v)_{A^k(\tau)} + (\mathop{\mathrm{d}}_{\tau}^k u, \mathop{\mathrm{d}}_{\tau}^k v)_{A^{k+1}(\tau)}. \end{align}\] Note that \(\langle \cdot, \cdot \rangle_{A^k(\tau)}\) is an inner product on \(\tilde{B}^k(\tau)\) owing to the exactness of 46 .

For \(v \in V^{k, h}\), property 40 means that for \(\tau \in \Delta(\mathcal{T})\), we may define the trace of \(v\) on \(\tau\) independent of the parent tetrahedron; i.e. \(\mathop{\mathrm{Tr}}^k_{\tau} v := \mathop{\mathrm{Tr}}^k_{\tau \mathop{\mathrm{\leftarrow}}K} v\), where \(K \in \mathcal{T}\) with \(\tau \mathop{\mathrm{\unlhd}}K\) is well-defined independent of the particular choice of \(K\). As shown in [18] and the remaining discussion on [18], a unisolvent set of degrees of freedom on \(V^{k, h}\) are \[\label{eq:global-dofs-bubble} \begin{alignat}{2} &\langle \mathop{\mathrm{Tr}}^k_{\tau} v, \phi \rangle_{_{A^k(\tau)}} \qquad & &\forall \phi \in \tilde{B}^k(\tau), \;\forall \tau \in \Delta(\mathcal{T}), \\ &\mathcal{I}_{\tau}^k(v) \qquad & &\forall \tau \in \Delta_k(\mathcal{T}). \end{alignat}\tag{47}\] Consequently, we define a lift of the modified bubble functions and the skeletal space as follows: \[\begin{align} \mathbb{B}^k(\tau) &:= \{ v \in V^{k, h} : \mathcal{I}^k_{\eta}(v) = 0 \;\forall \eta \in \Delta_k(\mathcal{T}) \text{ and} \\ &\qquad \qquad \langle \mathop{\mathrm{Tr}}_{\eta}^k v, w \rangle_{A^k(\eta)} = 0 \;\forall w \in \tilde{B}^k(\eta), \;\forall \eta \in \Delta(\mathcal{T}) \setminus \{\tau\} \} \qquad \forall \tau \in \Delta(\mathcal{T}),\\ \mathcal{S}^k &:= \{ v \in V^{k, h} : \langle \mathop{\mathrm{Tr}}_{\eta}^k v, w \rangle_{A^k(\eta)} = 0 \;\forall w \in \tilde{B}^k(\eta), \;\forall \eta \in \Delta(\mathcal{T}) \}. \end{align}\] The next lemma summarizes the support properties of these spaces.

Thanks to [18], each column of the following diagram is a direct sum decomposition, each row is a complex, and the final row is exact: \[\label{eq:stokes-complex-full-geom-decomp-nobcs} \begin{tikzcd}[ampersand replacement=\&] 0 \arrow[r] \& V^{0, h} \arrow[r, "\mathop{\mathrm{grad}}"] \& V^{1, h} \arrow[r, "\mathop{\mathrm{curl}}"] \& V^{2, h} \arrow[r, "\undefined"] \& V^{3, h} \arrow[r] \& 0 \\[-2em] \& \rotatebox{90}{=} \& \rotatebox{90}{=} \& \rotatebox{90}{=} \& \rotatebox{90}{=} \& \\[-2em] 0 \arrow[r] \& \mathcal{S}^0 \arrow[r, "\mathop{\mathrm{grad}}"] \& \mathcal{S}^1 \arrow[r, "\mathop{\mathrm{curl}}"] \& \mathcal{S}^2 \arrow[r, "\undefined"] \& \mathcal{S}^3 \arrow[r] \& 0 \\[-2em] \& \bigoplus\limits_{\tau \in \Delta(\mathcal{T})} \& \bigoplus\limits_{\tau \in \Delta(\mathcal{T})} \& \bigoplus\limits_{\tau \in \Delta(\mathcal{T})} \& \bigoplus\limits_{\tau \in \Delta(\mathcal{T})} \& \\[-2em] 0 \arrow[r] \& \mathbb{B}^0(\tau) \arrow[r, "\mathop{\mathrm{grad}}"] \& \mathbb{B}^1(\tau) \arrow[r, "\mathop{\mathrm{curl}}"] \& \mathbb{B}^2(\tau) \arrow[r, "\undefined"] \& \mathbb{B}^3(\tau) \arrow[r] \& 0, \end{tikzcd}\tag{50}\] In particular, the cohomology of the first row of 50 is isomorphic to the cohomology of the second row, which in turn is isomorphic to the de Rham cohomology (the cohomology of 6 with \(\Gamma_0 = \emptyset\)) [18].

3.6 Incorporating mixed BCs into spaces↩︎

Recall that \(\mathcal{U}\) is a conforming mesh of \(\Gamma_0\). The supersmoothness of the spaces \(V_{\Gamma_0}^{0, h}\) and \(V_{\Gamma_0}^{1, h}\) at mesh vertices introduces some subtleties that are not naturally reflected in the trace operators. In particular, denote the mesh vertices laying on the “flat” portion of the \(\bar{\Gamma}_{0}\) by \[\begin{align} \Delta_0^{\flat}(\mathcal{U}) := \{ z \in \Delta_0(\mathcal{U}) : \text{ all faces meeting at z are coplanar} \}. \end{align}\] Then, the following result shows that not all Hessian degrees of freedom for functions in \(V_{\Gamma_0}^{0, h}\) or gradient degrees of freedom vector fields in \(V_{\Gamma_0}^{1, h}\) vanish on \(\Delta_0^{\flat}\).

One consequence of 3 is that the “zero trace spaces” defined as the kernel of \(\mathop{\mathrm{Tr}}^k_{f}\) for \(f \in \mathcal{U}\) may not coincide with \(V_{\Gamma_0}^{k, h}\) for \(k=0, 1\); i.e., \[\begin{align} \{ v \in V^{k, h} : \mathop{\mathrm{Tr}}^k_{f} v = 0 \text{ if } f \in \mathcal{U}\} \subseteq V^{k, h}_{\Gamma_0}, \end{align}\] but the inclusion will generally be strict for \(k=0, 1\) if \(\Delta_0^{\flat}(\mathcal{U})\) is nonempty. To rectify this discrepancy, we modify the vertex bubble functions further for \(z \in \Delta_0^{\flat}(\mathcal{U})\): \[\begin{align} \mathbb{B}^{0, nn}(z) &:= \{ v \in \mathbb{B}^{0}(z) : v(z) = 0, \;\mathop{\mathrm{grad}}v(z) = 0, \; (I - \hat{n}_{\Gamma} \otimes \hat{n}_{\Gamma})\mathop{\mathrm{hess}}v(z) = 0 \}, \\ \mathbb{B}^{1, nn}(z) &:= \{ v \in \mathbb{B}^{1}(z) : v(z) = 0, \; (I - \hat{n}_{\Gamma} \otimes \hat{n}_{\Gamma}) \mathop{\mathrm{grad}}v(z) = 0\}. \end{align}\] The corresponding modified bubble spaces, which will be shown to satisfy the boundary conditions, are given by \[\begin{align} \mathbb{B}_{\Gamma_0}^k(\tau) := \begin{cases} \mathbb{B}^k(\tau) & \text{if } \tau \in \Delta(\mathcal{T}) \setminus \Delta(\mathcal{U}), \\ \mathbb{B}^{k, nn}(\tau) & \text{if } \tau \in \Delta_0^{\flat}(\mathcal{U}) \text{ and } k < 2, \\ \{0\} & \text{otherwise}, \end{cases} \qquad \forall \tau \in \Delta(\mathcal{T}), \;\forall k \in 0:3. \end{align}\] The skeletal spaces with boundary conditions are simply \[\begin{align} \mathcal{S}_{\Gamma_0}^k := \{ v \in \mathcal{S}^k : \mathcal{I}^k_{\tau}(v) = 0 \;\forall \tau \in \Delta_k(\mathcal{U}) \} \qquad \forall k \in 0:3. \end{align}\] The next result shows that these spaces do indeed satisfy the boundary conditions.

We also have the analog of the diagram 50 :

3.7 Cohomology via relative simplicial cochain complex↩︎

We now introduce the relative simplicial cochain complex, modifying the presentation in [35]. For \(k \in 0:3\), let \(C_k(\mathcal{V})\) denote the space of simplicial \(k\)-chains (i.e. formal linear combinations of subsimplices of dimension \(k\)) of a mesh \(\mathcal{V}\). Then, \(C_k(\mathcal{U})\) is a subspace of \(C_k(\mathcal{T})\), and we define \(C_k(\mathcal{T}, \mathcal{U}) := C_k(\mathcal{T}) / C_k(\mathcal{U})\). Let \({\partial_k : C_k(\mathcal{T}, \mathcal{U}) \to C_{k-1}(\mathcal{T}, \mathcal{U})}\) denote the standard simplicial boundary operator defined on the quotient space uniquely by the condition \[\begin{align} \partial_k (\tau + C_k(\mathcal{U})) = \sum_{\substack{\eta \in \Delta_{k-1}(\tau) \\ \eta \notin \Delta_{k-1}(\mathcal{U})}} \mathcal{O}(\eta, \tau) (\eta + C_{k-1}(\mathcal{U})) \qquad \forall \tau \in \Delta_k(\mathcal{T}) \setminus \Delta_k(\mathcal{U}). \end{align}\] Let \({\partial}'_k : {C_k(\mathcal{T}, \mathcal{U})}' \to {C_{k+1}(\mathcal{T}, \mathcal{U})}'\) denote the corresponding cochain map defined uniquely by the condition: \[\begin{align} ({\partial}'_k \ell_k)(\omega) = \ell_k( \partial_{k+1} \omega) \qquad \forall \omega \in C_{k+1}(\mathcal{T}, \mathcal{U}), \; \forall \ell_k \in {C_k(\mathcal{T}, \mathcal{U})}'. \end{align}\]

4 Construction of a “minimal” conforming complex↩︎

We now turn to the construction of a “minimal” conforming subcomplex of 15 . With \(\tilde{V}^{2, h}(K_A) = \mathop{\mathrm{GN}}(K_A)\) and \(\tilde{V}^{3, h}(K_A) = \mathop{\mathrm{DG}}^0(K)\) as in 2.2, we seek discrete spaces \(\tilde{V}^{k, h}(K_A) \subset V^{k, h}(K_A)\), \(k \in 0:1\), so that 25 is an exact complex. We want the spaces to be sufficiently large so that the inclusions \(\mathcal{P}_{3-k}(K) \otimes \mathbb{X}^k \subset \tilde{V}^{k, h}(K_A)\) hold, but also minimal in the sense that degrees of freedom in 27 for \(k=0\) and 28 for \(k=1\) are unisolvent. We achieve this by defining the spaces \(\tilde{V}^{k, h}(K_A)\) as subspaces of \(V^{k, h}(K_A)\) satisfying particular constraints.

4.1 Local \(H^2\)-conforming space \(\tilde{V}^{0, h}(K_A)\)↩︎

Let \(K \in \mathcal{T}\) and suppose that the bilinear form \({a_K(\cdot,\cdot) : H^1(K)^3 \times H^1(K)^3 \to \mathbb{R}}\) is continuous and satisfies: \[\tag{61} \begin{alignat}{2} \tag{62} a_K(u, \mathop{\mathrm{grad}}\phi) &= 0 \qquad & &\forall u \in \mathcal{P}_2(K)^3, \;\forall \phi \in V^{0, h}_0(K_A), \\ \tag{63} a_K(\mathop{\mathrm{grad}}\psi, \mathop{\mathrm{grad}}\psi) &\geq \gamma_a \|\psi\|_{H^2(K)}^2 \qquad & &\forall \psi \in V^{0, h}_0(K_A), \end{alignat}\] where \(\gamma_a\) is independent of \(K\) and we recall that \(V^{0, h}_0(K_A) = V^{0, h}(K_A) \cap H^2_0(K)\). One bilinear form satisfying 61 is \(a_K(\cdot,\cdot) = (\mathop{\mathrm{grad}}\cdot, \mathop{\mathrm{grad}}\cdot)_{L^2(K)}\). For each face \(f \in \Delta_2(K)\), let \(\ell_f \in \mathcal{P}_3(f)^*\) be any linear functional with the following properties: \[\begin{align} \label{eq:kill-cubic-bubble-dof} \ell_f(p) = 0 \qquad \forall p \in \mathcal{P}_2(f) \quad \text{and} \quad \ell_f(b_f) \neq 0. \end{align}\tag{64}\] For example, one could take \[\begin{align} \ell_f(p) = \int_f (I - \mathbb{P}_{2, f})b_f p \,\mathrm{d}{s} \qquad \forall p \in \mathcal{P}_3(f), \end{align}\] where \(\mathbb{P}_{2, f} : L^1(f) \to \mathcal{P}_2(f)\) is the \(L^2(f)\)-orthogonal projector onto \(\mathcal{P}_2(f)\).

Consider the following subspace of \(V^{0, h}(K_A)\): \[\begin{gather} \label{eq:local-h2-space-def} \tilde{V}^{0, h}(K_A) := \{ v \in V^{0, h}(K_A) : \partial_n v|_{f} \in \mathcal{P}_3(f) \text{ and } \ell_f(\partial_n v|_{f}) = 0 \;\forall f \in \Delta_2(K) \\ \text{and } a_K(\mathop{\mathrm{grad}}v, \mathop{\mathrm{grad}}w) = 0 \;\forall w \in V^{0, h}_0(K_A) \}. \end{gather}\tag{65}\] The following result summarizes the key properties of \(\tilde{V}^{0, h}(K_A)\).

4.2 Local \(H^1(\mathop{\mathrm{curl}})\)-conforming space \(\tilde{V}^{1, h}(K_A)\)↩︎

Note that \(\tilde{V}^{0, h}(K_A)\) chosen as in 65 appears to the left of \(\tilde{V}^{1, h}(K_A)\) in the complex 25 , while \(\mathop{\mathrm{GN}}(K_A)\) appears to the right. To satisfy the complex property, we simply define \(\tilde{V}^{1, h}(K_A)\) to be the subspace of \(V^{1, h}(K_A)\) satisfying the constraints imposed by these two spaces: \[\begin{gather} \label{eq:local-h1curl-space-def} \tilde{V}^{1, h}(K_A) := \{ v \in V^{1, h}(K_A) : v \cdot \hat{n}_f|_{f} \in \mathcal{P}_3(f) \text{ and } \ell_f(v \cdot \hat{n}_f|_{f}) = 0 \;\forall f \in \Delta_2(K), \\ \mathop{\mathrm{curl}}v \in \mathop{\mathrm{GN}}(K_A), \text{ and } a_K(v, \mathop{\mathrm{grad}}\phi) = 0 \;\forall \phi \in V^{0, h}_0(K_A) \}. \end{gather}\tag{66}\]

4.3 Global spaces and complexes↩︎

We begin with a simple result showing that reducing the degree of normal components on faces also reduces the degree of the normal components on edges.

We now proceed space-by-space in 17 and show that the global spaces 26 defined in terms of the local spaces 65 66 properly glue together.

Standard arguments also show that \(\tilde{V}_{\Gamma_0}^{2, h}\) may be characterized similarly by omitting degrees of freedom associated to \(\Delta(\mathcal{U})\):

5 Cohomology of the “minimal” complexes↩︎

We could proceed as in 3 and apply the framework in [18]. However, the degrees of freedom for the discrete spaces \(\tilde{V}^{k, h}\) are simple enough that we can prove the key results directly. To this end, we define the skeletal spaces for \(k \in 0:2\) by \[\begin{align} \tilde{\mathcal{S}}^{k}_{\Gamma_0} &:= \left\{ v \in \tilde{V}^{k, h}_{\Gamma_0} : D^{\alpha} v(z) = 0 \;\forall |\alpha| \in \max\{1-k, 0\} : 2-k, \;\forall z \in \Delta_0(\mathcal{T}) \right\}, \end{align}\] and set \(\tilde{\mathcal{S}}^3_{\Gamma_0} := \tilde{V}^{3, h}_{\Gamma_0}\). Owing to 9 10 [ lem:min-h1-space-dofs-bcs]{reference-type=“ref” reference=” lem:min-h1-space-dofs-bcs”}, the “currents" \(\mathcal{I}^k_{\tau}(\cdot)\) for \({\tau \in \Delta_k(\mathcal{T}) \setminus \Delta_k(\mathcal{U})}\) defined in 18 are a unisolvent set of degrees of freedom on \(\tilde{\mathcal{S}}^{k}_{\Gamma_0}\).

We also define the lifted (modified) bubble spaces associated with each vertex \(z \in \Delta_0(z)\) for \(k \in 0:2\) by \[\begin{gather} \tilde{\mathbb{B}}^{k}_{\Gamma_0}(z) := \left\{ v \in \tilde{V}^{k, h}_{\Gamma_0} : D^{\alpha} v(z') = 0 \;\forall |\alpha| \leq 2-k, \;\forall z' \in \Delta_0(\mathcal{T}) \setminus \{z\} \right. \\ \left. \vphantom{v \in \tilde{V}^{k, h}_{\Gamma_0}} \text{ and } \mathcal{I}_{\tau}^k(v) = 0 \;\forall \tau \in \Delta_k(\mathcal{T}) \right\}. \end{gather}\] Analogous to 60 , we have an associated diagram: \[\label{eq:stokes-complex-gn-geom-decomp} \begin{tikzcd}[ampersand replacement=\&] 0 \arrow[r] \& \tilde{V}^{0, h}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{grad}}"] \& \tilde{V}^{1, h}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{curl}}"] \& \tilde{V}^{2, h}_{\Gamma_0} \arrow[r, "\undefined"] \& \tilde{V}^{3, h}_{\Gamma_0} \arrow[r] \& 0 \\[-2em] \& \rotatebox{90}{=} \& \rotatebox{90}{=} \& \rotatebox{90}{=} \& \rotatebox{90}{=} \& \\[-2em] 0 \arrow[r] \& \tilde{\mathcal{S}}^{0}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{grad}}"] \& \tilde{\mathcal{S}}^{1}_{\Gamma_0} \arrow[r, "\mathop{\mathrm{curl}}"] \& \tilde{\mathcal{S}}^{2}_{\Gamma_0} \arrow[r, "\undefined"] \& \tilde{\mathcal{S}}^{3}_{\Gamma_0} \arrow[r] \& 0 \\[-2em] \& \bigoplus\limits_{z \in \Delta_0(\mathcal{T})} \& \bigoplus\limits_{z \in \Delta_0(\mathcal{T})} \& \bigoplus\limits_{z \in \Delta_0(\mathcal{T})} \& \& \\[-2em] 0 \arrow[r] \& \tilde{\mathbb{B}}^{0}_{\Gamma_0}(z) \arrow[r, "\mathop{\mathrm{grad}}"] \& \tilde{\mathbb{B}}^{1}_{\Gamma_0}(z) \arrow[r, "\mathop{\mathrm{curl}}"] \& \tilde{\mathbb{B}}^{2}_{\Gamma_0}(z) \arrow[r, "\undefined"] \& 0 \arrow[r] \& 0. \end{tikzcd}\tag{73}\] Each column is a direct sum decomposition thanks to 9 10 [ lem:min-h1-space-dofs-bcs]{reference-type=“ref” reference=” lem:min-h1-space-dofs-bcs”}. The complex properties are summarized as follows.

5.1 Proof of 3↩︎

For \(k \in 0:3\), let \[\begin{align} W^{k, h} := \{ v \in L^2(\Omega) \otimes \mathbb{X}^{k} : \mathop{\mathrm{d}}^k v \in L^2(\Omega) \otimes \mathbb{X}^{k+1} \text{ and } v|_{K} \in W^{k, h}(K) \;\forall K \in \mathcal{T}\} \end{align}\] denote the Whitney forms on \(\mathcal{T}\) equipped with the canonical degrees of freedom \(\mathcal{I}^k_{\tau}\), \(\tau \in \Delta_k(\mathcal{T})\). Set \(W_{\Gamma_0}^{k, h} := W^{k, h} \cap W^{k}_{\Gamma_0}\) so that \(\{ \mathcal{I}^k_{\tau} : \tau \in \Delta_k(\mathcal{T}) \setminus \Delta_k(\mathcal{U})\}\), are a unisolvent set of degrees of freedom on \(W_{\Gamma_0}^{k, h}\). Then, \(\tilde{\mathcal{S}}_{\Gamma_0}^{k}\) and \(W_{\Gamma_0}^{k, h}\) are isomorphic, and let \(\pi^k : \tilde{\mathcal{S}}_{\Gamma_0}^{k} \to W_{\Gamma_0}^{k, h}\) denote the isomorphism that maps an element of \(\tilde{\mathcal{S}}_{\Gamma_0}^{k}\) to the unique element in \(W_{\Gamma_0}^{k, h}\) with the same degrees of freedom. The generalized Stokes theorem 44 then shows that the following diagram commutes: \[\begin{tikzcd}[ampersand replacement=\&] 0 \arrow[r] \arrow[d] \& \tilde{\mathcal{S}}_{\Gamma_0}^0 \arrow[r, "\mathop{\mathrm{grad}}"] \arrow[d, "\pi^0"] \& \tilde{\mathcal{S}}_{\Gamma_0}^1 \arrow[r, "\mathop{\mathrm{curl}}"] \arrow[d, "\pi^1"] \& \tilde{\mathcal{S}}_{\Gamma_0}^2 \arrow[r, "\undefined"] \arrow[d, "\pi^2"] \& \tilde{\mathcal{S}}_{\Gamma_0}^3 \arrow[r] \arrow[d, "\pi^3"] \& 0 \arrow[d] \\ 0 \arrow[r] \& W_{\Gamma_0}^{0, h} \arrow[r, "\mathop{\mathrm{grad}}"] \& W_{\Gamma_0}^{1, h} \arrow[r, "\mathop{\mathrm{curl}}"] \& W_{\Gamma_0}^{2, h} \arrow[r, "\undefined"] \& W_{\Gamma_0}^{3, h} \arrow[r] \& 0. \end{tikzcd}\] Thus, the two sequences have isomorphic cohomologies, the second of which is isomorphic to 6 (see [35] for the precise application of [35]), which in turn is isomorphic to 5 thanks to 8 .

The exactness of 25 follows on taking \(\mathcal{T}= K_A\) and \(\Gamma_0 = \emptyset\) and noting that \(\ker (\mathop{\mathrm{grad}}: \tilde{V}^{0, h}(K_A) \to \tilde{V}^{1, h}(K_A)) = \mathbb{R}\) and \(b_{k}(K) = 0\) for \(k \in 1:3\) since \(K\) is contractible.

6 Proof of 5↩︎

For a face \(f\in \Delta_2(\mathcal{T})\) and positive integer \(p \in \mathbb{N}_0\), denote the \(L^2(f)\)-orthogonal projection operator onto \(\mathcal{P}_p(f) \otimes \mathbb{X}^k\) by \(\mathbb{P}^k_{p, f} : L^1(f) \otimes \mathbb{X}^k \to \mathcal{P}_p(f) \otimes \mathbb{X}^k\). Similarly, for \(K \in \mathcal{T}\), let \(\mathbb{P}^k_{p, K} : L^1(K) \otimes \mathbb{X}^k \to V^{k, h}(K_A)\) denote the \(L^2(K)\)-orthogonal projection operator onto \(V^{k, h}(K_A)\).
Step 1: Construction. Let \(z \in \Delta_0(\mathcal{T})\). We choose \(f_{z} \in \Delta_2(\mathcal{T})\) with \(z \mathop{\mathrm{\unlhd}}f_{z}\), \(\tau_{z} \in \Delta_2(\mathcal{T}) \cup \mathcal{T}\) with \(z \mathop{\mathrm{\unlhd}}\tau_{z}\), and a basis for \(\mathbb{R}^3\) \(\{ \hat{\nu}_{z, j} \}_{j=1}^{3}\) as follows:

1. If \(z \in \Delta_0^{\flat}(\mathcal{U})\), let \(f_{z} \in \Delta_2(\mathcal{U})\), \(\tau_z \in \mathcal{T}\), \(\hat{\nu}_{z, j}\), \(j \in 1:2\), span the tangent plane of \(\Gamma_0\) at \(z\), and \(\hat{\nu}_{z, 3}\) be the unit outward normal of \(\partial \Omega\) at \(z\).

2. If \(z \in \Delta_0(\mathcal{U}) \setminus \Delta_0^{\flat}(\mathcal{U})\), let \(f_{z}, \tau_{z} \in \Delta_2(\mathcal{U})\) be not coplanar, \(\hat{\nu}_{z, j}\), \(j \in 1:2\) span the tangent plane of \(f_z\), and \(\hat{\nu}_{z, 3}\) be in the tangent plane of \(\tau_z\) so that \(\{\hat{\nu}_{z, j}\}_{j=1}^{3}\) is a basis for \(\mathbb{R}^3\).

3. If \(z \in \Delta_0(\mathcal{T}) \setminus \Delta_0(\mathcal{U})\), let \(f_z, \tau_z \in \Delta_2(\mathcal{T})\) be not coplanar and let \(\{\hat{\nu}_{z, j}\}_{j=1}^{3}\) be as in (b).

We define \(S_z^k \in \mathbb{R}^{3\times 3}_{\mathop{\mathrm{sym}}}\), \(k \in 0:1\), according to \[\begin{align} S_z^k : \hat{\nu}_{z, i} \otimes \hat{\nu}_{z, j} &= (\mathop{\mathrm{sym}}\mathop{\mathrm{grad}}_{f_{z}} \mathbb{P}_{4, f_{z}}^1 \mathop{\mathrm{grad}}^{1-k} v^k)(z) : \hat{\nu}_{z, i} \otimes \hat{\nu}_{z, j}, \\ S_z^k : \hat{\nu}_{z, 3} \otimes \hat{\nu}_{z, 3} &= (\mathop{\mathrm{sym}}\mathop{\mathrm{grad}}_{\tau_z} \mathbb{P}_{4, \tau_{z}}^1 \mathop{\mathrm{grad}}^{1-k} v^k)(z) : \hat{\nu}_{z, 3} \otimes \hat{\nu}_{z, 3}, \end{align}\] for all \(i \in 1:3\) and \(j \in 1:2\), where we recall that \(\mathop{\mathrm{grad}}^0 = I\), \(\mathop{\mathrm{grad}}_{f}\) for \(f \in \Delta_2(\mathcal{T})\) is the surface gradient (here taking values in \(\mathbb{R}^{3 \times 3}\)), and we set \(\mathop{\mathrm{grad}}_K := \mathop{\mathrm{grad}}\) for \(K \in \mathcal{T}\).

We define \(\tilde{\Pi}^k v^k\), \(k \in 0:3\), by assigning the degrees of freedom in 23 28 27 as follows: \[\begin{align} \label{eq:proof:fortin-current-dofs} \mathcal{I}_{\tau}^{k}( \tilde{\Pi}^k v^k) = \mathcal{I}_{\tau}^{k}(v^k) \qquad \forall \tau \in \Delta_k(\mathcal{T}), \end{align}\tag{74}\] where we note that the above conditions are well-defined since \(H^2(\Omega)\) is continuously embedded into \(C(\bar{\Omega})\) and \(v \mapsto \int_{e} v \cdot \hat{t}_e \,\mathrm{d}{s}\) is a continuous linear functional on \(H^1(\mathop{\mathrm{curl}}; \Omega)\) [40]. The remaining vertex degrees of freedom for \(z \in \Delta_0(\mathcal{T})\) are given by \[\begin{align} {2} \mathop{\mathrm{grad}}\tilde{\Pi}^0 v^0(z) &= (\mathbb{P}_{4, f_z}^{1} \mathop{\mathrm{grad}}v^0)(z), \quad & \mathop{\mathrm{hess}}\tilde{\Pi}^0 v^0(z) &= S_z^{0}, \\ \tilde{\Pi}^1 v^1(z) &= (\mathbb{P}_{4, f_z}^{1} v^1)(z), \quad & \mathop{\mathrm{grad}}\tilde{\Pi}^1 v^1(z) &= S_z^{1} + \frac{1}{2} \mathop{\mathrm{mskw}}(\mathbb{P}_{3, f_z}^{2} \mathop{\mathrm{curl}}v^1)(z), \\ & \quad & \tilde{\Pi}^2 v^2(z) &= (\mathbb{P}_{3, f_z}^{2} v^2)(z), \end{align}\] where \[\begin{align} \mathop{\mathrm{mskw}}(u) := \begin{pmatrix} 0 & -u_3 & u_2 \\ u_3 & 0 & -u_1 \\ -u_2 & u_1 & 0 \end{pmatrix} \qquad \forall u \in \mathbb{R}^3. \end{align}\] In particular, we have the formal identity \(\mathop{\mathrm{grad}}v - (\mathop{\mathrm{grad}}v)^T = \mathop{\mathrm{mskw}}(\mathop{\mathrm{curl}}v)\). We also note that above the conditions are well-defined by the trace theorem.
Step 2: Commutativity. Thanks to 44 74 , we have \[\begin{align} \mathcal{I}_{\tau}^{k+1}( \mathop{\mathrm{d}}^k \tilde{\Pi}^k v^k) = \sum_{\eta \in \Delta_{k}(\tau)} \mathcal{O}(\eta, \tau) \mathcal{I}_{\tau}^{k}(\tilde{\Pi}^k v^k) &= \sum_{\eta \in \Delta_{k}(\tau)} \mathcal{O}(\eta, \tau) \mathcal{I}_{\tau}^{k}(v^k) \\ &= \mathcal{I}_{\tau}^{k+1}( \mathop{\mathrm{d}}^k v^k) = \mathcal{I}_{\tau}^{k+1}( \tilde{\Pi}^{k+1} \mathop{\mathrm{d}}^k v^k) \end{align}\] for all \(\tau \in \Delta_{k+1}(\mathcal{T})\) and \(k \in 0:2\). Moreover, we easily see from the choice of vertex degrees of freedom that for all \(z \in \Delta_0(\mathcal{T})\), there holds \[\begin{align} (\mathop{\mathrm{grad}}^{\ell} \tilde{\Pi}^0 v^0)(z) &= \mathop{\mathrm{grad}}^{\ell-1} (\tilde{\Pi}^1 \mathop{\mathrm{grad}}v^0)(z) \qquad \ell \in 1:2 \\ (\mathop{\mathrm{curl}}\tilde{\Pi}^1 v^1)(z) &= \tilde{\Pi}^2 \mathop{\mathrm{curl}}v^1 (z), \end{align}\] and so \(\tilde{\Pi}^{k+1} \mathop{\mathrm{d}}^k v^k = \mathop{\mathrm{d}}^k \tilde{\Pi}^{k} v^k\), \(k \in 0:2\).
Step 3: Trace preservation and projection. Suppose that \(v^k \in V^k_{\Gamma_0}\). Then, we have \(\mathcal{I}^k_{\tau}(\tilde{\Pi}^k v^k) = 0\) for all \(\tau \in \Delta_k(\mathcal{U})\). Moreover, the choice of vertex degrees of freedom ensure that \[\begin{align} (I - \hat{n}_{\Gamma}(z) \otimes \hat{n}_{\Gamma}(z)) S_z^k = 0 \text{ if } z \in \Delta_0^{\flat}(\mathcal{U}) \quad \text{and} \quad S^k_z = 0 \text{ if } z \in \Delta_0(\mathcal{U}) \setminus \Delta_0^{\flat}(\mathcal{U}) \end{align}\] for \(k \in 0:1\) and additionally \[\begin{align} \mathop{\mathrm{grad}}\tilde{\Pi}^0 v^0(z) = \tilde{\Pi}^1 v^1(z) = \tilde{\Pi}^1 \mathop{\mathrm{curl}}v^1(z) = \tilde{\Pi}^2 v^2(z) = 0 \qquad \forall z \in \Delta_0(\mathcal{U}). \end{align}\] Thus, the degrees of freedom in 9 vanish, and so \(\tilde{\Pi}^k v^k \in V^k_{\Gamma_0}\).

That \(\tilde{\Pi}^k\) is a projection readily follows from the choice of degrees and ranges of the operators \(\mathbb{P}^k_{p, f_z}\) and \(\mathbb{P}^k_{p, \tau_z}\).
Step 4: Continuity. Continuity of the operators follow from standard scaling and approximation arguments (e.g. analogous arguments to proof of [40] and [30]) and are omitted for brevity.

References↩︎

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1. PB acknowledges that this work has received funding through the UKRI Digital Research Infrastructure Programme through the Science and Technology Facilities Council’s Computational Science Centre for Research Communities (CoSeC)↩︎

2. YL was partly supported through a Royal Society University Research Fellowship (URF\R1\221398, RF\ERE\221047).↩︎

3. CP was supported by an appointment to the NRC Research Associateship Program at the U.S. Naval Research Laboratory, administered by the Fellowships Office of the National Academies of Sciences, Engineering, and Medicine. Distribution Statement A. Approved for public release: distribution is unlimited.↩︎

4. In [23], \(H^k_{\Gamma_0}(\Omega)\) is defined to be the completion of smooth functions whose support is disjoint from \(\Gamma_0\), which is equivalent to 1 thanks to [25] on noting that \(\bar{\Gamma}_0\) is a closed 2-Ahlfors regular set.↩︎