1 Introduction↩︎

Finite element complexes provide a structural framework for constructing conforming finite element spaces whose unknowns are linked by differential operators. For linear elasticity in three space dimensions, the relevant continuous complex is \[\label{elascomplex3d} {\rm RM} \xrightarrow{\subset} H^1(\Omega;\mathbb{R}^3) \xrightarrow{\operatorname{def}} H(\operatorname{inc},\Omega;\mathbb{S}) \xrightarrow{\operatorname{inc}} H(\operatorname{div},\Omega;\mathbb{S}) \xrightarrow{\operatorname{div}} L^2(\Omega;\mathbb{R}^3) \to0,\tag{1}\] where \({\rm RM}\) is the space of infinitesimal rigid motions, \(\operatorname{def}=\operatorname{sym}\operatorname{grad}\) is the linearized strain, and \(\operatorname{inc}\) is the incompatibility operator. The tensor-valued Sobolev spaces in 1 are \[\begin{align} H(\operatorname{inc},\Omega;\mathbb{S}) &:=\{\boldsymbol{\tau}\in L^2(\Omega;\mathbb{S}): \operatorname{inc}\boldsymbol{\tau}\in L^2(\Omega;\mathbb{S})\}, \\ H(\operatorname{div},\Omega;\mathbb{S}) &:=\{\boldsymbol{\tau}\in L^2(\Omega;\mathbb{S}): \operatorname{div}\boldsymbol{\tau}\in L^2(\Omega;\mathbb{R}^3)\}. \end{align}\] Thus \(H(\operatorname{inc},\Omega;\mathbb{S})\) is the space for symmetric tensor fields with square-integrable incompatibility, while \(H(\operatorname{div},\Omega;\mathbb{S})\) is the natural space for symmetric stress tensors. The complex is the linear elasticity analogue of the de Rham complex and plays an important role in mixed elasticity and structure-preserving discretizations [1], [2], intrinsic elasticity and Saint-Venant compatibility conditions [3], [4], and models of defects and incompatibility [5], [6]. It also gives explicit descriptions of kernels and ranges, which are useful in stability analysis, preconditioning, and the construction of commuting projections; see, for example, [2], [7], [8].

Constructing finite element subcomplexes of 1 is delicate for two related reasons. First, the stress space must enforce both symmetry and \(H(\operatorname{div})\) conformity. Classical polynomial symmetric stress elements are stable, but they typically involve vertex degrees of freedom and relatively high polynomial degrees. Second, the preceding \(H(\operatorname{inc})\) space has nonstandard traces: tangential–tangential components and second-order surface differential information enter the Green identity for the incompatibility operator. Consequently, conforming \(H(\operatorname{inc})\) elements are substantially more constrained than standard \(H(\operatorname{curl})\)- or \(H(\operatorname{div})\)-conforming elements.

The literature contains both constructions of stable symmetric stress spaces and constructions of full finite element elasticity complexes. On simplicial meshes, the two-dimensional Arnold–Winther element [9], together with its interpretation through finite element exterior calculus and the Bernstein–Gelfand–Gelfand (BGG) construction [2], gives a conforming discretization of the elasticity complex on triangular meshes. More systematic two-dimensional BGG constructions, including elasticity and divdiv complexes with several smoothness levels, were developed in [10]. In three dimensions, stable conforming symmetric stress elements on tetrahedral meshes were constructed in [11]–[15]. A full finite element elasticity complex on tetrahedral meshes, involving an \(H(\operatorname{inc})\)-conforming tensor element, was given in [16]. Systematic three-dimensional BGG constructions, which derive finite element complexes from existing complexes and include the elasticity complex as a central example, were developed in [8].

Alfeld-type macroelements provide another route to exact elasticity sequences. In two dimensions, the Clough–Tocher split is the two-dimensional Alfeld split. It underlies the Johnson–Mercier and Arnold–Douglas–Gupta stress elements [17], [18], and also appears in the finite element system approach to elasticity and curvature [19]. In three dimensions, a complete discrete elasticity complex on tetrahedral Alfeld splits was constructed in [7], using smooth finite element de Rham complexes on Alfeld refinements [20]. This is the closest predecessor of the present work and provides the first three-dimensional Alfeld elasticity complex. Its construction, however, is tied to smoother Alfeld de Rham spaces and to a stress space whose degrees of freedom include vertex data. Related three-dimensional exact elasticity sequences on the Worsey–Farin split were developed in [21].

The Alfeld-split setting is retained here, but the terminal stress spaces are replaced by the hybridizable \(H(\operatorname{div};\mathbb{S})\)-conforming symmetric stress spaces on the Alfeld split introduced in [22], for all polynomial orders \(k\ge1\). At the lowest order, the terminal stress space together with the corresponding discontinuous displacement space recovers the Johnson–Mercier–Křížek stress–displacement pair [17], [23], [24]. These stress spaces have degrees of freedom only on faces and in element interiors. After static condensation, the globally coupled stress variables are therefore facet variables rather than vertex stress data. Relative to [7], this replacement changes not only the stress space in the sequence but also the regularity and polynomial degree required of the preceding spaces. The displacement and incompatibility spaces must therefore be chosen so that the image of \(\operatorname{inc}\) is exactly the divergence-free subspace of the hybridizable symmetric stress space and so that commuting interpolants can be defined.

At the Sobolev level, two local models are used on each tetrahedron \(T\). The first is close to the standard elasticity sequence and is used to construct the global \(H^1\)–\(H(\operatorname{inc})\) family. The second lowers the displacement regularity and leads to the global \(H^1(\operatorname{curl})\)–\(H(\operatorname{inc}^+)\) family. The smoother local sequence is \[\label{elascomplex:localH2H1inc} {\rm RM} \xrightarrow{\subset} H^2(T;\mathbb{R}^3) \xrightarrow{\operatorname{def}} H^1(\operatorname{inc},T;\mathbb{S}) \xrightarrow{\operatorname{inc}} H(\operatorname{div},T;\mathbb{S}) \xrightarrow{\operatorname{div}} L^2(T;\mathbb{R}^3) \to0,\tag{2}\] where \(H^1(\operatorname{inc},T;\mathbb{S})=H^1(T;\mathbb{S})\cap H(\operatorname{inc},T;\mathbb{S})\). The lower-regularity local sequence is \[\label{elascomplex:localH1curlinc43} {\rm RM} \xrightarrow{\subset} H^1(\operatorname{curl},T) \xrightarrow{\operatorname{def}} H(\operatorname{inc}^+,T;\mathbb{S}) \xrightarrow{\operatorname{inc}} H(\operatorname{div},T;\mathbb{S}) \xrightarrow{\operatorname{div}} L^2(T;\mathbb{R}^3) \to0,\tag{3}\] where \(H(\operatorname{inc}^+,T;\mathbb{S})=H(\operatorname{inc},T;\mathbb{S})\cap H(\operatorname{curl},T;\mathbb{S})\). The smoother local sequence gives a global finite element subcomplex of 1 , for which the assembled spaces impose the \(H^1\) and \(H(\operatorname{inc})\) traces. The lower-regularity local sequence gives a global finite element subcomplex of \[\label{elascomplex:H1curlinc43} {\rm RM} \xrightarrow{\subset} H^1(\operatorname{curl},\Omega) \xrightarrow{\operatorname{def}} H(\operatorname{inc}^+,\Omega;\mathbb{S}) \xrightarrow{\operatorname{inc}} H(\operatorname{div},\Omega;\mathbb{S}) \xrightarrow{\operatorname{div}} L^2(\Omega;\mathbb{R}^3) \to0,\tag{4}\] where \(H(\operatorname{inc}^+,\Omega;\mathbb{S})=H(\operatorname{inc},\Omega;\mathbb{S})\cap H(\operatorname{curl},\Omega;\mathbb{S})\). Thus both the \(H^1\)–\(H(\operatorname{inc})\) elasticity complex and the lower-regularity \(H^1(\operatorname{curl})\)–\(H(\operatorname{inc}^+)\) elasticity complex are discretized.

The main contributions are as follows. First, using the BGG mechanism [25] and the polynomial de Rham complexes on the Alfeld split [20], polynomial analogues of the two Sobolev elasticity sequences above, together with their bubble subcomplexes, are derived; bubble exactness and dimension formulas are established for both. Second, finite element degrees of freedom are given for the \(H^1(\operatorname{curl})\)-conforming displacement space and the \(H(\operatorname{inc}^+;\mathbb{S})\)-conforming tensor space, and a global finite element subcomplex of 4 is proved for all \(k\ge1\). In the lowest-order case, the \(H(\operatorname{inc}^+;\mathbb{S})\)-conforming tensor space is piecewise cubic, and the terminal stress–displacement pair recovers the Johnson–Mercier–Křížek element. Third, a second family is constructed for the standard \(H^1\)–\(H(\operatorname{inc})\) elasticity sequence: an \(H(\operatorname{inc};\mathbb{S})\)-conforming tensor element is introduced, its commuting properties are proved, and a global finite element subcomplex of 1 is obtained for all \(k\ge2\). Compared with the earlier three-dimensional Alfeld elasticity complex [7], these constructions lower the Sobolev regularity of the preceding spaces, reduce the polynomial degree, and end in a symmetric \(H(\operatorname{div};\mathbb{S})\)-conforming stress space with no vertex degrees of freedom; consequently, the stress space is naturally hybridizable.

Several ingredients enter the construction. The face degrees of freedom for the \(H(\operatorname{inc})\) spaces are dictated by the Green identity for the incompatibility operator and by two trace complexes on each triangular face. These trace elements are related to two-dimensional \(H(\operatorname{rot}\operatorname{rot})\), \(H(\operatorname{rot})\), \(H(\operatorname{div}\operatorname{div})\), and \(H(\operatorname{div})\) tensor elements; see [10], [13], [14], [26]. The interior degrees of freedom are tied to bubble elasticity complexes. The commuting projections use modified degrees of freedom for the hybridizable \(H(\operatorname{div};\mathbb{S})\) stress element. This organization preserves global exactness while lowering both the regularity and the polynomial degree compared with the earlier three-dimensional elasticity complex on Alfeld splits in [7].

The paper is organized as follows. Section 2 fixes notation, algebraic conventions, differential operators, and the trace identities for the incompatibility operator. Section 3 proves the local elasticity complexes on a tetrahedral Alfeld split and records the bubble exactness and dimension formulas. Section 4 constructs the finite element complex for the \(H^1(\operatorname{curl})\)–\(H(\operatorname{inc}^+)\) sequence, including unisolvence, global exactness, and commuting interpolation operators. Section 5 constructs the lower-regularity \(H^1\)–\(H(\operatorname{inc})\) finite element complex and proves the corresponding commuting diagram. The appendix contains the proofs of the bubble exactness results used in the main text.

2 Preliminaries↩︎

This section fixes the notation and conventions used throughout the paper. We first introduce the function spaces, polynomial spaces, and geometric notation for tetrahedra and their Alfeld splits. We then specify the algebraic conventions for vector–matrix products and define the differential operators appearing in the elasticity complex. Finally, we record the trace identities and Green’s formula used in the construction of the local finite element spaces.

2.1 Geometric and polynomial notation↩︎

Let \(\Omega\subset\mathbb{R}^3\) be a bounded polyhedral domain with boundary \(\partial\Omega\). For a subdomain \(D\subseteq\Omega\), we use the standard Sobolev spaces \(H^m(D)\) and \(H_0^m(D)\) and write \(L^2(D):=H^0(D)\). The \(L^2\) inner product over \(D\) is denoted by \((\cdot,\cdot)_D\), and \(L_0^2(D)\) denotes the subspace of \(L^2(D)\) consisting of functions with vanishing mean. If \(U,V\subseteq L^2(D)\), then \[U/V:=\{u\in U:(u,v)_D=0\text{ for all }v\in V\}.\] Thus \(U/V\) denotes the \(L^2(D)\)-orthogonal complement of \(V\) in \(U\), not an abstract quotient space.

For an integer \(k\ge0\), let \(\mathbb{P}_k(D)\) denote the space of polynomials on \(D\) of total degree at most \(k\), with the convention \(\mathbb{P}_k(D)=\{0\}\) for \(k<0\). The outward unit normal to \(\partial D\) is denoted by \(\boldsymbol{n}_{\partial D}\), or simply by \(\boldsymbol{n}\) when the domain is clear.

Let \(\mathcal{T}_h\) be a tetrahedral mesh of \(\Omega\), with mesh size \(h\). If \(T\) is a \(d\)-simplex, \(d=2,3\), then \(\Delta(T)\) denotes the set of all subsimplices of \(T\), and \(\Delta_\ell(T)\) denotes the set of \(\ell\)-dimensional subsimplices, \(0\le \ell\le d\). Thus \(\Delta_0(T)=\{\texttt{v}_0,\ldots,\texttt{v}_d\}\) is the vertex set and \(\Delta_d(T)=\{T\}\). Similarly, \(\Delta_\ell(\mathcal{T}_h)\) denotes the set of all \(\ell\)-dimensional subsimplices of the mesh. For a tetrahedron \(T\) with vertices \(\texttt{v}_0,\ldots,\texttt{v}_3\) and \(0\leq i\leq 3\), \(F_i\) denotes the face opposite to \(\texttt{v}_i\), and \(\boldsymbol{n}_{F_i}\) denotes the outward unit normal to \(F_i\). We write \(\lambda_i\) for the barycentric coordinate associated with \(\texttt{v}_i\), and \(\boldsymbol{t}_{i,j}\) for the tangent vector from \(\texttt{v}_i\) to \(\texttt{v}_j\).

Orientations of lower-dimensional subsimplices are fixed once and for all. For each edge \(e\), choose a unit tangent vector \(\boldsymbol{t}_e\) and two unit normal vectors \(\boldsymbol{n}_1^e\) and \(\boldsymbol{n}_2^e\). For each face \(F\), choose a unit normal vector \(\boldsymbol{n}_F\) and two tangential vectors \(\boldsymbol{t}_1^F\) and \(\boldsymbol{t}_2^F\). When no confusion can arise, these vectors are abbreviated by \(\boldsymbol{t}\), \(\boldsymbol{n}_1\) and \(\boldsymbol{n}_2\), and \(\boldsymbol{n}\), \(\boldsymbol{t}_1\) and \(\boldsymbol{t}_2\), respectively. On a conforming mesh, edge and face orientations are chosen globally rather than elementwise. In expressions such as \(\partial_n u\), the roman letter \(n\) indicates differentiation in the normal direction. If \(e\in\Delta_1(F)\), let \(\boldsymbol{n}_{F,e}\) be the unit vector tangent to \(F\) and normal to \(e\) induced by the orientation of \(F\), and set \[\boldsymbol{t}_{F,e}:=\boldsymbol{n}\times\boldsymbol{n}_{F,e}.\]

We will use the Alfeld split of a tetrahedron throughout. Let \(\texttt{v}_c\) be the barycenter of a tetrahedron \(T\). The Alfeld split \(T^{\rm R}\) is obtained by joining \(\texttt{v}_c\) to all vertices of \(T\). We denote by \(T_i\) the subtetrahedron whose vertices are \(\texttt{v}_c\) together with all vertices of \(T\) except \(\texttt{v}_i\); thus \[T^{\rm R}=\{T_i:0\le i\le3\}.\] The corresponding global Alfeld refinement of \(\mathcal{T}_h\) is denoted by \(\mathcal{T}_h^{\rm R}\). Given a collection \(\mathcal{S}\) of tetrahedra, let \(\omega_{\mathcal{S}}:=\bigcup_{T\in\mathcal{S}}T\) and define the broken polynomial space \[\mathbb{P}_k^{-1}(\mathcal{S}) :=\{q\in L^2(\omega_{\mathcal{S}}): q|_T\in\mathbb{P}_k(T)\text{ for each }T\in\mathcal{S}\}.\] The superscript \(-1\) indicates that no continuity is imposed across interfaces between elements of \(\mathcal{S}\). We also set \[\mathbb{P}_k^{\operatorname{grad}}(\mathcal{S}) :=H^1(\omega_{\mathcal{S}})\cap\mathbb{P}_k^{-1}(\mathcal{S}) =\{q\in H^1(\omega_{\mathcal{S}}): q|_T\in\mathbb{P}_k(T)\text{ for each }T\in\mathcal{S}\}.\] For a face \(F\in\Delta_2(T)\), denote by \(b_F\) the cubic face bubble function.

2.2 Algebraic conventions↩︎

We next specify the conventions for products involving vectors and matrices. Following [16], products between a vector and a matrix are interpreted according to the side on which the vector appears. For a vector \(\boldsymbol{b}\) and a matrix \(\boldsymbol{A}\), the products \[\boldsymbol{b}\cdot\boldsymbol{A}, \qquad \boldsymbol{b}\times\boldsymbol{A}\] are taken column-wise. Conversely, the products \[\boldsymbol{A}\cdot\boldsymbol{b}, \qquad \boldsymbol{A}\times\boldsymbol{b}\] are taken row-wise. Equivalently, \(\boldsymbol{b}\) is regarded as a column vector when it acts from the left and as the row vector \(\boldsymbol{b}^{\intercal}\) when it acts from the right. This convention is purely notational and avoids repeated transposes.

Since column-wise and row-wise products act on different indices, the order of mixed products is unambiguous. For example, \[\boldsymbol{b}\times\boldsymbol{A}\times\boldsymbol{c} := (\boldsymbol{b}\times\boldsymbol{A})\times\boldsymbol{c} = \boldsymbol{b}\times(\boldsymbol{A}\times\boldsymbol{c}).\] The same convention applies to mixed products such as \(\boldsymbol{b}\cdot\boldsymbol{A}\cdot\boldsymbol{c}\) and \(\boldsymbol{b}\cdot\boldsymbol{A}\times\boldsymbol{c}\), and parentheses will usually be omitted. Transposition reverses the order of the factors; moreover, a cross product changes sign. Thus \[(\boldsymbol{b}\cdot\boldsymbol{A})^{\intercal} = \boldsymbol{A}^{\intercal}\cdot\boldsymbol{b}, \qquad (\boldsymbol{b}\times\boldsymbol{A})^{\intercal} =-\boldsymbol{A}^{\intercal}\times\boldsymbol{b}.\]

For column vectors \(\boldsymbol{u}\) and \(\boldsymbol{v}\), their tensor product is \[\boldsymbol{u}\otimes\boldsymbol{v} :=\boldsymbol{u}\boldsymbol{v}^{\intercal}.\] We also write \(\boldsymbol{u}\boldsymbol{v}\) for the same rank-one matrix. With this notation, row-wise and column-wise products with another vector \(\boldsymbol{x}\) act on the adjacent factor: \[\begin{align} \boldsymbol{x}\cdot(\boldsymbol{u}\boldsymbol{v}) &= (\boldsymbol{x}\cdot\boldsymbol{u})\boldsymbol{v}^{\intercal}, & (\boldsymbol{u}\boldsymbol{v})\cdot\boldsymbol{x} &= \boldsymbol{u}(\boldsymbol{v}\cdot\boldsymbol{x}), \\ \boldsymbol{x}\times(\boldsymbol{u}\boldsymbol{v}) &= (\boldsymbol{x}\times\boldsymbol{u})\boldsymbol{v}, & (\boldsymbol{u}\boldsymbol{v})\times\boldsymbol{x} &= \boldsymbol{u}(\boldsymbol{v}\times\boldsymbol{x}). \end{align}\]

We denote by \(\mathbb{M}\) the space of all \(3\times3\) matrices, by \(\mathbb{S}\) the subspace of symmetric matrices, and by \(\mathbb{K}\) the subspace of skew-symmetric matrices. Every \(\boldsymbol{B}\in\mathbb{M}\) admits the decomposition \[\boldsymbol{B} = \operatorname{sym}\boldsymbol{B}+\operatorname{skw}\boldsymbol{B} := \frac{1}{2}(\boldsymbol{B}+\boldsymbol{B}^{\intercal}) +\frac{1}{2}(\boldsymbol{B}-\boldsymbol{B}^{\intercal}).\] The map \(\operatorname{mskw}:\mathbb{R}^3\to\mathbb{K}\) is defined by \[\operatorname{mskw}\boldsymbol{\omega} := \begin{pmatrix} 0 & -\omega_3 & \omega_2 \\ \omega_3 & 0 & -\omega_1\\ -\omega_2 & \omega_1 & 0 \end{pmatrix}, \qquad \boldsymbol{\omega}=(\omega_1,\omega_2,\omega_3)^{\intercal}.\] This map is an isomorphism. We define \[\operatorname{vskw}:\mathbb{M}\to\mathbb{R}^3, \qquad \operatorname{vskw}:= \operatorname{mskw}^{-1}\circ \operatorname{skw}.\] Finally, for a scalar function space \(B(D)\), we use the compact notation \[B(D;\mathbb{X}):=B(D)\otimes\mathbb{X}, \qquad \mathbb{X}\in\{\mathbb{R}^d,\mathbb{M},\mathbb{S},\mathbb{K}\}.\]

2.3 Differential operators and function spaces↩︎

Let \(\nabla=(\partial_1,\partial_2,\partial_3)^{\intercal}\). For a vector field \(\boldsymbol{v}\), we use \[\nabla\boldsymbol{v}:=\nabla\otimes\boldsymbol{v}, \qquad \operatorname{grad}\boldsymbol{v}:=(\nabla\boldsymbol{v})^{\intercal}, \qquad \operatorname{curl}\boldsymbol{v}:=\nabla\times\boldsymbol{v}, \qquad \operatorname{div}\boldsymbol{v}:=\nabla\cdot\boldsymbol{v}.\] The symmetric gradient, also denoted by \(\varepsilon(\boldsymbol{v})\) in elasticity, is \[\operatorname{def}\boldsymbol{v} :=\operatorname{sym}\operatorname{grad}\boldsymbol{v} =\frac{1}{2}\big(\operatorname{grad}\boldsymbol{v}+(\operatorname{grad}\boldsymbol{v})^{\intercal}\big).\] With the above convention for \(\operatorname{mskw}\), the gradient decomposes as \[\label{eq:gradudecomp} \operatorname{grad}\boldsymbol{v} =\operatorname{def}\boldsymbol{v}+\frac{1}{2}\operatorname{mskw}(\nabla\times\boldsymbol{v}).\tag{5}\]

For matrix-valued fields, \(\operatorname{curl}\) and \(\operatorname{div}\) are applied row-wise: \[\operatorname{curl}\boldsymbol{\tau}=(\nabla\times\boldsymbol{\tau}^{\intercal})^{\intercal}, \qquad \operatorname{div}\boldsymbol{\tau}=(\nabla\cdot\boldsymbol{\tau}^{\intercal})^{\intercal}.\] We use the following form of the incompatibility operator: \[\operatorname{inc}\boldsymbol{\tau}:=\operatorname{curl}S^{-1}(\operatorname{curl}\boldsymbol{\tau}),\] where \[S\boldsymbol{\sigma}:=\boldsymbol{\sigma}^{\intercal}-(\mathsf{t}\mathsf{r}\boldsymbol{\sigma})\boldsymbol{I}, \qquad S^{-1}\boldsymbol{\sigma}:=\boldsymbol{\sigma}^{\intercal} -\frac{1}{2}(\mathsf{t}\mathsf{r}\boldsymbol{\sigma})\boldsymbol{I}.\] The operator \(\operatorname{inc}\) depends only on the symmetric part of its argument: \[\operatorname{inc}\boldsymbol{\tau}=\operatorname{inc}(\operatorname{sym}\boldsymbol{\tau}).\] In particular, if \(\boldsymbol{\tau}\) is symmetric, then \[\label{eq:incandcott} \operatorname{inc}\boldsymbol{\tau} =\operatorname{curl}(\operatorname{curl}\boldsymbol{\tau})^{\intercal}.\tag{6}\]

We also require surface differential operators. Let \(F\) be a face with unit normal vector \(\boldsymbol{n}\). The tangential projection onto the plane of \(F\) is \[\Pi_F\boldsymbol{v} :=(\boldsymbol{n}\times\boldsymbol{v})\times\boldsymbol{n} =\boldsymbol{n}\times(\boldsymbol{v}\times\boldsymbol{n}) =-\boldsymbol{n}\times(\boldsymbol{n}\times\boldsymbol{v}) =(\boldsymbol{I}-\boldsymbol{n}\boldsymbol{n}^{\intercal})\boldsymbol{v}.\] Define \[\nabla_F:=\Pi_F\nabla, \qquad \nabla_F^{\bot}:=-\boldsymbol{n}\times\nabla.\] For a scalar function \(v\), \[\begin{align} \operatorname{grad}_F v=\nabla_F v &=\Pi_F(\nabla v) =-\boldsymbol{n}\times(\boldsymbol{n}\times\nabla v), \\ \operatorname{curl}_F v=\nabla_F^{\bot}v &=-\boldsymbol{n}\times\nabla v =-\boldsymbol{n}\times\nabla_F v. \end{align}\] For a vector field \(\boldsymbol{v}\), the surface divergence and surface rotation are \[\operatorname{div}_F\boldsymbol{v}:=\nabla_F\cdot\boldsymbol{v} =\nabla_F\cdot(\Pi_F\boldsymbol{v}),\] and \[\operatorname{rot}_F\boldsymbol{v} :=-\nabla_F^{\bot}\cdot\boldsymbol{v} =(\boldsymbol{n}\times\nabla)\cdot\boldsymbol{v} =\boldsymbol{n}\cdot(\nabla\times\boldsymbol{v}).\] Thus \(\operatorname{rot}_F\boldsymbol{v}\) is the normal component of \(\operatorname{curl}\boldsymbol{v}\). Define the surface deformation operator by \[\operatorname{def}_F(\boldsymbol{v}) :=\Pi_F\operatorname{def}(\boldsymbol{v})\Pi_F= \operatorname{sym}(\operatorname{grad}_F(\Pi_F\boldsymbol{v})).\]

We use the following Sobolev spaces associated with these differential operators. For a domain \(D\subset\Omega\), define \[\begin{align} H(\operatorname{div},D) &:=\{\boldsymbol{v}\in L^2(D;\mathbb{R}^3):\operatorname{div}\boldsymbol{v}\in L^2(D)\}, \\ H(\operatorname{curl},D) &:=\{\boldsymbol{v}\in L^2(D;\mathbb{R}^3): \operatorname{curl}\boldsymbol{v}\in L^2(D;\mathbb{R}^3)\}, \\ H^1(\operatorname{curl},D) &:=\{\boldsymbol{v}\in H^1(D;\mathbb{R}^3): \operatorname{curl}\boldsymbol{v}\in H^1(D;\mathbb{R}^3)\}. \end{align}\] We denote by \(H_0(\operatorname{div},D)\) and \(H_0(\operatorname{curl},D)\) the subspaces with vanishing normal and tangential traces on \(\partial D\), respectively. We further define \[H_0^1(\operatorname{curl},D):=\{\boldsymbol{v}\in H^1(\operatorname{curl},D): \boldsymbol{v}=\operatorname{curl}\boldsymbol{v}=0 \;\;\textrm{ on } \partial D\}.\] For \(\mathbb{X}\in\{\mathbb{M},\mathbb{S}\}\), the corresponding matrix-valued spaces are \[\begin{align} H(\operatorname{div},D;\mathbb{X}) &:=\{\boldsymbol{\tau}\in L^2(D;\mathbb{X}): \operatorname{div}\boldsymbol{\tau}\in L^2(D;\mathbb{R}^3)\}, \\ H(\operatorname{curl},D;\mathbb{X}) &:=\{\boldsymbol{\tau}\in L^2(D;\mathbb{X}): \operatorname{curl}\boldsymbol{\tau}\in L^2(D;\mathbb{M})\}, \\ H(\operatorname{inc},D;\mathbb{S}) &:=\{\boldsymbol{\tau}\in L^2(D;\mathbb{S}): \operatorname{inc}\boldsymbol{\tau}\in L^2(D;\mathbb{S})\}. \end{align}\] Denote by \(H_0(\operatorname{div},D;\mathbb{X})\) the subspace of \(H(\operatorname{div},D;\mathbb{X})\) with vanishing normal trace \(\boldsymbol{\tau}\boldsymbol{n}=\boldsymbol{0}\) on \(\partial D\), and by \(H_0(\operatorname{curl},D;\mathbb{X})\) the subspace of \(H(\operatorname{curl},D;\mathbb{X})\) with vanishing row-wise tangential trace on \(\partial D\). Finally, we set \[\begin{align} H^1(\operatorname{curl},D;\mathbb{M})&:=\mathbb{R}^3\otimes H^1(\operatorname{curl},D), \\ H(\operatorname{inc}^+,D;\mathbb{S})&:=H(\operatorname{inc},D;\mathbb{S})\cap H(\operatorname{curl},D;\mathbb{S}), \\ H^1(\operatorname{inc},D;\mathbb{S})&:=H(\operatorname{inc},D;\mathbb{S})\cap H^1(D;\mathbb{S}). \end{align}\]

2.4 Trace operators and Green’s identities↩︎

We now record the trace operators for \(\operatorname{inc}\) used in the construction and analysis of the local complexes. For a smooth symmetric tensor \(\boldsymbol{\tau}\) and a face \(F\), define \[\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau}):=\Pi_F\boldsymbol{\tau}\Pi_F, \qquad \mathsf{t}\mathsf{r}_2(\boldsymbol{\tau}):=2\operatorname{def}_F(\boldsymbol{n}\cdot\boldsymbol{\tau}\Pi_F)-\Pi_F\partial_n\boldsymbol{\tau}\Pi_F.\] The second trace has the following equivalent form [16]: \[\label{eq:20240319tr2inc} \mathsf{t}\mathsf{r}_2(\boldsymbol{\tau}) =\boldsymbol{n}\times(\operatorname{curl}\boldsymbol{\tau})^{\intercal}\Pi_F +\operatorname{grad}_F(\Pi_F\boldsymbol{\tau}\boldsymbol{n}) =-\Pi_F(\operatorname{curl}\boldsymbol{\tau})\times\boldsymbol{n} +\nabla_F(\boldsymbol{n}\cdot\boldsymbol{\tau}\,\Pi_F).\tag{7}\] Moreover, [16] and [7] imply \[\begin{align} \boldsymbol{n}\cdot(\operatorname{inc}\boldsymbol{\tau})\cdot\boldsymbol{n} &=\operatorname{rot}_F\operatorname{rot}_F(\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau})), \tag{8}\\ \boldsymbol{n}\times(\operatorname{inc}\boldsymbol{\tau})\cdot\boldsymbol{n} &=\operatorname{rot}_F\mathsf{t}\mathsf{r}_2(\boldsymbol{\tau}). \tag{9} \end{align}\] For a smooth vector field \(\boldsymbol{v}\), the traces of the symmetric gradient are [16] \[\begin{align} \label{eq:trdef} \mathsf{t}\mathsf{r}_1(\operatorname{def}(\boldsymbol{v})) &=\operatorname{def}_F(\Pi_F\boldsymbol{v}), & \mathsf{t}\mathsf{r}_2(\operatorname{def}(\boldsymbol{v})) &=\nabla_F^2(\boldsymbol{v}\cdot\boldsymbol{n}). \end{align}\tag{10}\]

The following edge traces enter the two-dimensional Green identity on each face. For a smooth tensor \(\boldsymbol{\tau}\) and an edge \(e\subset\partial F\), define \[\begin{align} \mathsf{t}\mathsf{r}_1^{F}(\boldsymbol{\tau}) &:=\boldsymbol{t}_e^{\intercal}\boldsymbol{\tau}\boldsymbol{t}_e,\\ \mathsf{t}\mathsf{r}_2^{F}(\boldsymbol{\tau}) &:=-\partial_{\boldsymbol{t}_e} (\boldsymbol{t}_e^{\intercal}\boldsymbol{\tau}\boldsymbol{n}_{F,e}) +\boldsymbol{t}_{F,e}^{\intercal}\operatorname{rot}_F\boldsymbol{\tau},\\ \mathsf{t}\mathsf{r}^{F}(\boldsymbol{\tau}) &:=\boldsymbol{\tau}\boldsymbol{t}_{F,e}. \end{align}\] On each edge \(e\in\Delta_1(F)\), the face trace \(\mathsf{t}\mathsf{r}_1\) is compatible with these edge traces: \[\label{eq:trrotFrotFtr1inc} \mathsf{t}\mathsf{r}_1^{F}(\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau}))=\mathsf{t}\mathsf{r}_1^{F}(\boldsymbol{\tau}), \qquad \mathsf{t}\mathsf{r}_2^{F}(\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau}))=\mathsf{t}\mathsf{r}_2^{F}(\boldsymbol{\tau}).\tag{11}\] Similarly, using 7 , one obtains \[\label{eq:trrotFtr2inc} \mathsf{t}\mathsf{r}^{F}(\mathsf{t}\mathsf{r}_2(\boldsymbol{\tau})) =\boldsymbol{n}\times(\operatorname{curl}\boldsymbol{\tau})^{\intercal}\boldsymbol{t}_{F,e} +\partial_{t_{F,e}}(\Pi_F\boldsymbol{\tau}\boldsymbol{n}).\tag{12}\]

We conclude this section with the Green identity for the operator \(\operatorname{rot}_F\operatorname{rot}_F\) on a polygonal face. This identity is the rotated counterpart of the Green identity for \(\operatorname{div}_F\operatorname{div}_F\); see [27].

3 Local complexes on the Alfeld split↩︎

This section establishes the local polynomial elasticity complexes that provide the local algebraic input for the global finite element complexes constructed below. Let \(T\) be a tetrahedron and let \(T^{\rm R}\) denote its Alfeld split. For \(k\geq1\), the smoother, \(H^2\)–\(H^1(\operatorname{inc})\) type, sequence is \[\label{eq:localelascomplex3d} {\rm RM} \xrightarrow{\subset} V_{k+3}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \xrightarrow{\operatorname{def}} \Sigma_{k+2}^{1,\operatorname{inc}}(T;\mathbb{S}) \xrightarrow{\operatorname{inc}} \Sigma_k^{\operatorname{div}}(T;\mathbb{S}) \xrightarrow{\operatorname{div}} \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3) \to 0,\tag{14}\] whereas the lower-regularity, \(H^1(\operatorname{curl})\)–\(H(\operatorname{inc}^+)\) type, sequence is \[\label{eq:localelascomplex13d} {\rm RM} \xrightarrow{\subset} V_{k+3}^{1,\operatorname{curl}}(T^{\rm R}) \xrightarrow{\operatorname{def}} \Sigma_{k+2}^{\operatorname{inc}^+}(T;\mathbb{S}) \xrightarrow{\operatorname{inc}} \Sigma_k^{\operatorname{div}}(T;\mathbb{S}) \xrightarrow{\operatorname{div}} \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3) \to 0.\tag{15}\] The two sequences have the same last two spaces and differ only in the regularity imposed on the first two nontrivial spaces. The local spaces are defined by \[\begin{align} V_{k+3}^{\operatorname{hess}}(T^{\rm R}) &:=\mathbb{P}_{k+3}^{-1}(T^{\rm R})\cap H^2(T), \\ V_{k+3}^{1,\operatorname{curl}}(T^{\rm R}) &:= \mathbb{P}_{k+3}^{-1}(T^{\rm R};\mathbb{R}^3) \cap H^1(\operatorname{curl},T), \\ \Sigma_{k+2}^{1,\operatorname{inc}}(T;\mathbb{S}) &:=\{ \boldsymbol{\tau}\in \mathbb{P}_{k+2}^{-1}(T^{\rm R};\mathbb{S}) \cap H^1(\operatorname{inc},T;\mathbb{S}): \textrm{ \boldsymbol{\tau} is C^1-continuous} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \textrm{at all vertices of T^{\rm R}} \}, \\ \Sigma_{k+2}^{\operatorname{inc}^+}(T;\mathbb{S}) &:=\{ \boldsymbol{\tau}\in \mathbb{P}_{k+2}^{-1}(T^{\rm R};\mathbb{S}) \cap H(\operatorname{inc}^+,T;\mathbb{S}): \textrm{ \boldsymbol{\tau} is continuous} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \textrm{at all vertices of T^{\rm R}} \}, \\ \Sigma_k^{\operatorname{div}}(T;\mathbb{S}) &:= \Sigma_k^{\operatorname{div}}(T;\mathbb{M})\cap\ker(\operatorname{vskw}) = \mathbb{P}_k^{-1}(T^{\rm R};\mathbb{S}) \cap H(\operatorname{div}, T;\mathbb{S}), \end{align}\] where \[\Sigma_k^{\operatorname{div}}(T;\mathbb{M}) :=\mathbb{R}^3\otimes V_k^{\operatorname{div}}(T^{\rm R}), \quad V_k^{\operatorname{div}}(T^{\rm R}) := \mathbb{P}_k^{-1}(T^{\rm R};\mathbb{R}^3) \cap H(\operatorname{div},T).\] Complexes 14 and 15 are polynomial discretizations of the local Sobolev complexes 2 and 3 , respectively.

The following dimension formulas are used in the finite element construction: \[\begin{align} \tag{16} \dim V_{k+3}^{\operatorname{hess}}(T^{\rm R})&= \binom{k+6}{3}+3\binom{k+2}{3}=\frac{2}{3}(k^3+6k^2+20k+30), \\ \tag{17} \dim V_{k+3}^{1,\operatorname{curl}}(T^{\rm R})&= (k+3)(2k^2+9k+22), \\ \tag{18} \dim\Sigma_{k+2}^{1,\operatorname{inc}}(T;\mathbb{S})& = 4k^3+21k^2+53k+60, \\ \tag{19} \dim\Sigma_{k+2}^{\operatorname{inc}^+}(T;\mathbb{S})& = 4k^3+24k^2+62k+66, \\ \tag{20} \dim \Sigma_k^{\operatorname{div}}(T;\mathbb{S}) &=(4k+3)(k+1)(k+2). \end{align}\] The first two formulas, 16 and 17 , are taken from [20]. The dimensions of the two incompatibility spaces, 18 and 19 , are derived in Lemmas 3 and 6 from the exactness of 14 and 15 , respectively. Formula 20 follows from Corollary 1.

3.1 Exactness of the smoother local complex↩︎

We first prove the exactness of 14 . The row-wise de Rham complexes give the following local BGG diagram: \[\label{eq:localbggdiagram951} \begin{tikzcd}[column sep=small] V_{k+3}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \arrow{r}{\operatorname{grad}} & \Sigma_{k+2}^{1,\operatorname{curl}}(T;\mathbb{M}) \arrow{r}{\operatorname{curl}} & \mathbb{P}_{k+1}^{\operatorname{grad}}(T^{\rm R};\mathbb{M}) \arrow{r}{\operatorname{div}} & \mathbb{P}_{k}^{-1}(T^{\rm R};\mathbb{R}^3) \to 0 \\ V_{k+2}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3)\arrow[ur,swap,"\operatorname{mskw}"'] \arrow{r}{\operatorname{grad}} & \mathbb{P}_{k+1}^{\operatorname{grad}}(T^{\rm R};\mathbb{M}) \arrow[ur,swap,"S"'] \arrow{r}{\operatorname{curl}} & \Sigma_k^{\operatorname{div}}(T;\mathbb{M}) \arrow[ur,swap,"-2\operatorname{vskw}"'] \arrow{r}{\operatorname{div}} \arrow[r] & \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3)\to0, \end{tikzcd}\tag{21}\] where \[\Sigma_{k+2}^{1,\operatorname{curl}}(T;\mathbb{M}) :=\mathbb{R}^3\otimes V_{k+2}^{1,\operatorname{curl}}(T^{\rm R}).\] Both rows are exact; see [20]. Moreover, functions in \(\Sigma_{k+2}^{1,\operatorname{curl}}(T;\mathbb{M})\) are \(C^1\)-continuous at all vertices of \(T^{\rm R}\); see [20]. The diagonal arrows encode the algebraic identities [7], [8] \[\begin{align} \notag \operatorname{div}(S\boldsymbol{\tau})&=2\operatorname{vskw}(\operatorname{curl}\boldsymbol{\tau}), \qquad \forall\,\boldsymbol{\tau}\in H(\operatorname{curl},T;\mathbb{M}), \\ \label{eq:anticommutprop2461} S \operatorname{grad}\boldsymbol{v}&=-\operatorname{curl}(\operatorname{mskw}\boldsymbol{v}), \quad\;\; \forall\,\boldsymbol{v} \in H^1(T;\mathbb{R}^3), \end{align}\tag{22}\] so the diagram 21 is anticommutative. It is the polynomial analogue of the corresponding Sobolev BGG diagram \[\begin{tikzcd} H^2(T;\mathbb{R}^3) \arrow{r}{\operatorname{grad}} & H^1(\operatorname{curl},T;\mathbb{M}) \arrow{r}{\operatorname{curl}} & H^1(T;\mathbb{M}) \arrow{r}{\operatorname{div}} & L^2(T;\mathbb{R}^3) \to 0 \\ H^2(T;\mathbb{R}^3)\arrow[ur,swap,"\operatorname{mskw}"'] \arrow{r}{\operatorname{grad}} &H^1(T;\mathbb{M}) \arrow[ur,swap,"S"'] \arrow{r}{\operatorname{curl}} & H(\operatorname{div},T;\mathbb{M}) \arrow[ur,swap,"-2\operatorname{vskw}"'] \arrow{r}{\operatorname{div}} \arrow[r] & L^2(T;\mathbb{R}^3)\to0. \end{tikzcd}\]

Applying Proposition 2.3 of [7] to this anticommutative diagram yields the following exact sequence for \(k\geq1\): \[\label{163} \begin{align} \begin{bmatrix} V_{k+3}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \\ V_{k+2}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \end{bmatrix} &\xrightarrow{[\operatorname{grad}, -\!\operatorname{mskw}]} \Sigma_{k+2}^{1,\operatorname{curl}}(T;\mathbb{M}) \xrightarrow{\operatorname{curl}S^{-1} \operatorname{curl}} \Sigma_k^{\operatorname{div}}(T;\mathbb{M}) \\ &\xrightarrow{ \begin{bmatrix} 2 \operatorname{vskw}\\ \operatorname{div} \end{bmatrix}} \begin{bmatrix} \mathbb{P}_{k}^{-1}(T^{\rm R};\mathbb{R}^3) \\ \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3) \end{bmatrix} \to 0. \end{align}\tag{23}\]

We first record two consequences.

The dimension of the incompatibility space \(\Sigma_{k+2}^{1,\operatorname{inc}}(T;\mathbb{S})\) now follows from the exact sequence 14 , thereby justifying formula 18 .

3.2 Exactness of the lower-regularity local complex↩︎

We next prove the exactness of 15 . The lower-regularity complex is obtained from an analogous local BGG diagram. The row-wise de Rham complexes now give \[\label{eq:localbggdiagram952} \begin{tikzcd}[column sep=small] V_{k+3}^{1,\operatorname{curl}}(T^{\rm R}) \arrow{r}{\operatorname{grad}} & \Sigma_{k+2}^{\operatorname{curl},\operatorname{skw}}(T;\mathbb{M}) \arrow{r}{\operatorname{curl}} & \mathbb{P}_{k+1}^{\operatorname{grad}}(T^{\rm R};\mathbb{M}) \arrow{r}{\operatorname{div}} & \mathbb{P}_{k}^{-1}(T^{\rm R};\mathbb{R}^3) \to 0 \\ V_{k+2}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3)\arrow[ur,swap,"\operatorname{mskw}"'] \arrow{r}{\operatorname{grad}} & \mathbb{P}_{k+1}^{\operatorname{grad}}(T^{\rm R};\mathbb{M}) \arrow[ur,swap,"S"'] \arrow{r}{\operatorname{curl}} & \Sigma_k^{\operatorname{div}}(T;\mathbb{M}) \arrow[ur,swap,"-2\operatorname{vskw}"'] \arrow{r}{\operatorname{div}} \arrow[r] & \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3)\to0, \end{tikzcd}\tag{27}\] where \[\begin{align} \Sigma_{k+2}^{\operatorname{curl},\operatorname{skw}}(T;\mathbb{M}) := & \{\boldsymbol{\tau}\in \mathbb{P}_{k+2}^{-1}(T^{\rm R};\mathbb{M}) \cap H(\operatorname{curl}, T;\mathbb{M}): \operatorname{vskw}\boldsymbol{\tau}\in H^1(T;\mathbb{R}^3), \\ &\quad \operatorname{curl}\boldsymbol{\tau}\in H^1(T; \mathbb{M}),\; \boldsymbol{\tau}\text{ is continuous at all vertices of }T^{\rm R}\}. \end{align}\]

Applying Proposition 2.3 of [7] to 27 yields the following exact sequence for \(k\geq1\): \[\label{263} \begin{align} \begin{bmatrix} V_{k+3}^{1,\operatorname{curl}}(T^{\rm R}) \\ V_{k+2}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \end{bmatrix} &\xrightarrow{[\operatorname{grad}, -\!\operatorname{mskw}]} \Sigma_{k+2}^{\operatorname{curl},\operatorname{skw}}(T;\mathbb{M}) \xrightarrow{\operatorname{curl}S^{-1} \operatorname{curl}} \Sigma_k^{\operatorname{div}}(T;\mathbb{M}) \\ &\xrightarrow{ \begin{bmatrix} 2 \operatorname{vskw}\\ \operatorname{div} \end{bmatrix}} \begin{bmatrix} \mathbb{P}_{k}^{-1}(T^{\rm R};\mathbb{R}^3) \\ \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3) \end{bmatrix} \to 0. \end{align}\tag{28}\]

3.3 Bubble complexes and dimension formulas↩︎

We collect the bubble exactness and dimension results needed for the finite element constructions below. The proofs are based on bubble de Rham complexes and the local BGG construction, and are given in Appendix 6.

For \(k\geq1\), define the symmetric \(H(\operatorname{div})\) bubble space by \[\mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{S}) :=H_0(\operatorname{div},T;\mathbb{S})\cap\Sigma_k^{\operatorname{div}}(T;\mathbb{S}) =H_0(\operatorname{div},T;\mathbb{S})\cap\mathbb{P}_k^{-1}(T^{\rm R};\mathbb{S}).\]

We first record the bubble results associated with the smoother local complex 14 . Define \[\begin{align} \mathbb{B}_{k+3}^{\rm herm}(T^{\rm R}) &:=V_{k+3}^{\operatorname{hess}}(T^{\rm R})\cap H_0^1(T),\\ \mathbb{B}_{k+2}^{\operatorname{inc}}(T^{\rm R};\mathbb{S}) &:=\{\boldsymbol{\tau}\in\Sigma_{k+2}^{1,\operatorname{inc}}(T;\mathbb{S}): \mathsf{t}\mathsf{r}_1(\boldsymbol{\tau})=0,\;\mathsf{t}\mathsf{r}_2(\boldsymbol{\tau})=0,\\ &\qquad \boldsymbol{\tau}\text{ and }\nabla\boldsymbol{\tau} \text{ vanish at the vertices of }T,\\ &\qquad \boldsymbol{\tau}\text{ and } (\operatorname{curl}\boldsymbol{\tau})^{\intercal}\boldsymbol{t} \text{ vanish on the edges of }T\}. \end{align}\]

We next record the corresponding results for the lower-regularity local complex 15 . Define \[\begin{align} \mathbb{B}_{k+3}^{1,\operatorname{curl}}(T^{\rm R}) &:=\{\boldsymbol{v}\in V_{k+3}^{1,\operatorname{curl}}(T^{\rm R}): \boldsymbol{v}\text{ and }\operatorname{curl}\boldsymbol{v} \text{ vanish on }\partial T\},\\ \mathbb{B}_{k+2}^{\operatorname{inc}^+}(T^{\rm R};\mathbb{S}) &:=\{\boldsymbol{\tau}\in\Sigma_{k+2}^{\operatorname{inc}^+}(T;\mathbb{S}): \boldsymbol{\tau}\text{ vanishes at all vertices of }T,\\ &\qquad \boldsymbol{\tau}\times\boldsymbol{n} \text{ and }(\operatorname{curl}\boldsymbol{\tau})^{\intercal}\times\boldsymbol{n} \text{ vanish on }\partial T\}. \end{align}\]

4 Finite element complex for the \(H^1(\operatorname{curl})\)–\(H(\operatorname{inc}^+)\) elasticity sequence↩︎

This section constructs a finite element elasticity complex for the \(H^1(\operatorname{curl})\)–\(H(\operatorname{inc}^+)\) elasticity sequence on the Alfeld refinement of a tetrahedral mesh. For \(k\geq1\), the discrete sequence is \[\label{eq:elascomplex13d} {\rm RM} \xrightarrow{\subset} V_h^{1,\operatorname{curl}} \xrightarrow{\operatorname{def}} \Sigma_h^{\operatorname{inc}^+} \xrightarrow{\operatorname{inc}} \Sigma_{k,h}^{\operatorname{div}}\xrightarrow{\operatorname{div}} V_{k-1,h}^{L^2} \to 0.\tag{37}\] Here \[V_{k-1,h}^{L^2} :=\{\boldsymbol{v}_h\in L^2(\Omega;\mathbb{R}^3): \boldsymbol{v}_h|_T\in\mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3) \textrm{ for each } T\in\mathcal{T}_h\},\] and the finite element spaces \(V_h^{1,\operatorname{curl}}\), \(\Sigma_h^{\operatorname{inc}^+}\), and \(\Sigma_{k,h}^{\operatorname{div}}\) are defined in 68 , 85 , and 53 , respectively. The sequence 37 is a finite element subcomplex of the continuous elasticity complex 4 . It should be distinguished from the finite element elasticity complex of [7], which is a subcomplex of the smoother elasticity complex \[{\rm RM} \xrightarrow{\subset} H^2(\Omega;\mathbb{R}^3) \xrightarrow{\operatorname{def}} H^1(\operatorname{inc},\Omega;\mathbb{S}) \xrightarrow{\operatorname{inc}} H(\operatorname{div},\Omega;\mathbb{S}) \xrightarrow{\operatorname{div}} L^2(\Omega;\mathbb{R}^3) \to 0.\] Commuting interpolation operators for 37 are also constructed.

4.1 Finite elements for tensors on faces↩︎

Two face finite elements are used as trace elements in the three-dimensional construction.

Let \(F\) be a triangular face and identify tangential tensors on \(F\) with two-dimensional matrices in a fixed tangential frame. The first face element controls \(\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau})=\Pi_F\boldsymbol{\tau}\Pi_F\). For \(k\geq1\), it has shape space \(\mathbb{P}_{k+2}(F;\mathbb{S}_F)\) and is conforming for \(H(\operatorname{rot}_F\operatorname{rot}_F,F;\mathbb{S}_F)\), where \(\mathbb{S}_F:=\Pi_F\mathbb{S}\Pi_F\). Its degrees of freedom are \[\tag{38} \begin{align} \boldsymbol{\tau}(\delta), & \quad\delta\in \Delta_0(F), \tag{39}\\ (\boldsymbol{\tau}\boldsymbol{t}, \boldsymbol{q})_e, & \quad\boldsymbol{q}\in\mathbb{P}_{k}(e;\mathbb{R}^2), e\in\Delta_1(F),\tag{40}\\ (\boldsymbol{t}^{\intercal}\operatorname{rot}_F\boldsymbol{\tau}, q)_e, & \quad q\in\mathbb{P}_{k+1}(e), e\in\Delta_1(F),\tag{41}\\ (\operatorname{rot}_F\operatorname{rot}_F\boldsymbol{\tau}, q)_F, & \quad q\in\mathbb{P}_{k}(F)/\mathbb{P}_{1}(F),\tag{42}\\ (\operatorname{rot}_F(\boldsymbol{\tau}), \boldsymbol{q})_F, & \quad \boldsymbol{q}\in \boldsymbol{x}(\mathbb{P}_{k-1}(F)/\mathbb{R}), \tag{43}\\ (\boldsymbol{\tau}, \boldsymbol{q})_F, & \quad \boldsymbol{q}\in (\boldsymbol{x}\otimes\boldsymbol{x})\mathbb{P}_{k-2}(F). \tag{44} \end{align}\]

The second face element controls \(\Pi_F((\operatorname{curl}\boldsymbol{\tau})^{\intercal})\Pi_F\). It is an \(H(\operatorname{rot}_F,F;\mathbb{M}_F)\) element with \(\mathbb{M}_F:=\Pi_F\mathbb{M}\Pi_F\). Its shape function space is \(\mathbb{P}_{k+1}(F;\mathbb{M}_F)\), and its degrees of freedom are \[\tag{45} \begin{align} (\boldsymbol{\tau}\boldsymbol{t}, \boldsymbol{q})_e, & \quad\boldsymbol{q}\in\mathbb{P}_{k+1}(e; \mathbb{R}^2), e\in\Delta_1(F),\tag{46}\\ (\operatorname{rot}_F\boldsymbol{\tau}, \boldsymbol{q})_F, & \quad\boldsymbol{q}\in\mathbb{P}_{k}(F;\mathbb{R}^2)/\mathbb{R}^2,\tag{47}\\ (\boldsymbol{\tau}, \boldsymbol{q})_F, & \quad\boldsymbol{q}\in \mathbb{P}_{k-1}(F;\mathbb{R}^2)\otimes\boldsymbol{x}. \tag{48} \end{align}\]

4.2 \(H(\operatorname{div};\mathbb{S})\)-conforming finite elements↩︎

We first define the final stress space in 37 . For \(k\geq1\), the local \(H(\operatorname{div})\)-conforming space for symmetric tensor fields on each tetrahedron \(T\) is \(\Sigma_k^{\operatorname{div}}(T;\mathbb{S})\); see [22]–[24]. The degrees of freedom are given by \[\tag{49} \begin{align} (\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{q})_F, & \quad \boldsymbol{q}\in\mathbb{P}_k(F;\mathbb{R}^3),\quad F\in\Delta_2(T),\tag{50}\\ (\operatorname{div}\boldsymbol{\tau},\boldsymbol{q})_T, & \quad \boldsymbol{q}\in\mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3)/{\rm RM},\tag{51}\\ (\boldsymbol{\tau},\boldsymbol{q})_T, & \quad \boldsymbol{q}\in\mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{S})\cap\ker(\operatorname{div}). \tag{52} \end{align}\] These degrees of freedom differ from those in [22]; they are chosen to facilitate the construction of a commuting projection operator.

The global \(H(\operatorname{div};\mathbb{S})\)-conforming finite element space is defined by \[\label{eq:Sigmahdiv} \Sigma_{k,h}^{\operatorname{div}} =\{\boldsymbol{\tau}_h\in H(\operatorname{div},\Omega;\mathbb{S}): \boldsymbol{\tau}_h|_T\in\Sigma_k^{\operatorname{div}}(T;\mathbb{S}) \text{ for all }T\in\mathcal{T}_h\}.\tag{53}\] Let \(I_h^{\operatorname{div}}:H^1(\Omega;\mathbb{S})\to\Sigma_{k,h}^{\operatorname{div}}\) be the interpolation operator determined by 49 , and let \(Q_h\) denote the \(L^2\) projection onto \(V_{k-1,h}^{L^2}\).

4.3 \(H^1(\operatorname{curl})\)-conforming finite elements for vector fields↩︎

The first nontrivial space in 37 is the \(H^1(\operatorname{curl})\)-conforming vector space. On each \(T\), we take the local shape function space \(V_{k+3}^{1,\operatorname{curl}}(T^{\rm R})\). Its degrees of freedom are \[\tag{56} \begin{align} \boldsymbol{v}(\delta),\;\nabla \boldsymbol{v}(\delta), & \quad \delta\in \Delta_0(T), \tag{57}\\ (\boldsymbol{v}, \boldsymbol{q})_e, & \quad \boldsymbol{q}\in\mathbb{P}_{k-1}(e;\mathbb{R}^3),\; e\in\Delta_1(T), \tag{58}\\ (\operatorname{curl}\boldsymbol{v}, \boldsymbol{q})_e, & \quad \boldsymbol{q}\in\mathbb{P}_{k}(e;\mathbb{R}^3),\; e\in\Delta_1(T), \tag{59}\\ (\operatorname{grad}_F(\Pi_F(\operatorname{curl}\boldsymbol{v})), \boldsymbol{q})_F, & \quad \boldsymbol{q}\in \mathbb{P}_{k-1}(F;\mathbb{R}^2)\otimes\boldsymbol{x},\; F\in\Delta_2(T), \tag{60}\\ (\operatorname{grad}_F(\operatorname{rot}_F\boldsymbol{v}), \boldsymbol{q})_F, & \quad \boldsymbol{q}\in \boldsymbol{x}(\mathbb{P}_{k-1}(F)/\mathbb{R}),\; F\in\Delta_2(T), \tag{61}\\ (\operatorname{grad}_F(\Pi_F\boldsymbol{v}), \boldsymbol{q})_F, & \quad \boldsymbol{q}\in (\boldsymbol{x}\otimes\boldsymbol{x})\mathbb{P}_{k-2}(F),\; F\in\Delta_2(T), \tag{62}\\ (\Pi_F\operatorname{def}(\boldsymbol{v})\boldsymbol{n}, \boldsymbol{q})_F, & \quad \boldsymbol{q}\in \boldsymbol{x} \mathbb{P}_{k}(F),\; F\in\Delta_2(T), \tag{63}\\ (\operatorname{def}(\boldsymbol{v}), \boldsymbol{q})_T, & \quad \boldsymbol{q}\in \operatorname{def}\bigl(\mathbb{B}_{k+3}^{1,\operatorname{curl}}(T^{\rm R})\bigr). \tag{64} \end{align}\] The degrees of freedom in 56 differ from those given in [20].

The global vector finite element space is defined by \[\label{eq:Vh1curlk1} \begin{align} V_h^{1,\operatorname{curl}} &:= \bigl\{ \boldsymbol{v}_h\in H^1(\Omega;\mathbb{R}^3): \;\boldsymbol{v}_h|_T\in V_{k+3}^{1,\operatorname{curl}}(T^{\rm R}) \text{ for all }T\in\mathcal{T}_h,\\ &\qquad\qquad\qquad \text{the degrees of freedom } \eqref{H1femk1dof1}\text{--}\eqref{H1femk1dof7} \text{ are single-valued} \bigr\}. \end{align}\tag{68}\] We have \(V_h^{1,\operatorname{curl}}\subseteq H^1(\operatorname{curl},\Omega)\). Let \(I_h^{1,\operatorname{curl}}:H^3(\Omega;\mathbb{R}^3)\to V_h^{1,\operatorname{curl}}\) be the interpolation operator determined by 56 .

4.4 \(H(\operatorname{inc}^+;\mathbb{S})\)-conforming finite elements↩︎

We next construct the middle space. The local shape function space is \(\Sigma_{k+2}^{\operatorname{inc}^+}(T;\mathbb{S})\). Its degrees of freedom are \[\tag{69} \begin{align} \boldsymbol{\tau}(\delta), & \quad \delta\in \Delta_0(T), \tag{70}\\ (\boldsymbol{\tau}\boldsymbol{t}, \boldsymbol{q})_e, & \quad \boldsymbol{q}\in\mathbb{P}_{k}(e;\mathbb{R}^3),\; e\in\Delta_1(T), \tag{71}\\ ((\operatorname{curl}\boldsymbol{\tau})^{\intercal}\boldsymbol{t}, \boldsymbol{q})_e, & \quad \boldsymbol{q}\in\mathbb{P}_{k+1}(e;\mathbb{R}^3),\; e\in\Delta_1(T), \tag{72}\\ (\operatorname{rot}_F(\Pi_F((\operatorname{curl}\boldsymbol{\tau})^{\intercal})\Pi_F), \boldsymbol{q})_F, & \quad \boldsymbol{q}\in\mathbb{P}_{k}(F;\mathbb{R}^2)/{\rm RM}(F),\; F\in\Delta_2(T), \tag{73}\\ (\Pi_F((\operatorname{curl}\boldsymbol{\tau})^{\intercal})\Pi_F, \boldsymbol{q})_F, & \quad \boldsymbol{q}\in \mathbb{P}_{k-1}(F;\mathbb{R}^2)\otimes\boldsymbol{x},\; F\in\Delta_2(T), \tag{74}\\ (\operatorname{rot}_F\operatorname{rot}_F(\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau})), q)_F, & \quad q\in\mathbb{P}_{k}(F)/\mathbb{P}_{1}(F),\; F\in\Delta_2(T), \tag{75}\\ (\operatorname{rot}_F(\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau})), \boldsymbol{q})_F, & \quad \boldsymbol{q}\in \boldsymbol{x}(\mathbb{P}_{k-1}(F)/\mathbb{R}),\; F\in\Delta_2(T), \tag{76}\\ (\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau}), \boldsymbol{q})_F, & \quad \boldsymbol{q}\in (\boldsymbol{x}\otimes\boldsymbol{x})\mathbb{P}_{k-2}(F),\; F\in\Delta_2(T), \tag{77}\\ (\Pi_F\boldsymbol{\tau}\boldsymbol{n}, \boldsymbol{q})_F, & \quad \boldsymbol{q}\in \boldsymbol{x}\mathbb{P}_{k}(F),\; F\in\Delta_2(T), \tag{78}\\ (\operatorname{inc}\boldsymbol{\tau}, \boldsymbol{q})_T, & \quad \boldsymbol{q}\in \mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{S})\cap\ker(\operatorname{div}), \tag{79}\\ (\boldsymbol{\tau}, \boldsymbol{q})_T, & \quad \boldsymbol{q}\in \operatorname{def}\bigl(\mathbb{B}_{k+3}^{1,\operatorname{curl}}(T^{\rm R})\bigr). \tag{80} \end{align}\]

The global \(H(\operatorname{inc}^+;\mathbb{S})\)-conforming finite element space is defined by \[\label{eq:Sigmahinc43k1} \begin{align} \Sigma_h^{\operatorname{inc}^+} &:= \bigl\{ \boldsymbol{\tau}_h\in L^2(\Omega;\mathbb{S}): \;\boldsymbol{\tau}_h|_T\in\Sigma_{k+2}^{\operatorname{inc}^+}(T;\mathbb{S}) \text{ for all }T\in\mathcal{T}_h,\\ &\qquad\qquad \text{the degrees of freedom } \eqref{Hincfem13ddof1}\text{--}\eqref{Hincfem13ddof9} \text{ are single-valued} \bigr\}. \end{align}\tag{85}\] By construction, \(\Sigma_h^{\operatorname{inc}^+} \subset H(\operatorname{inc}^+,\Omega;\mathbb{S})\). Let \(I_h^{\operatorname{inc}^+}:H^3(\Omega;\mathbb{S})\to\Sigma_h^{\operatorname{inc}^+}\) be the interpolation operator determined by the degrees of freedom 69 .

4.5 The finite element elasticity complex↩︎

We now prove the global exactness of the finite element elasticity complex.

The construction also admits the following less regular variant.

5 Finite element complex for the \(H^1\)–\(H(\operatorname{inc})\) elasticity sequence↩︎

This section constructs a finite element elasticity complex for the \(H^1\)–\(H(\operatorname{inc})\) elasticity sequence on the Alfeld refinement of a tetrahedral mesh. Throughout this section, we assume \(k\geq2\). The discrete complex is \[\label{eq:elascomplex3d} {\rm RM} \xrightarrow{\subset} V_h^{\rm herm} \xrightarrow{\operatorname{def}} \Sigma_h^{\operatorname{inc}} \xrightarrow{\operatorname{inc}} \Sigma_{k,h}^{\operatorname{div}} \xrightarrow{\operatorname{div}} V_{k-1,h}^{L^2} \to0.\tag{90}\] Here \(V_h^{\rm herm}\), \(\Sigma_h^{\operatorname{inc}}\), and \(\Sigma_{k,h}^{\operatorname{div}}\) are defined in 108 , 119 , and 53 , respectively. The sequence 90 is a finite element subcomplex of the continuous elasticity complex 1 . We also construct commuting interpolation operators for 90 .

5.1 Finite elements for symmetric tensors on faces↩︎

The \(H(\operatorname{inc})\) trace of a symmetric tensor has two components. On each face \(F\), the trace \(\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau})\) is governed by the scalar operator \(\operatorname{rot}_F\operatorname{rot}_F\), whereas \(\mathsf{t}\mathsf{r}_2(\boldsymbol{\tau})\) is governed by \(\operatorname{rot}_F\); see 8 –9 . We therefore begin with the two face finite elements that will be used as trace elements in the three-dimensional construction.

Let \(F\) be a triangular face and identify tangential symmetric tensors on \(F\) with two-dimensional symmetric matrices in a fixed tangential frame. For \(k\geq2\), the first trace element has shape space \(\mathbb{P}_{k+2}(F;\mathbb{S}_F)\) and is conforming for \(H(\operatorname{rot}_F\operatorname{rot}_F,F;\mathbb{S}_F)\). Its degrees of freedom are \[\tag{91} \begin{align} \boldsymbol{\tau}(\delta), \nabla_F\boldsymbol{\tau}(\delta), & \quad\delta\in \Delta_0(F), \tag{92}\\ (\boldsymbol{\tau}, \boldsymbol{q})_e, & \quad\boldsymbol{q}\in\mathbb{P}_{k-2}(e;\mathbb{S}_F), e\in\Delta_1(F),\tag{93}\\ (\boldsymbol{t}^{\intercal}\operatorname{rot}_F\boldsymbol{\tau}, q)_e, & \quad q\in\mathbb{P}_{k-1}(e), e\in\Delta_1(F),\tag{94}\\ (\operatorname{rot}_F\operatorname{rot}_F\boldsymbol{\tau}, q)_F, & \quad q\in\mathbb{P}_{k}(F)/\mathbb{P}_{1}(F),\tag{95}\\ (\boldsymbol{\tau}, \boldsymbol{q})_F, & \quad\boldsymbol{q}\in\operatorname{sym}(\boldsymbol{x}\otimes\mathbb{P}_{k-3}(F;\mathbb{R}^2)). \tag{96} \end{align}\]

This \(H(\operatorname{rot}_F\operatorname{rot}_F,F;\mathbb{S}_F)\) element is the rotated form of the \(H(\operatorname{div}_F\operatorname{div}_F,F;\mathbb{S}_F)\) element in [10], with smoothness vectors \(\boldsymbol{r}_1=(1,0)^{\intercal}\) and \(\boldsymbol{r}_2=-1\).

The second face element controls \(\mathsf{t}\mathsf{r}_2\). It is an \(H(\operatorname{rot}_F,F;\mathbb{S}_F)\) element with shape space \(\mathbb{P}_{k+1}(F;\mathbb{S}_F)\) and degrees of freedom \[\tag{97} \begin{align} \boldsymbol{\tau}(\delta), & \quad\delta\in \Delta_0(F), \tag{98}\\ (\boldsymbol{\tau}\boldsymbol{t}, \boldsymbol{q})_e, & \quad\boldsymbol{q}\in\mathbb{P}_{k-1}(e; \mathbb{R}^2), e\in\Delta_1(F),\tag{99}\\ (\operatorname{rot}_F\boldsymbol{\tau}, \boldsymbol{q})_F, & \quad\boldsymbol{q}\in\mathbb{P}_{k}(F; \mathbb{R}^2)/{\rm RT}(F),\tag{100}\\ (\boldsymbol{\tau}, \boldsymbol{q})_F, & \quad\boldsymbol{q}\in\boldsymbol{x}\boldsymbol{x}^{\intercal}\mathbb{P}_{k-3}(F). \tag{101} \end{align}\] This is the rotated form of the two-dimensional Hu–Zhang element; see [13] and [14]. In particular, the functionals 97 are unisolvent for \(\mathbb{P}_{k+1}(F;\mathbb{S}_F)\).

5.2 Hermite-type vector finite elements↩︎

For \(k\geq2\), we take \(V_{k+3}^{\operatorname{hess}}(T^{\rm R})\) as the scalar local shape function space. Its degrees of freedom are (cf. [20]) \[\tag{102} \begin{align} v (\delta), \nabla v (\delta), \nabla^2 v (\delta), & \quad \delta\in \Delta_0(T), \tag{103}\\ (v, q)_e, & \quad q\in\mathbb{P}_{k-3}(e), e\in\Delta_1(T),\tag{104}\\ (\partial_{n_i}v, q)_e, & \quad q\in\mathbb{P}_{k-2}(e), e\in\Delta_1(T), i=1,2,\tag{105}\\ (v, q)_F, & \quad q\in\mathbb{P}_{k-3}(F), F\in\Delta_2(T),\tag{106}\\ (v, q)_T, & \quad q\in \mathbb{B}_{k+3}^{\rm herm}(T^{\rm R}). \tag{107} \end{align}\]

Define the global vector finite element space by \[\label{eq:Vhherm} \begin{align} V_h^{\rm herm} :=\{\boldsymbol{v}_h\in H^1(\Omega;\mathbb{R}^3)&: \boldsymbol{v}_h|_T\in V_{k+3}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \text{ for all } T\in\mathcal{T}_h, \\ &\quad \text{the degrees of freedom } \eqref{H1femdof1}\text{--}\eqref{H1femdof4} \text{ are single-valued} \}. \end{align}\tag{108}\]

Let \(I_h^{\rm herm}:H^4(\Omega;\mathbb{R}^3)\to V_h^{\rm herm}\) denote the interpolation operator defined by the degrees of freedom 102 .

5.3 \(H(\operatorname{inc};\mathbb{S})\)-conforming finite elements↩︎

We now define the middle tensor space. The local shape space is \(\Sigma_{k+2}^{1,\operatorname{inc}}(T;\mathbb{S})\) and the degrees of freedom are \[\tag{109} \begin{align} \boldsymbol{\tau} (\delta), \nabla\boldsymbol{\tau} (\delta), & \quad \delta\in \Delta_0(T), \tag{110}\\ (\boldsymbol{\tau}, \boldsymbol{q})_e, & \quad \boldsymbol{q}\in\mathbb{P}_{k-2}(e;\mathbb{S}),\; e\in\Delta_1(T), \tag{111}\\ ((\operatorname{curl}\boldsymbol{\tau})^{\intercal}\boldsymbol{t}, \boldsymbol{q})_e, & \quad \boldsymbol{q}\in\mathbb{P}_{k-1}(e;\mathbb{R}^3),\; e\in\Delta_1(T), \tag{112}\\ (\operatorname{rot}_F\operatorname{rot}_F\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau}), q)_F, & \quad q\in\mathbb{P}_{k}(F)/\mathbb{P}_{1}(F),\; F\in\Delta_2(T), \tag{113}\\ (\mathsf{t}\mathsf{r}_1(\boldsymbol{\tau}), \boldsymbol{q})_F, & \quad \boldsymbol{q}\in \operatorname{sym}(\boldsymbol{x}\otimes\mathbb{P}_{k-3}(F;\mathbb{R}^2)),\; F\in\Delta_2(T), \tag{114}\\ (\operatorname{rot}_F\mathsf{t}\mathsf{r}_2(\boldsymbol{\tau}), \boldsymbol{q})_F, & \quad \boldsymbol{q}\in\mathbb{P}_{k}(F; \mathbb{R}^2)/{\rm RT}(F),\; F\in\Delta_2(T), \tag{115}\\ (\mathsf{t}\mathsf{r}_2(\boldsymbol{\tau}), \boldsymbol{q})_F, & \quad \boldsymbol{q}\in \boldsymbol{x}\boldsymbol{x}^{\intercal}\mathbb{P}_{k-3}(F),\; F\in\Delta_2(T), \tag{116}\\ (\operatorname{inc}\boldsymbol{\tau}, \boldsymbol{q})_T, & \quad \boldsymbol{q}\in \mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{S})\cap\ker(\operatorname{div}), \tag{117}\\ (\boldsymbol{\tau}, \boldsymbol{q})_T, & \quad \boldsymbol{q}\in \operatorname{def}\bigl(\mathbb{B}_{k+3}^{\rm herm}(T^{\rm R};\mathbb{R}^3)\bigr). \tag{118} \end{align}\]

The global \(H(\operatorname{inc};\mathbb{S})\)-conforming finite element space is \[\label{eq:Sigmahinc} \begin{align} \Sigma_h^{\operatorname{inc}} :=\{\boldsymbol{\tau}_h\in L^2(\Omega;\mathbb{S})&: \boldsymbol{\tau}_h|_T\in\Sigma_{k+2}^{1,\operatorname{inc}}(T;\mathbb{S}) \text{ for all }T\in\mathcal{T}_h, \\ &\quad \text{the degrees of freedom } \eqref{Hincfem3ddof1}\text{--}\eqref{Hincfem3ddof7} \text{ are single-valued} \}. \end{align}\tag{119}\] Lemma 22, the unisolvence of the degrees of freedom 97 , and the Green identity 13 show that \(\Sigma_h^{\operatorname{inc}}\subset H(\operatorname{inc},\Omega;\mathbb{S})\).

Let \(I_h^{\operatorname{inc}}:H^3(\Omega;\mathbb{S})\to\Sigma_h^{\operatorname{inc}}\) denote the interpolation operator defined by the degrees of freedom 109 .

5.4 Finite element elasticity complex↩︎

Combining the commuting properties 54 , 120 , and 121 , we obtain the following commuting diagram. \[\begin{array}{c} \xymatrix{ {\rm RM}\ar[r]^-{} &H^4(\Omega;\mathbb{R}^3)\ar[r]^-{\operatorname{def}} \ar[d]^{I^{\rm herm}_h}& H^3(\Omega;\mathbb{S}) \ar[r]^-{\operatorname{inc}} \ar[d]^{I^{\operatorname{inc}}_h}& H^1(\Omega;\mathbb{S}) \ar[r]^-{\operatorname{div}} \ar[d]^{I^{\operatorname{div}}_h}& L^{2}(\Omega;\mathbb{R}^3) \ar[r]^-{} \ar[d]^{Q_h}& 0\\ {\rm RM}\ar[r]^-{} & V_h^{\rm herm}\ar[r]^-{\operatorname{def}} & \Sigma_h^{\operatorname{inc}}\ar[r]^-{\operatorname{inc}} & \Sigma_{k,h}^{\operatorname{div}}\ar[r]^-{\operatorname{div}} & V_{k-1,h}^{L^2} \ar[r]^-{} & 0. } \end{array}\]

6 Proofs of the bubble exactness results↩︎

This appendix proves the bubble exactness and dimension results collected in Subsection 3.3. The argument applies the local BGG construction to suitable bubble de Rham complexes.

6.1 The symmetric divergence bubbles and the smoother complex↩︎

Introduce the matrix-valued bubble spaces \[\begin{align} \mathbb{B}_{k+2}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) &:=V_{k+2}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3)\cap H_0^2(T;\mathbb{R}^3),\\ \mathbb{B}_{k+2}^{1,\operatorname{curl}}(T^{\rm R}) &:=V_{k+2}^{1,\operatorname{curl}}(T^{\rm R})\cap H_0^1(\operatorname{curl},T),\\ \mathbb{B}_{k+2}^{1,\operatorname{curl}}(T^{\rm R};\mathbb{M}) &:=\mathbb{R}^3\otimes\mathbb{B}_{k+2}^{1,\operatorname{curl}}(T^{\rm R}),\\ \mathbb{B}_{k+1}^{\operatorname{grad}}(T^{\rm R};\mathbb{M}) &:=\mathbb{P}_{k+1}^{\operatorname{grad}}(T^{\rm R};\mathbb{M})\cap H_0^1(T;\mathbb{M}),\\ \mathbb{B}_{k+2}^{\operatorname{grad}}(T^{\rm R};\mathbb{X}) &:=\mathbb{P}_{k+2}^{\operatorname{grad}}(T^{\rm R};\mathbb{X})\cap H_0^1(T;\mathbb{X}), \qquad \mathbb{X}\in\{\mathbb{R}^3,\mathbb{M}\},\\ \mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{M}) &:=\{\boldsymbol{\tau}\in\Sigma_k^{\operatorname{div}}(T;\mathbb{M}) \cap H_0(\operatorname{div},T;\mathbb{M}): \int_T\operatorname{vskw}\boldsymbol{\tau}\,\,{\rm d}x=0\}. \end{align}\] Functions in \(\mathbb{B}_{k+2}^{1,\operatorname{curl}}(T^{\rm R})\) have vanishing first-order derivatives at the vertices of \(T\). The relevant bubble de Rham complexes form the diagram \[\label{eq:localbggbubblediagram951} \begin{tikzcd}[column sep=small] \mathbb{B}_{k+3}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \arrow{r}{\operatorname{grad}} & \mathbb{B}_{k+2}^{1,\operatorname{curl}}(T^{\rm R};\mathbb{M}) \arrow{r}{\operatorname{curl}} & \mathbb{B}_{k+1}^{\operatorname{grad}}(T^{\rm R};\mathbb{M}) \arrow{r}{\operatorname{div}} & \mathbb{P}_k^{-1}(T^{\rm R};\mathbb{R}^3)/\mathbb{R}^3 \to0 \\ \mathbb{B}_{k+2}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \arrow[ur,swap,"\operatorname{mskw}"'] \arrow{r}{\operatorname{grad}} & \mathbb{B}_{k+1}^{\operatorname{grad}}(T^{\rm R};\mathbb{M}) \arrow[ur,swap,"S"'] \arrow{r}{\operatorname{curl}} & \mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{M}) \arrow[ur,swap,"-2\operatorname{vskw}"'] \arrow{r}{\operatorname{div}} & \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3)/{\rm RM} \to0. \end{tikzcd}\tag{122}\]

Applying Proposition 2.3 of [7] to 122 gives the exact sequence \[\label{bubble2631} \begin{align} \begin{bmatrix} \mathbb{B}_{k+3}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3)\\ \mathbb{B}_{k+2}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \end{bmatrix} &\xrightarrow{[\operatorname{grad},-\!\operatorname{mskw}]} \mathbb{B}_{k+2}^{1,\operatorname{curl}}(T^{\rm R};\mathbb{M}) \xrightarrow{\operatorname{curl}S^{-1}\operatorname{curl}} \mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{M})\\ &\xrightarrow{\left[\begin{smallmatrix}2\operatorname{vskw}\\ \operatorname{div}\end{smallmatrix}\right]} \begin{bmatrix} \mathbb{P}_k^{-1}(T^{\rm R};\mathbb{R}^3)/\mathbb{R}^3\\ \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3)/{\rm RM} \end{bmatrix} \to0. \end{align}\tag{124}\]

The BGG construction also gives the exact complex \[\label{eq:bubbleelascomplex3d95lemma} \begin{align} \mathbb{B}_{k+3}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) &\xrightarrow{\operatorname{def}} \operatorname{sym}(\mathbb{B}_{k+2}^{1,\operatorname{curl}}(T^{\rm R};\mathbb{M})) \xrightarrow{\operatorname{inc}} \mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{S}) \xrightarrow{\operatorname{div}} \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3)/{\rm RM} \to0. \end{align}\tag{125}\] Set \[\begin{align} \mathbb{B}_{k+2}^{1,\operatorname{inc}}(T^{\rm R};\mathbb{S}) :=\{\boldsymbol{\tau}\in\Sigma_{k+2}^{1,\operatorname{inc}}(T;\mathbb{S}): &\;\operatorname{grad}\boldsymbol{\tau}\text{ vanishes at all vertices of }T,\\ &\;\boldsymbol{\tau}\text{ and } (\operatorname{curl}\boldsymbol{\tau})^{\intercal}\times\boldsymbol{n} \text{ vanish on }\partial T\}. \end{align}\]

6.2 The less regular bubble complex↩︎

Define \[\begin{align} \mathbb{B}_{k+2}^{\operatorname{curl},\operatorname{skw}}(T^{\rm R};\mathbb{M}) :=\{\boldsymbol{\tau}\in &\Sigma_{k+2}^{\operatorname{curl},\operatorname{skw}}(T;\mathbb{M}): \boldsymbol{\tau}\text{ vanishes at all vertices of }T,\\ &\boldsymbol{\tau}\times\boldsymbol{n},\;\operatorname{curl}\boldsymbol{\tau}, \text{ and }\operatorname{vskw}\boldsymbol{\tau}\text{ vanish on }\partial T\}. \end{align}\] These spaces form the diagram \[\label{eq:localbggbubblediagram952} \begin{tikzcd}[column sep=tiny] \mathbb{B}_{k+3}^{1,\operatorname{curl}}(T^{\rm R}) \arrow{r}{\operatorname{grad}} & \mathbb{B}_{k+2}^{\operatorname{curl},\operatorname{skw}}(T^{\rm R};\mathbb{M}) \arrow{r}{\operatorname{curl}} & \mathbb{B}_{k+1}^{\operatorname{grad}}(T^{\rm R};\mathbb{M}) \arrow{r}{\operatorname{div}} & \mathbb{P}_k^{-1}(T^{\rm R};\mathbb{R}^3)/\mathbb{R}^3 \to0 \\ \mathbb{B}_{k+2}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \arrow[ur,swap,"\operatorname{mskw}"'] \arrow{r}{\operatorname{grad}} & \mathbb{B}_{k+1}^{\operatorname{grad}}(T^{\rm R};\mathbb{M}) \arrow[ur,swap,"S"'] \arrow{r}{\operatorname{curl}} & \mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{M}) \arrow[ur,swap,"-2\operatorname{vskw}"'] \arrow{r}{\operatorname{div}} & \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3)/{\rm RM} \to0. \end{tikzcd}\tag{127}\]

Proposition 2.3 of [7] gives \[\label{bubble263} \begin{align} \begin{bmatrix} \mathbb{B}_{k+3}^{1,\operatorname{curl}}(T^{\rm R})\\ \mathbb{B}_{k+2}^{\operatorname{hess}}(T^{\rm R};\mathbb{R}^3) \end{bmatrix} &\xrightarrow{[\operatorname{grad},-\!\operatorname{mskw}]} \mathbb{B}_{k+2}^{\operatorname{curl},\operatorname{skw}}(T^{\rm R};\mathbb{M}) \xrightarrow{\operatorname{curl}S^{-1}\operatorname{curl}} \mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{M})\\ &\xrightarrow{\left[\begin{smallmatrix}2\operatorname{vskw}\\ \operatorname{div}\end{smallmatrix}\right]} \begin{bmatrix} \mathbb{P}_k^{-1}(T^{\rm R};\mathbb{R}^3)/\mathbb{R}^3\\ \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3)/{\rm RM} \end{bmatrix} \to0. \end{align}\tag{128}\] Consequently, \[\label{eq:bubbleelascomplex13d95lemma} \begin{align} \mathbb{B}_{k+3}^{1,\operatorname{curl}}(T^{\rm R}) &\xrightarrow{\operatorname{def}} \operatorname{sym}(\mathbb{B}_{k+2}^{\operatorname{curl},\operatorname{skw}}(T^{\rm R};\mathbb{M})) \xrightarrow{\operatorname{inc}} \mathbb{B}_k^{\operatorname{div}}(T^{\rm R};\mathbb{S}) \xrightarrow{\operatorname{div}} \mathbb{P}_{k-1}^{-1}(T^{\rm R};\mathbb{R}^3)/{\rm RM} \to0 \end{align}\tag{129}\] is exact.

Acknowledgments↩︎

The authors thank Professor Long Chen (University of California, Irvine) for helpful discussions at the Workshop on Finite Element Tensor Calculus held at the Tsinghua Sanya International Mathematics Forum. The first author thanks Professor Jeonghun Lee (Baylor University) for several insightful conversations. Part of this work was also discussed during the thematic programme “Differential Complexes: Theory, Discretization, and Applications” at the Erwin Schrödinger International Institute for Mathematics and Physics. The authors gratefully acknowledge the hospitality of both institutes.

References↩︎