Korn’s inequalities for piecewise $H^1$ vector fields
By Susanne C. Brenner
Abstract
Korn’s inequalities for piecewise $H^1$ vector fields are established. They can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.
1. Introduction
In this paper we use a boldface italic lower-case Roman letter such as $\boldsymbol{v}$ to denote a vector (or vector function) with components $v_j$($1\leq j\leq d$) and a boldface lower-case Greek letter such as $\boldsymbol{\eta }$ to denote a $d\times d$ matrix (or matrix function) with components $\eta _{ij}$($1\leq i,j\leq d$). The Euclidean norm of the vector $\boldsymbol{v}$ (resp. the Frobenius norm of the matrix $\boldsymbol{\eta }$) will be denoted by $|\boldsymbol{v}|$ (resp. $|\boldsymbol{\eta }|$).
Let $\Omega$ be a bounded connected open polyhedral domain in $\mathbb{R}^d$($d=2$ or $3$). The classical Korn inequality (cf. Reference 8, Reference 14, Reference 5 and the references therein) states that there exists a (generic) positive constant $C_{\Omega }$ such that
where $\boldsymbol{x}=(x_1,\ldots ,x_d)^t$ is the position vector function on $\Omega$ and $\mathfrak{so}(d)$ is the Lie algebra of anti-symmetric $d\times d$ matrices. The space $\mathbf{RM}(\Omega )$ is precisely the kernel of the strain tensor; i.e., for $\boldsymbol{v}\in [H^1(\Omega )]^d$,
for all $\boldsymbol{u}\in [H^1(\Omega )]^d$, where $ds$ is the infinitesimal $(d-1)$-dimensional volume and $\Gamma$ is a measurable subset of $\partial \Omega$ with a positive $(d-1)$-dimensional volume; and
In this paper we establish analogs of Equation 1.7, Equation 1.8 and Equation 1.9 for piecewise $H^1$ vector fields (functions) with respect to a partition $\mathcal{P}$ of $\Omega$ consisting of nonoverlapping polyhedral subdomains, which is not necessarily a triangulation of $\Omega$. In other words, we only assume that
$$\begin{equation*} \text{$D\cap D'=\emptyset $ if $D$ and $D'$ are distinct members of $\mathcal{P}$, and $\displaystyle \overline{\Omega }=\bigcup _{D\in \mathcal{P}} \overline{D}$.} \end{equation*}$$
Typical two- and three-dimensional examples of partitions are depicted in Figure 1, where the square is partitioned into 7 subdomains and the cube is partitioned into 5 subdomains.
The space $[H^1(\Omega ,\mathcal{P})]^d$ of piecewise $H^1$ vector fields (functions) is defined by
Let $S(\mathcal{P},\Omega )$ be the set of all the (open) sides (i.e., edges ($d=2$) or faces ($d=3$)) common to two subdomains in $\mathcal{P}$. For example, there are 10 such edges in the two-dimensional example in Figure 1 and 8 such faces in the three-dimensional example. (Precise definitions of $S(\mathcal{P},\Omega )$ will be given in Section 4 and Section 5.) For $\sigma \in S(\mathcal{P},\Omega )$, we denote by $\pi _\sigma$ the orthogonal projection operator from $[L_2(\sigma )]^d$ onto $[P_1(\sigma )]^d$, the space of vector polynomial functions on $\sigma$ of degree $\leq 1$.
The following are analogs of the classical Korn inequalities for $\boldsymbol{u}\in [H^1(\Omega ,\mathcal{P})]^d$:
where $[\boldsymbol{u}]_\sigma$ is the jump of $\boldsymbol{u}$ across the side $\sigma$ and the positive constant $C$ depends only on the shape regularity of the partition $\mathcal{P}$. In particular these inequalities are valid for partitions that are not quasi-uniform. (More details on the shape regularity assumptions are given in Section 4 and Section 5.)
for all $\boldsymbol{u}\in [H^1(\Omega ,\mathcal{P})]^d$. These estimates are useful for the analysis of discontinuous Galerkin methods for elasticity problems (cf. Reference 15, Reference 6, Reference 12 and the references therein).
The rest of the paper is organized as follows. First we derive Korn’s inequalities for piecewise linear and piecewise $H^1$ vector fields with respect to simplicial triangulations of $\Omega$. These are carried out in Section 2 and Section 3. Korn’s inequalities for piecewise $H^1$ vector fields with respect to general partitions are then established in Section 4 for two-dimensional domains and in Section 5 for three-dimensional domains. A generalization of the result in Section 2 to piecewise polynomial vector fields is given in Section 6, which can be used to derive Equation 1.14–Equation 1.16 for some nonconforming finite elements that violate Equation 1.17. The appendix contains a discussion of the dependence of the constant in Korn’s second inequality Equation 1.9 on the underlying domain, which is used in Section 3 and Section 5.
Throughout this paper we use $|S|$ to denote the $k$-dimensional volume of a $k$-dimensional geometric object $S$ in a Euclidean space.
2. A generalized Korn’s inequality for piecewise linear vector fields with respect to simplicial triangulations
In this and the next two sections we restrict our attention to the case where the partition is actually a triangulation $\mathcal{T}$ of $\Omega$ by simplexes (i.e., triangles for $d=2$ and tetrahedra for $d=3$). The intersection of the closures of any two simplexes in $\mathcal{T}$ is therefore either empty, a vertex, a closed edge or a closed face. In this case $S(\mathcal{T},\Omega )$ coincides with the set of interior open edges ($d=2$) or open faces ($d=3$). The minimum angle of the triangles or tetrahedra in $\mathcal{T}$ will be denoted by $\theta _\mathcal{T}$.
To avoid the proliferation of constants, we henceforth use the notation $A\lesssim B$ to represent the statement $A\leq \kappa (\theta _\mathcal{T})B$, where the (generic) function $\kappa :\mathbb{R}_+\longrightarrow \mathbb{R}_+$ is continuous and independent of $\mathcal{T}$. The notation $A\approx B$ is equivalent to $A\lesssim B$ and $B\lesssim A$.
Let $V_\mathcal{T}=\{\boldsymbol{v}\in [L_2(\Omega )]^d:\,\boldsymbol{v}_{_T}=\boldsymbol{v}\big |_{T}\in [P_1(T)]^d\;$$\forall \,T\in \mathcal{T}\}$ be the space of piecewise linear vector fields and $W_\mathcal{T}=\{\boldsymbol{w}\in [H^1(\Omega )]^d:\,\boldsymbol{w}_{_T}=\boldsymbol{w}\big |_{T}\in [P_1(T)]^d\;$$\forall \,T\in \mathcal{T}\}$ be the space of continuous piecewise linear vector fields. We define a linear map $E:V_\mathcal{T}\longrightarrow W_\mathcal{T}$ as follows. Let $\mathcal{V}(\mathcal{T})$ be the set of all the vertices of $\mathcal{T}$. Then $E\boldsymbol{v}$ is defined by
3. Korn’s inequalities for piecewise $H^1$ vector fields with respect to simplicial triangulations
Let $\mathcal{T}$ be a simplicial triangulation of $\Omega$. First we define on each $T\in \mathcal{T}$ an interpolation operator $\Pi _T$ from $[H^1(T)]^d$ onto $\mathbf{RM}(T)$ (the space of the rigid motions restricted to $T$) by the following conditions:
for all $T\in \mathcal{T},\;\boldsymbol{v}\in [H^1(T)]^d$, and Equation 3.1 together with the classical Poincaré-Friedrichs inequality (with scaling) yields
Let $\Pi :[H^1(\Omega ,\mathcal{T})]^d\longrightarrow V_\mathcal{T}$, the space of piecewise linear vector fields with respect to $\mathcal{T}$, be defined by
We can now prove a generalized Korn’s inequality for functions in $[H^1(\Omega ,\mathcal{T})]^d$.
Finally we observe that the seminorms in Examples 2.3–2.5 satisfy the condition Equation 3.6. In view of Equation 3.2, this is trivial for $\Phi _3$. Using Equation 3.3 and Equation 3.4, the case of $\Phi _1$ can be established as follows:
4. Korn’s inequalities for $[H^1(\Omega ,\mathcal{P})]^2$ on a two-dimensional $\Omega$
First we need a precise definition of the set $S(\mathcal{P},\Omega )$ of interior (open) edges for a general partition $\mathcal{P}$, which in turn requires the concept of a vertex of $\mathcal{P}$. We define a vertex of $\mathcal{P}$ to be a vertex of any of the subdomains in $\mathcal{P}$. (For example, the partition of the square in Figure 1 has 14 vertices.) We then define an open edge of $\mathcal{P}$ to be an open line segment on the boundary of a subdomain in $\mathcal{P}$ bounded between two of the vertices of $\mathcal{P}$. The set $S(\mathcal{P},\Omega )$ consists of the open edges of $\mathcal{P}$ that are common to the boundaries of two subdomains in $\mathcal{P}$.
In order to apply Theorem 3.1 we introduce the set
$$\begin{align} \mathfrak{T}_\mathcal{P}=\{\mathcal{T}:\,&\text{$\mathcal{T}$ is a simplicial triangulation of $\Omega $}\cssId{texmlid57}{\tag{4.1}}\\ &\text{and each member of $S(\mathcal{P},\Omega )$ is also an edge of $\mathcal{T}\}$.} \end{align}$$
By definition Equation 4.1, $[H^1(\Omega ,\mathcal{P})]^2$ is a subspace of $[H^1(\Omega ,\mathcal{T})]^2$ for every $\mathcal{T}\in \mathfrak{T}_\mathcal{P}$. Since functions in $[H^1(\Omega ,\mathcal{P})]^2$ are continuous on the edges of $\mathcal{T}$ that are not in $S(\mathcal{P},\Omega )$, the following result is an immediate consequence of Theorem 3.1.
The set $\{\theta _\mathcal{T}:\,\mathcal{T}\in \mathfrak{T}_\mathcal{P}\}$ provides an abstract measure of the shape regularity of the partition $\mathcal{P}$ and the number $\inf _{\mathcal{T}\in \mathfrak{T}_\mathcal{P}}\kappa (\theta _\mathcal{T})$ can be viewed as a constant depending on the shape regularity of $\mathcal{P}$. However, in applications one may want to relate the abstract estimate Equation 4.2 to a concrete description of the shape regularity of $\mathcal{P}$ given in terms of (i) the shape regularity of individual subdomains and (ii) the relative positions of subdomains that share a common edge of $\mathcal{P}$.
We can measure the shape regularity of a polygon (or a polyhedron in 3D) by using an affine homeomorphism between $D$ and a reference domain and by using the aspect ratio of $D$ defined by (diameter of $D$)/(radius of the largest disc (or ball) in the closure of $D$).
The relative positions between subdomains sharing a common edge of $\mathcal{P}$ can be measured in terms of the quantity
The following corollary gives an application of Theorem 4.2 to a fairly general class of two-dimensional partitions.
An example of a family of partitions satisfying the assumptions of Corollary 4.3 is depicted in Figure 4, where a square is being refined successively towards the upper right corner.
5. Korn’s inequalities for $[H^1(\Omega ,\mathcal{P})]^3$ on a three-dimensional $\Omega$
In order to give a precise definition of $S(\mathcal{P},\Omega )$, we first introduce the concept of an edge of $\mathcal{P}$, which is just an edge of any of the subdomains in $\mathcal{P}$. We then define an open face of $\mathcal{P}$ to be an open subset of the boundary of a subdomain in $\mathcal{P}$ enclosed by edges of $\mathcal{P}$. The set $S(\mathcal{P},\Omega )$ consists of open faces of $\mathcal{P}$ common to the boundaries of two subdomains in $\mathcal{P}$.
As in the two-dimensional case, we would like to derive a generalized Korn’s inequality for partitions from Theorem 3.1. But here the situation is more complicated since the faces in $S(\mathcal{P},\Omega )$ may not be triangles. Accordingly we introduce the following family of triangulations:
$$\begin{align} \mathfrak{T}_\mathcal{P}=\{\mathcal{T}:&\,\text{$\mathcal{T}$ is a simplicial triangulation of $\Omega $ such that each face}\cssId{texmlid65}{\tag{5.1}}\\ &\text{in $S(\mathcal{P},\Omega )$ is triangulated by the (triangular) faces in $S(\mathcal{T},\Omega )\}$.} \end{align}$$
Since a face in $S(\mathcal{P},\Omega )$ may not be a face in $S(\mathcal{T},\Omega )$ for $\mathcal{T}\in \mathfrak{T}_\mathcal{P}$, we cannot immediately derive an analog of Theorem 4.2. We need to introduce two more parameters related to the shape regularity of $\mathcal{P}$ in addition to the parameter $\rho (\mathcal{P})$ already defined in Equation 4.3.
Let $\mathcal{T}\in \mathfrak{T}_\mathcal{P}$. For $\sigma \in S(\mathcal{P},\Omega )$ we will denote by $\mathcal{T}_\sigma$ the triangulation of $\sigma$ by faces of $S(\mathcal{T},\Omega )$, i.e., $\mathcal{T}_\sigma =\{{\tilde{\sigma }}\in S(\mathcal{T},\Omega ):\,{\tilde{\sigma }}\subseteq \sigma \}$, and define the parameter
where $D$ is any subdomain in $\mathcal{P}$,$\boldsymbol{v}$ is any function in $[H^1(D)]^3$ and $\Pi _{D}:[H^1(D)]^3 \longrightarrow \mathbf{RM}(D)$ (the space of rigid motions restricted to $D$) is defined by the conditions
The existence of $\lambda (\mathcal{P})$ is a consequence of Equation 1.4, Equation 5.6, the trace theorem, the Poincaré-Friedrichs inequality and Korn’s second inequality Equation 1.9.
We can now state and prove a generalized Korn’s inequality.
The set $\{\big (\rho (\mathcal{P}),\lambda (\mathcal{P}),\rho (\mathcal{P},\mathcal{T}),\theta _\mathcal{T}\big ):\,\mathcal{T}\in \mathfrak{T}_\mathcal{P}\}$ provides an abstract measure of the shape regularity of the partition $\mathcal{P}$ and we can think of
as a constant depending on the shape regularity of $\mathcal{P}$. Under appropriate concrete shape regularity assumptions one can also obtain from Theorem 5.2 Korn’s inequalities for a family of partitions with a uniform constant. For simplicity we only give an analog of Corollary 4.3 for partitions by convex polyhedra.
Since a face of a partition $\mathcal{P}$ consisting of convex polyhedra is a convex polygon, it can be triangulated by connecting its center to the vertices of $\mathcal{P}$ on its boundary by straight lines. Such a triangulation will be referred to as the canonical triangulation of the face.
An example of a family of partitions satisfying the assumptions of Corollary 5.3 is depicted in Figure 5, where a cube is being refined successively towards the upper left front corner.
6. Korn’s inequalities for piecewise polynomial vector fields with respect to triangulations by polyhedral subdomains
Attentive readers may have already noticed that the inequality Equation 2.9 for piecewise linear vector fields is different from the inequalities Equation 3.7, Equation 4.2 and Equation 5.7 for piecewise $H^1$ vector fields. Since pointwise evaluation is not well defined for functions in $[H^1(T)]^d$ and $d\geq 2$, the formulation of Korn’s inequalities in Equation 3.7, Equation 4.2 and Equation 5.7 is the appropriate one for piecewise $H^1$ vector fields. However, for piecewise polynomial vector fields associated with a triangulation $\mathcal{T}$ (simplicial or otherwise), pointwise evaluation of the jump across $\sigma \in S(\mathcal{T},\Omega )$ is possible. The following theorem generalizes Lemma 2.2 to such vector fields.
for all $\boldsymbol{v}\in [H^1(D)]^d$. In this appendix we briefly discuss the behavior of $k(D)$ under affine homeomorphisms. More precisely, we assume that $D$ is homeomorphic to a reference domain $\hat{D}$ under the affine transformation $\alpha :\hat{D}\longrightarrow D$ defined by $\alpha (\hat{x})=B\hat{x}+b$, and we consider the dependence of $k(D)$ on $B\in GL(d)$, the Lie group of nonsingular $d\times d$ matrices.
Without loss of generality, we may assume $b=0$ and write the constant $k(D)$ in Equation A.1 as $k(B)$. We have,
Observe that, since $|\hat{\boldsymbol{v}}|_{H^1(\hat{D})}=1$, the quotients on the right-hand side of Equation A.2 form a family of equicontinuous functions on $GL(d)$. Therefore $k(\cdot )$, defined as the supremum of this equicontinuous family, is continuous on $GL(d)$ (cf. Reference 7).
For a simplex $T$ we can also control the constant $k(T)$ in terms of the minimum angle $\theta _T$ of $T$.
$$\begin{align} \mathfrak{T}_\mathcal{P}=\{\mathcal{T}:\,&\text{$\mathcal{T}$ is a simplicial triangulation of $\Omega $}\cssId{texmlid57}{\tag{4.1}}\\ &\text{and each member of $S(\mathcal{P},\Omega )$ is also an edge of $\mathcal{T}\}$.} \end{align}$$
$$\begin{align} \mathfrak{T}_\mathcal{P}=\{\mathcal{T}:&\,\text{$\mathcal{T}$ is a simplicial triangulation of $\Omega $ such that each face}\cssId{texmlid65}{\tag{5.1}}\\ &\text{in $S(\mathcal{P},\Omega )$ is triangulated by the (triangular) faces in $S(\mathcal{T},\Omega )\}$.} \end{align}$$
$$\begin{equation} S_\theta =\{k(T):\, \text{$T$ is a simplex and the minimum angle of $T$ is $\geq \theta $}\} \cssId{texmlid95}{\tag{A.5}} \end{equation}$$
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