Orthogonality relations of Crouzeix–Raviart and Raviart–Thomas finite element spaces

Abstract

Identities that relate projections of Raviart–Thomas finite element vector fields to discrete gradients of Crouzeix–Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix–Ravaiart space onto elementwise constant functions are deduced.

1 Introduction

Recent developments in the numerical analysis of total variation regularized and related nonsmooth minimization problems show that nonconforming and discontinuous finite element methods lead to optimal convergence rates under suitable regularity conditions [3, 4, 10]. This is in contrast to standard conforming methods which often perform suboptimally [6]. A key ingredient in the derivation of quasi-optimal error estimates are discrete convex duality results which exploit relations between Crouzeix–Raviart and Raviart–Thomas finite element spaces introduced in [11] and [13]. In particular, assume that $\Omega \subset {\mathbb {R}}^d$ is a bounded Lipschitz domain with a partitioning of the boundary into subsets ${\Gamma _N},{\Gamma _D}\subset \partial \Omega$, and let ${\mathcal {T}}_h$ be a regular triangulation of $\Omega$. For a function $v_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ and a vector field $y_h\in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ we then have the integration-by-parts formula

$$\begin{aligned} \int _\Omega \nabla _{\! h}v_h \cdot y_h \,{\mathrm d}x = - \int _\Omega v_h {{\,\mathrm{div}\,}}y_h \,{\mathrm d}x. \end{aligned}$$

Important aspects here are that despite the possible discontinuity of $v_h$ and $y_h$ no terms occur that are related to interelement sides and that the vector field $y_h$ and the function $v_h$ can be replaced by their elementwise averages on the left- and right-hand side, respectively. In combination with Fenchel’s inequality this implies a weak discrete duality relation.

The validity of a strong discrete duality principle has been established in [4, 10] under certain differentiability or more generally approximability properties of minimization problems using the orthogonality relation

$$\begin{aligned} \big (\Pi _h {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)\big )^\perp = \nabla _{\! h}\big (\ker \Pi _h |_{{\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)}\big ), \end{aligned}$$

(1)

within the space of piecewise constant vector fields ${\mathcal {L}}^0({\mathcal {T}}_h)^d$ equipped with the $L^2$ inner product and with $\nabla _{\! h}$ and $\Pi _h$ denoting the elementwise application of the gradient and orthogonal projection onto ${\mathcal {L}}^0({\mathcal {T}}_h)^d$, respectively, $\ker$ denotes the kernel of an operator. The identity implies that if a vector field $w_h\in {\mathcal {L}}^0({\mathcal {T}}_h)^d$ satisfies

$$\begin{aligned} \int _\Omega w_h \cdot \nabla _{\! h}v_h \,{\mathrm d}x = 0 \end{aligned}$$

for all $v_h \in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ with $\Pi _h v_h=0$ then there exists a vector field $z_h \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ such that

$$\begin{aligned} w_h = \Pi _h z_h. \end{aligned}$$

Note that this is a stronger implication then the well known result that if $w_h$ is orthogonal to discrete gradients of all Crouzeix–Raviart functions then it belongs to the Raviart–Thomas finite element space. Although strong duality is not required in the error analysis, it reveals a compatibility property of discretizations and indicates optimality of estimates. Moreover, it is related to postprocessing procedures that provide the solution of computationally expensive discretized dual problems via simple postprocessing procedures of numerical solutions of less expensive primal problems, cf. [1, 4, 9, 12].

The proof of (1) given in [10] makes use of a discrete Poincaré lemma which is valid if the Dirichlet boundary ${\Gamma _D}\subset \partial \Omega$ is empty or if $d=2$ and ${\Gamma _D}$ is connected. In this note we show that (1) can be established for general boundary partitions by avoiding the use of the discrete Poincaré lemma. The new proof is based on the surjectivity property of the discrete divergence operator

$$\begin{aligned} {{\,\mathrm{div}\,}}: {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)\rightarrow {\mathcal {L}}^0({\mathcal {T}}_h). \end{aligned}$$

This is a fundamental property for the use of the Raviart–Thomas method for discretizing saddle-point problems, cf. [5, 13]. It is an elementary consequence of a projection property of a quasi-interpolation operator ${\mathcal {I}}_{{\mathcal {R}}T}:H^s(\Omega ;{\mathbb {R}}^d)\rightarrow {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ and the surjectivity of the divergence operator onto the space $L^2(\Omega )$.

Our arguments also provide a dual version of the orthogonality relation (1) which states that

$$\begin{aligned} {{\,\mathrm{div}\,}}\big (\ker \Pi _h |_{{{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)}\big ) = \big (\Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)\big )^\perp . \end{aligned}$$

(2)

Unless ${\Gamma _D}= \partial \Omega$ we have that the left-hand side is trivial and hence the identity yields that

$$\begin{aligned} \Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)= {\mathcal {L}}^0({\mathcal {T}}_h), \end{aligned}$$

i.e., that the projection of Crouzeix–Raviart functions onto elementwise constant functions is a surjection. If ${\Gamma _D}=\partial \Omega$ then depending on the triangulation both equality or strict inclusion occur. This observation reveals that the discretizations of total-variation regularized problems devised in [4, 10] can be seen as discretizations using elementwise constant functions with suitable nonconforming discretizations of the total variation functional.

The most important consequence of (2) is the strong duality relation for the discrete primal problem defined by minimizing the functional

$$\begin{aligned} I_h(u_h) = \int _\Omega \phi (\nabla _{\! h}u_h) + \psi _h(x,\Pi _h u_h) \,{\mathrm d}x \end{aligned}$$

in the space ${\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ and the discrete dual problem consisting in maximizing the functional

$$\begin{aligned} D_h(z_h) = - \int _\Omega \phi ^*(\Pi _h z_h) + \psi _h^*(x,{{\,\mathrm{div}\,}}z_h) \,{\mathrm d}x \end{aligned}$$

in the space ${{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$. The functions $\phi$ and $\psi _h$ are suitable convex functions with convex conjugates $\phi ^*$ and $\psi _h^*$, we refer the reader to [4] for details.

This article is organized as follows. In Sect. 2 we define the required finite element spaces along with certain projection operators. Our main results are contained in Sect. 3, where we prove the identities (1) and (2) and deduce various corollaries. In the Appendix A we provide a proof of the discrete Poincaré lemma that leads to an alternative proof of the main result under certain restrictions.

2 Preliminaries

2.1 Triangulations

Throughout what follows we let $({\mathcal {T}}_h)_{h>0}$ be a sequence of regular triangulations of the bounded polyhedral Lipschitz domain $\Omega \subset {\mathbb {R}}^d$, cf. [7, 8]. We let $P_k(T)$ denote the set of polynomials of maximal total degree k on $T\in {\mathcal {T}}_h$ and define the set of elementwise polynomial functions or vector fields

$$\begin{aligned} {\mathcal {L}}^k({\mathcal {T}}_h)^\ell = \{ w_h \in L^\infty (\Omega ;{\mathbb {R}}^\ell ): w_h|_T \in P_k(T) \text { for all }T\in {\mathcal {T}}_h\}. \end{aligned}$$

The parameter $h>0$ refers to the maximal mesh-size of the triangulation ${\mathcal {T}}_h$. The set of sides of elements is denoted by ${\mathcal {S}}_h$. We let $x_S$ and $x_T$ denote the midpoints (barycenters) of sides and elements, respectively. The $L^2$ projection onto piecewise constant functions or vector fields is denoted by

$$\begin{aligned} \Pi _h : L^1(\Omega ;{\mathbb {R}}^\ell ) \rightarrow {\mathcal {L}}^0({\mathcal {T}}_h)^\ell . \end{aligned}$$

For $v_h\in {\mathcal {L}}^1({\mathcal {T}}_h)^\ell$ we have $\Pi _h v_h|_T = v_h(x_T)$ for all $T\in {\mathcal {T}}_h$. We repeatedly use that $\Pi _h$ is self-adjoint, i.e.,

$$\begin{aligned} (\Pi _h v,w) = (v,\Pi _h w) \end{aligned}$$

for all $v,w\in L^1(\Omega ;{\mathbb {R}}^\ell )$ with the $L^2$ inner product $(\cdot ,\cdot )$.

2.2 Crouzeix–Raviart finite elements

A particular instance of a larger class of nonconforming finite element spaces introduced in [11] is the Crouzeix–Raviart finite element space which consists of piecewise affine functions that are continuous at the midpoints of sides of elements, i.e.,

$$\begin{aligned} {\mathcal {S}}^{1,{cr}}({\mathcal {T}}_h)= \{v_h \in {\mathcal {L}}^1({\mathcal {T}}_h): v_h \text { continuous in}\ x_S \text {for all }\ S\in {\mathcal {S}}_h \}. \end{aligned}$$

The elementwise application of the gradient operator to a function $v_h\in {\mathcal {S}}^{1,{cr}}({\mathcal {T}}_h)$ defines an elementwise constant vector field $\nabla _{\! h}v_h$ via

$$\begin{aligned} \nabla _{\! h}v_h|_T = \nabla (v_h|_T) \end{aligned}$$

for all $T\in {\mathcal {T}}_h$. For $v\in W^{1,1}(\Omega )$ we have $\nabla _{\! h}v = \nabla v$. Functions vanishing at midpoints of boundary sides on ${\Gamma _D}$ are contained in

$$\begin{aligned} {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)= \{v_h\in {\mathcal {S}}^{1,{cr}}({\mathcal {T}}_h): v_h(x_S)=0 \hbox { for all} S\in {\mathcal {S}}_h \hbox {with} S\subset {\Gamma _D}\}. \end{aligned}$$

A basis of the space ${\mathcal {S}}^{1,{cr}}({\mathcal {T}}_h)$ is given by the functions $\varphi _S \in {\mathcal {S}}^{1,{cr}}({\mathcal {T}}_h)$, $S\in {\mathcal {S}}_h$, satisfying the Kronecker property

$$\begin{aligned} \varphi _S(x_{S'}) = \delta _{S,S'} \end{aligned}$$

for all $S,S'\in {\mathcal {S}}_h$; a basis for the subspace ${\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ is obtained by eliminating those functions $\varphi _S$ that correspond to sides $S\subset {\Gamma _D}$. The function $\varphi _S$ vanishes on elements that do not contain the side S and is continuous with value 1 along S. A quasi-interpolation operator is for $v\in W^{1,1}(\Omega )$ defined via

$$\begin{aligned} {\mathcal {I}}_{cr}v = \sum _{S\in {\mathcal {S}}_h} v_S \varphi _S, \quad v_S = |S|^{-1} \int _S v \,{\mathrm d}s, \end{aligned}$$

We have that ${\mathcal {I}}_{cr}$ preserves averages of gradients, i.e.,

$$\begin{aligned} \nabla _{\! h}{\mathcal {I}}_{cr}v = \Pi _h \nabla v, \end{aligned}$$

which follows from an integration by parts, cf. [2, 5].

2.3 Raviart–Thomas finite elements

The Raviart–Thomas finite element space of [13] is defined as

$$\begin{aligned}\begin{aligned} {{\mathcal {R}}T}^0({\mathcal {T}}_h)= \{y_h\in&H({{\,\mathrm{div}\,}};\Omega ): y_h|_T(x) = a_T + b_T (x-x_T), \\&a_T\in {\mathbb {R}}^d, \, b_T\in {\mathbb {R}}\hbox { for all}\ T\in {\mathcal {T}}_h \}, \end{aligned}\end{aligned}$$

where $H({{\,\mathrm{div}\,}};\Omega )$ is the set of $L^2$ vector fields whose distributional divergence belongs to $L^2(\Omega )$. Vector fields in ${{\mathcal {R}}T}^0({\mathcal {T}}_h)$ have continuous constant normal components on element sides. The subset of vector fields with vanishing normal component on the Neumann boundary ${\Gamma _N}$ is defined as

$$\begin{aligned} {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)= \{ y_h\in {{\mathcal {R}}T}^0({\mathcal {T}}_h): y_h \cdot n = 0 \hbox { on}\ {\Gamma _N}\}, \end{aligned}$$

where n is the outer unit normal on $\partial \Omega$. A basis of the space ${{\mathcal {R}}T}^0({\mathcal {T}}_h)$ is given by vector fields $\psi _S$ associated with sides $S\in {\mathcal {S}}_h$; the subspace ${{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ is the span of functions $\psi _S$ with $S\not \subset {\overline{\Gamma }}_N$. Each vector field $\psi _S$ is supported on adjacent elements $T_\pm \in {\mathcal {T}}_h$ with

$$\begin{aligned} \psi _S(x) = \pm \frac{|S|}{d |T_\pm |} (z_{S,T_\pm } - x) \end{aligned}$$

(3)

for $x\in T_\pm$ with opposite vertex $z_{S,T_\pm }$ to $S\subset \partial T_\pm$. We have the Kronecker property

$$\begin{aligned} \psi _S|_{S'} \cdot n_{S'} = \delta _{S,S'} \end{aligned}$$

for all sides $S'$ with unit normal vector $n_{S'}$, if $S'=S$ we assume that $n_S$ points from $T_-$ into $T_+$. A quasi-interpolation operator is for vector fields $z\in W^{1,1}(\Omega ;{\mathbb {R}}^d)$ given by

$$\begin{aligned} {\mathcal {I}}_{{\mathcal {R}}T}z = \sum _{S\in {\mathcal {S}}_h} z_S \psi _S, \quad z_S = |S|^{-1} \int _S z \cdot n_S \,{\mathrm d}s. \end{aligned}$$

For the operator ${\mathcal {I}}_{{\mathcal {R}}T}$ we have the projection property

$$\begin{aligned} {{\,\mathrm{div}\,}}{\mathcal {I}}_{{\mathcal {R}}T}z = \Pi _h {{\,\mathrm{div}\,}}z, \end{aligned}$$

which is a consequence of an integration by parts, cf. [2, 5]. This identity implies that the divergence operator defines a surjection from ${{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ into ${\mathcal {L}}^0({\mathcal {T}}_h)$, provided that constants are eliminated from ${\mathcal {L}}^0({\mathcal {T}}_h)$ if ${\Gamma _D}= \emptyset$.

2.4 Integration by parts

An elementwise integration by parts implies that for $v_h\in {\mathcal {S}}^{1,{cr}}({\mathcal {T}}_h)$ and $y_h\in {{\mathcal {R}}T}^0({\mathcal {T}}_h)$ we have the integration-by-parts formula

$$\begin{aligned} \int _\Omega \nabla _h v_h \cdot y_h \,{\mathrm d}x + \int _\Omega v_h {{\,\mathrm{div}\,}}y_h \,{\mathrm d}x = \int _{\partial \Omega } v_h \, y_h \cdot n \,{\mathrm d}s. \end{aligned}$$

(4)

Here we used that $y_h$ has continuous constant normal components on inner element sides and that jumps of $v_h$ have vanishing integral mean. If an elementwise constant vector field $w_h\in {\mathcal {L}}^0({\mathcal {T}}_h)^d$ satisfies

$$\begin{aligned} \int _\Omega w_h \cdot \nabla _{\! h}v_h \,{\mathrm d}x = 0 \end{aligned}$$

for all $v_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ then by choosing $v_h = \varphi _S$ for $S\in {\mathcal {S}}_h\setminus {\Gamma _D}$ one finds that its normal components are continuous on inner element sides and vanish on the ${\Gamma _N}$, so that $w_h \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$. We thus have the decomposition

$$\begin{aligned} {\mathcal {L}}^0({\mathcal {T}}_h)^d = \ker ({{\,\mathrm{div}\,}}|_{{{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)}) \oplus \nabla _{\! h}{\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h), \end{aligned}$$

where we used that $\ker ({{\,\mathrm{div}\,}}|_{{{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)}) = {\mathcal {L}}^0({\mathcal {T}}_h)^d\cap {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$.

3 Orthogonality relations

The following identities and in particular their proofs and corollaries are the main contributions of this article.

Theorem 3.1

(Orthogonality relations) Within the sets of elementwise constant vector fields and functions ${\mathcal {L}}^0({\mathcal {T}}_h)^\ell$ equipped with the $L^2$ inner product we have

$$\begin{aligned}\begin{aligned} \big (\Pi _h {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)\big )^\perp&= \nabla _{\! h}\big (\ker \Pi _h|_{{\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)} \big ), \\ {{\,\mathrm{div}\,}}\big (\ker \Pi _h |_{{{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)} \big )&= \big (\Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)\big )^\perp . \end{aligned}\end{aligned}$$

Proof

(i) The integration-by-parts formula (4) implies

$$\begin{aligned} (\nabla _{\! h}v_h,\Pi _h y_h) = - (v_h, {{\,\mathrm{div}\,}}y_h) = - (\Pi _h v_h,{{\,\mathrm{div}\,}}y_h) = 0 \end{aligned}$$

if $\Pi _h v_h =0$ and hence $\Pi _h {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)\subset \big [\nabla _{\! h}\big (\ker \Pi _h|_{{\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)} \big )\big ]^\perp$. To prove the converse inclusion let $y_h \in {\mathcal {L}}^0({\mathcal {T}}_h)^d$ be orthogonal to $\nabla _{\! h}\big (\ker \Pi _h|_{{\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)} \big )$. We show that there exists ${\widetilde{y}}_h\in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ with $\Pi _h {\widetilde{y}}_h = y_h$. For this, let $Z_h = \big (\ker \Pi _h|_{{\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)} \big )^\perp \subset {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ and $r_h \in Z_h$ be the uniquely defined function with

$$\begin{aligned} (\Pi _h r_h,\Pi _h v_h) = (y_h,\nabla _{\! h}v_h) \end{aligned}$$

for all $v_h\in Z_h$. The identity holds for all $v_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ since $y_h$ is orthogonal to discrete gradients of functions $v_h \in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ with $\Pi _h v_h = 0$. In particular, $\Pi _h r_h$ is orthogonal to constant functions if ${\Gamma _D}= \emptyset$. We choose $z_h \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ with $-{{\,\mathrm{div}\,}}z_h = \Pi _h r_h$ and verify that

$$\begin{aligned} (y_h - z_h,\nabla _{\! h}v_h) = (\Pi _h r_h,\Pi _h v_h) + ({{\,\mathrm{div}\,}}z_h,v_h) = 0 \end{aligned}$$

for all $v_h \in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$. We next define ${\widetilde{y}}_h|_T = y_h|_T + d^{-1} (x-x_T) {{\,\mathrm{div}\,}}z_h|_T$ for all $T\in {\mathcal {T}}_h$ and note that

$$\begin{aligned} ({\widetilde{y}}_h - z_h,\nabla _{\! h}v_h) = (y_h-z_h,\nabla _{\! h}v_h) = 0 \end{aligned}$$

for all $v_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$. Since ${\widetilde{y}}_h -z_h$ is elementwise constant, it follows that ${\widetilde{y}}_h-z_h\in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ and in particular ${\widetilde{y}}_h \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$. By definition of ${\widetilde{y}}_h$ we have $\Pi _h {\widetilde{y}}_h = y_h$ which proves the first asserted identity.

(ii) For the second statement we first note that if $\Pi _h y_h = 0$ for $y_h\in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ then

$$\begin{aligned} (\Pi _h v_h,{{\,\mathrm{div}\,}}y_h) = (v_h, {{\,\mathrm{div}\,}}y_h) = - (\nabla _{\! h}v_h,\Pi _h y_h) = 0 \end{aligned}$$

for all $v_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ and hence ${{\,\mathrm{div}\,}}y_h \in \big (\Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h) \big )^\perp$. It remains to show that

$$\begin{aligned} \big (\Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h) \big )^\perp \subset {{\,\mathrm{div}\,}}\big (\ker \Pi _h |_{{{\mathcal {R}}T}^0_N({\mathcal {T}}_h)}\big ). \end{aligned}$$

If $w_h \in {\mathcal {L}}^0({\mathcal {T}}_h)$ is orthogonal to $\Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ we choose $z_h \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ with ${{\,\mathrm{div}\,}}z_h = w_h$ and note that

$$\begin{aligned} (\Pi _h z_h ,\nabla _{\! h}v_h) = (z_h,\nabla _{\! h}v_h) = -(w_h, v_h) = -(w_h,\Pi _h v_h) = 0 \end{aligned}$$

for all $v_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$. This implies that $\Pi _h z_h\in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ and hence also $y_h = z_h -\Pi _h z_h\in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$. Since $\Pi _h y_h = 0$ and ${{\,\mathrm{div}\,}}y_h = w_h$ we deduce the second identity. $\square$

An implication is a surjectivity property of the mapping $\Pi _h:{\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)\rightarrow {\mathcal {L}}^0({\mathcal {T}}_h)$ if ${\Gamma _D}\ne \partial \Omega$.

Corollary 3.2

(Surjectivity) If ${\Gamma _D}\ne \partial \Omega$ then we have

$$\begin{aligned} \Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)= {\mathcal {L}}^0({\mathcal {T}}_h). \end{aligned}$$

Otherwise, the subspace $\Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)\subset {\mathcal {L}}^0({\mathcal {T}}_h)$ has codimension at most one.

Proof

(i) From Theorem 3.1 we deduce that the asserted identity holds if and only if ${{\,\mathrm{div}\,}}\ker \Pi _h|_{{{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)} = \{0\}$. Since

$$\begin{aligned} \ker \Pi _h|_{{{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)} = \{y_h\in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h): y_h|_T = b_T (x-x_T) \text { f.a. } T\in {\mathcal {T}}_h\}, \end{aligned}$$

the latter condition is equivalent to $\ker \Pi _h|_{{{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)} = \{0\}$. Let $T\in {\mathcal {T}}_h$ such that a side $S_0 \subset \partial T$ belongs to ${\Gamma _N}$, i.e., we have $y_h|_T(x) = \sum _{j=0}^d \alpha _j (x-z_{S_j})$, where $z_{S_j}$ is the vertex of T opposite to the side $S_j\subset \partial T$, and with $\alpha _0 = 0$. If $y_h(x_T) =0$ then it follows that $\alpha _j = 0$ for $j=1,\dots ,d$ since the vectors $x_T-z_{S_j}$ are linearly independent. Starting from this element we may successively consider neighboring elements to deduce that $y_h|_T=0$ for all $T\in {\mathcal {T}}_h$.

(ii) If ${\Gamma _D}=\partial \Omega$ we may argue as in (i) by removing one side $S\in {\mathcal {S}}_h\cap {\Gamma _D}$ from ${\Gamma _D}$, define ${\Gamma _D}'= {\Gamma _D}\setminus S$, and using the larger space ${\mathcal {S}}^{1,cr}_{D'}({\mathcal {T}}_h)$. We then have $\Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)\subset \Pi _h {\mathcal {S}}^{1,cr}_{D'}({\mathcal {T}}_h)= {\mathcal {L}}^0({\mathcal {T}}_h)$. The difference is trivial if and only if $\Pi _h \varphi _S$ belongs to $\Pi _h {\mathcal {S}}^{1,cr}_{D'}({\mathcal {T}}_h)$. $\square$

The following examples show that both equality or strict inequality can occur if ${\Gamma _D}=\partial \Omega$.

Examples 3.3

(i) Let $L\in \{1,2\}$, ${\mathcal {T}}_h = \{T_1,\dots ,T_L\}$, ${\overline{\Omega }}=T_1\cup \dots \cup T_L$, ${\Gamma _D}= \partial \Omega$. Then $\Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)\simeq {\mathbb {R}}^{L-1}$ while ${\mathcal {L}}^0({\mathcal {T}}_h) \simeq {\mathbb {R}}^L$.

(ii) Let ${\mathcal {T}}_h = \{T_1,T_2,T_3\}$ be a triangulation consisting of the subtriangles obtained by connecting the vertices of a macro triangle T with its midpoint $x_T$. Let ${\overline{\Omega }}=T_1\cup T_2 \cup T_3$ and ${\Gamma _D}= \partial \Omega$. We then have $\Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)={\mathcal {L}}^0({\mathcal {T}}_h)$.

The second implication concerns discrete versions of convex duality relations. We let

$$\begin{aligned} \phi ^*(s) = \sup _{r\in {\mathbb {R}}^\ell } s\cdot r - \phi (r) \end{aligned}$$

be the convex conjugate of a given convex function $\phi \in C({\mathbb {R}}^d)$.

Corollary 3.4

(Convex conjugation) Let ${\overline{u}}_h\in \Pi _h {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ and $\phi \in C({\mathbb {R}}^d)$ be convex. We then have

$$\begin{aligned}\begin{aligned} \inf&\Big \{ \int _\Omega \phi (\nabla _{\! h}u_h) \,{\mathrm d}x: u_h \in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h), \, \Pi _h u_h = {\overline{u}}_h \Big \} \\&\ge \sup \Big \{ - \int _\Omega \phi ^*(\Pi _h z_h) \,{\mathrm d}x - ({\overline{u}}_h,{{\,\mathrm{div}\,}}z_h): z_h \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)\Big \}. \end{aligned}\end{aligned}$$

If $\phi \in C^1({\mathbb {R}}^d)$ and the infimum is finite then equality holds.

Proof

An integration by parts and Fenchel’s inequality show that

$$\begin{aligned} - (\Pi _h u_h,{{\,\mathrm{div}\,}}z_h) = (\nabla _{\! h}u_h, \Pi _h z_h) \le \phi (\nabla _{\! h}u_h) + \phi ^*(\Pi _h z_h). \end{aligned}$$

(5)

This implies that the left-hand side is an upper bound for the right-hand side. If $\phi$ is differentiable $u_h \in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ is optimal in the infimum then we have the optimality condition

$$\begin{aligned} \int _\Omega \phi '(\nabla _{\! h}u_h) \cdot \nabla _{\! h}v_h \,{\mathrm d}x = 0 \end{aligned}$$

for all $v_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ with $\Pi _h v_h = 0$. Theorem 3.1 yields that $\phi '(\nabla _{\! h}u_h) = \Pi _h z_h$ for some $z_h\in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$. This identity implies equality in (5) and hence

$$\begin{aligned} \int _\Omega \phi (\nabla _{\! h}u_h) \,{\mathrm d}x = -\int _\Omega \phi ^*(\Pi _h z_h) \,{\mathrm d}x - ({\overline{u}}_h,{{\,\mathrm{div}\,}}z_h) \end{aligned}$$

so that the asserted equality follows. $\square$

Remark 3.5

For nondifferentiable functions $\phi$, the strong duality relation can be established if there exists a sequence of continuously differentiable functions $\phi _\varepsilon$ such that the corresponding discrete primal and dual problems $I_{h,\varepsilon }$ and $D_{h,\varepsilon }$ are $\Gamma$-convergent to $I_h$ and $D_h$ as $\varepsilon \rightarrow 0$, respectively. An example is the approximation of $\phi (s)= |s|$ by functions $\phi _\varepsilon (s) = \min \{|s|-\varepsilon /2,|s|^2/(2\varepsilon )\}$ for $\varepsilon >0$.

With the conjugation formula we obtain a canonical definition of a discrete dual variational problem.

Corollary 3.6

(Discrete duality) Assume that $\phi \in C({\mathbb {R}}^d)$ is convex and $\psi _h:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\cup \{+\infty \}$ is elementwise constant in the first argument and convex with respect to the second argument. For $u_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ and $z_h \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ define

$$\begin{aligned}\begin{aligned} I_h(u_h)&= \int _\Omega \phi (\nabla _{\! h}u_h) + \psi _h(x,\Pi _h u_h) \,{\mathrm d}x, \\ D_h(z_h)&= - \int _\Omega \phi ^*(\Pi _h z_h) + \psi _h^*(x,{{\,\mathrm{div}\,}}z_h) \,{\mathrm d}x. \end{aligned}\end{aligned}$$

We then have

$$\begin{aligned} \inf _{u_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)} I_h(u_h) \ge \sup _{z_h\in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)} D_h(z_h). \end{aligned}$$

Proof

Using the result of Corollary 3.4 and exchanging the order of the extrema we find that

$$\begin{aligned}\begin{aligned} \inf _{u_h} I_h(u_h)&\ge \inf _{u_h} \sup _{z_h} -\int _\Omega \phi ^*(\Pi _h z_h) \,{\mathrm d}x - (\Pi _h u_h,{{\,\mathrm{div}\,}}z_h) + \int _\Omega \psi _h(x,\Pi _h u_h) \,{\mathrm d}x \\&\ge \sup _{z_h} -\int _\Omega \phi ^*(\Pi _h z_h)\,{\mathrm d}x + \inf _{u_h} - (\Pi _h u_h,{{\,\mathrm{div}\,}}z_h) + \int _\Omega \psi _h(x,\Pi _h u_h) \,{\mathrm d}x \\&= \sup _{z_h} -\int _\Omega \phi ^*(\Pi _h z_h)\,{\mathrm d}x - \sup _{u_h} \, (\Pi _h u_h,{{\,\mathrm{div}\,}}z_h) - \int _\Omega \psi _h(x,\Pi _h u_h) \,{\mathrm d}x \\&\ge \sup _{z_h} -\int _\Omega \phi ^*(\Pi _h z_h)\,{\mathrm d}x - \int _\Omega \psi _h^*(x,\Pi _h u_h) \,{\mathrm d}x \\&= \sup _{z_h} D_h(z_h). \end{aligned}\end{aligned}$$

This proves the asserted inequality. $\square$

The fourth implication concerns the postprocessing of solutions of the primal problem to obtain a solution of the dual problem. This also implies a strong discrete duality relation.

Corollary 3.7

(Strong discrete duality) In addition to the conditions of Corollary 3.6 assume that $\phi \in C^1({\mathbb {R}}^d)$ and $\psi _h :\Omega \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}$ is finite and differentiable with respect to the second argument. If $u_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ is minimal for $I_h$ then the vector field

$$\begin{aligned} z_h = \phi '(\nabla _{\! h}u_h) + \psi _h'(x,\Pi _h u_h) d^{-1} (1-\Pi _h) {{\,\mathrm{id}\,}}\end{aligned}$$

is maximal for $D_h$ with $I_h(u_h) = D_h(z_h)$.

Proof

The optimal $u_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ solves the optimality condition

$$\begin{aligned} \int _\Omega \phi '(\nabla _{\! h}u_h) \cdot \nabla _{\! h}v_h + \psi _h'(\cdot ,\Pi _h u_h) \Pi _h v_h \,{\mathrm d}x = 0 \end{aligned}$$

(6)

for all $v_h \in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$. By restricting to functions satisfying $\Pi _h v_h =0$ we deduce with Theorem 3.1 that there exists $z_h \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ with

$$\begin{aligned} \Pi _h z_h = \phi '(\nabla _{\! h}u_h). \end{aligned}$$

The optimality condition (6) implies that ${{\,\mathrm{div}\,}}z_h = \psi _h' (\cdot ,\Pi _h u_h)$. Hence, $z_h$ satisfies the asserted identity. With the resulting Fenchel identities

$$\begin{aligned}\begin{aligned} \nabla _{\! h}u_h \cdot \Pi _h z_h&= \phi (\nabla _{\! h}u_h) + \phi ^*(\Pi _h z_h),\\ \Pi _h u_h \cdot {{\,\mathrm{div}\,}}z_h&= \psi _h(\cdot , \Pi _h u_h) + \psi _h^*(\cdot ,{{\,\mathrm{div}\,}}z_h), \end{aligned}\end{aligned}$$

and by choosing $v_h = u_h$ in (6) we find that

$$\begin{aligned} I_h(u_h) = D_h(z_h) \end{aligned}$$

which in view of the weak duality relation $\inf _{u_h} I_h(u_h) \ge \sup _{z_h} D_h(z_h)$ implies that $z_h$ is optimal. $\square$

Appendix A: Explicit construction

For completeness we provide a derivation of (1) based on an explicit construction of a discrete scalar potential. We say that ${\Gamma _D}$ is connnected if its relative interior has at most one connectivity component.

Proposition A.1

(Crouzeix–Raviart potential) Assume that ${\Gamma _D}=\emptyset$ or $d=2$ and ${\Gamma _D}$ is connected. A vector field $w_h\in {\mathcal {L}}^0({\mathcal {T}}_h)^d$ satisfies $w_h = \nabla _{\! h}v_h$ for a function $v_h \in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ if and only if

$$\begin{aligned} \int _\Omega w_h \cdot y_h \,{\mathrm d}x = 0 \end{aligned}$$

for all $y_h \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ with ${{\,\mathrm{div}\,}}y_h = 0$.

Proof

If $w_h = \nabla _{\! h}v_h$ then the orthogonality relation follows from the integration-by-parts formula (4). Conversely, if $w_h$ is orthogonal to vector fields $y_h \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ with vanishing divergence then we can construct a function $v_h$ by integrating $w_h$ along a path connecting midpoints of sides, i.e., choosing a side $S_0$ at which some value is assigned to $v_h$, e.g., $v_h(x_{S_0})=0$. If ${\Gamma _D}\ne \emptyset$ then we choose $S_0\subset {\Gamma _D}$. The values at other sides are obtained via

$$\begin{aligned} v_h(x_S) = v_h(x_{S_0}) + \sum _{j=1}^J w_h|_{T_j} \cdot (x_{S_j}- x_{S_{j-1}}), \end{aligned}$$

where $(T_j)_{j=1,\dots ,J}$ is a chain of (unique) elements connecting $S_0 \subset T_1$ with $S=S_J\subset T_J$ via the shared sides $S_1,\dots ,S_{J-1}$, i.e., $T_j \cap T_{j+1} = S_j$ for $j=1,\dots ,J-1$. To see that this is well defined it suffices to show that for every closed path with $S_J=S_0$ the sum equals zero. To verify this we define the Raviart–Thomas vector field

$$\begin{aligned} y_h = \sum _{j=1}^J |S_j|^{-1} \psi _{S_j}, \end{aligned}$$

where we assume that the plus sign in

$$\begin{aligned} \psi _S(x) = \pm \frac{|S|}{d |T_\pm |} (z_{S,T_\pm } - x) \end{aligned}$$

occurs for $\psi _{S_j}$ on $T_{j+1}$ and the minus sign on $T_j$. For the element $T_j$ we then have that

$$\begin{aligned}\begin{aligned} \int _{T_j} w_h \cdot y_h \,{\mathrm d}x&= w_h|_{T_j} \cdot \int _{T_j} \big ( |S_{j-1}|^{-1} \psi _{S_{j-1}} + |S_j|^{-1} \psi _{S_j}\big ) \,{\mathrm d}x \\&= w_h|_{T_j} \cdot |T_j| \big (|S_{j-1}|^{-1} \psi _{S_{j-1}}(x_{T_j}) + |S_j|^{-1} \psi _{S_j}(x_{T_j})\big ) \\&= w_h|_{T_j} \cdot d^{-1} \big ((z_{S_{j-1}}) - x_{T_j}) - ( z_{S_j} - x_{T_j})\big ) \\&= w_h|_{T_j} \cdot (x_{S_j} - x_{S_{j-1}}), \end{aligned}\end{aligned}$$

where we used that $(z_{S,T} -x_T) = d (x_T-x_S)$ for $S\subset \partial T$. Moreover, we have

$$\begin{aligned} {{\,\mathrm{div}\,}}y_h|_{T_j} = {{\,\mathrm{div}\,}}\big ( |S_{j-1}|^{-1} \psi _{S_{j-1}} + |S_j|^{-1} \psi _{S_j} \big ) = 0. \end{aligned}$$

This implies that ${{\,\mathrm{div}\,}}y_h=0$ and hence by the assumed orthogonality

$$\begin{aligned} \sum _{j=1}^J w_h|_{T_j} \cdot (x_{S_j}- x_{S_{j-1}}) = \sum _{j=1}^J \int _{T_j} w_h \cdot y_h \,{\mathrm d}x = \int _\Omega w_h \cdot y_h \,{\mathrm d}x =0 \end{aligned}$$

for every closed path of elements. Hence, the function $v_h$ is well defined with $\nabla _h v_h = w_h$. If $d=2$ and ${\Gamma _D}$ is connected then by letting $\varphi _z \in C({\overline{\Omega }})$ be an elementwise affine nodal basis function associated with an inner node $z\in {\mathcal {N}}_h\cap {\Gamma _D}$, i.e., $z=S_1\cap S_2$ for $S_1,S_2\in {\mathcal {S}}_h \cap {\Gamma _D}$, and choosing $y_h = (\nabla \varphi _z)^\perp \in {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$, where $(a_1,a_2)^\perp = (-a_2,a_1)$, it follows that

$$\begin{aligned}\begin{aligned} 0&= \int _\Omega \nabla _{\! h}v_h \cdot (\nabla \varphi _z)^\perp \,{\mathrm d}x = \int _{S_1 \cup S_2} v_h (\nabla \varphi _z)^\perp \cdot n \,{\mathrm d}s \\&= \pm \big ( v_h(x_{S_2}) - v_h(x_{S_1})\big ), \end{aligned}\end{aligned}$$

i.e., that $v_h$ is constant on ${\Gamma _D}$. Here we used that $(\nabla \varphi _z)^\perp \cdot n$ is the tangential derivative on ${\Gamma _D}$ given by $\pm 1/|S_j|$ for $j=1,2$. $\square$

To deduce (1) from the proposition we argue as in [10] and let $w_h\in {\mathcal {L}}^0({\mathcal {T}}_h)^d$ be orthogonal to $\Pi _h {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ and hence also to ${{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$. Proposition A.1 implies that $w_h = \nabla _h v_h$ and it remains to show that $v_h$ has the same value at all element midpoints. This follows from

$$\begin{aligned} 0 = \int _\Omega \nabla _{\! h}v_h \cdot \psi _S \,{\mathrm d}x = -\int _\Omega v_h {{\,\mathrm{div}\,}}\psi _S \,{\mathrm d}x = |S| \big (v_h(x_{T_+}) - v_h(x_{T_-})\big ). \end{aligned}$$

Hence, $w_h\in \nabla _{\! h}\big (\ker \Pi _h|_{{\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)}\big )$. Note that a nontrivial $v_h$ only exists on triangulations that can be partitioned by two colors, e.g., consisting of halved squares with the same diagonal. Conversely, if $v_h\in {\mathcal {S}}^{1,{cr}}_D({\mathcal {T}}_h)$ with $\Pi _h v_h =0$ then the integration-by-parts formula (4) yields that $\nabla _{\! h}v_h$ is orthogonal to ${{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$ and in particular to $\Pi _h {{\mathcal {R}}T}^0_{\!N}({\mathcal {T}}_h)$.