The conforming virtual element space introduced above satisfies optimal properties for approximating sufficiently smooth functions. In particular, the theory in [17] for star-shaped domains may be used to prove the following theorem regarding the approximation properties of the \(L^2(E)\)-orthogonal projection to polynomials.
Theorem 7
(Approximation using polynomials) Suppose that Assumption 1 is satisfied. Let \(E\in {\mathcal {T}}_h\) and let \(\varPi ^0_{\ell } : L^2(E) \rightarrow \mathcal {P}_{\ell }(E)\), for \(\ell \ge 0\), denote the \(L^2(E)\)-orthogonal projection onto the polynomial space \(\mathcal {P}_{\ell }(E)\). Then, for any \(w\in H^{m}(E)\), with \(1 \le m\le \ell +1\), it holds
$$\begin{aligned}{||w- \varPi ^0_{\ell }w||}_{0,E} + h_{E} {|w- \varPi ^0_{\ell }w|}_{1,E} \le C_{{\text {proj}}}h_{E}^m{|w|}_{m,E}. \end{aligned}$$
The positive constant \(C_{{\text {proj}}}\) depends only on the polynomial degree \(\ell \) and the mesh regularity.
We shall make use of standard bubble functions on polygons/polyhedra below. A bubble function \(\psi _E\in H^1_0(E)\) for a polygon/polyhedron \(E\) can be constructed piecewise as the sum of the (polynomial) barycentric bubble functions (cf. [3, 41]) on each \(d\)-simplex of the shape-regular sub-triangulation of the mesh element \(E\) discussed in Remark 2.
Lemma 8
(Interior bubble functions) Let \(E\in {\mathcal {T}}_h\) and let \(\psi _{E}\) be the corresponding bubble function. There exists a constant \(C_{{\text {bub}}}\), independent of \(h_{E}\) such that for all \(q \in \mathcal {P}_{p}(E)\)
$$\begin{aligned}C_{{\text {bub}}}^{-1} {||q||}_{0,E}^2 \le \int _E\psi _{E}q^2 {\text {d}}\varvec{x}\le C_{{\text {bub}}}{||q||}_{0,E}^2, \end{aligned}$$
and
$$\begin{aligned}C_{{\text {bub}}}^{-1} {||q||}_{0,E} \le {||\psi _{E}q||}_{0,E} + h_{E} {||\nabla (\psi _{E}q)||}_{0,E} \le C_{{\text {bub}}}{||q||}_{0,E}. \end{aligned}$$
Lemma 9
(Edge bubble functions) For \(E\in {\mathcal {T}}_h\), let \(s\subset \partial {E}\) be a mesh interface and let \(\psi _{s}\) be the corresponding interface bubble function. There exists a constant \(C_{{\text {bub}}}\), independent of \(h_{E}\) such that for all \(q \in \mathcal {P}_{p}(s)\)
$$\begin{aligned}C_{{\text {bub}}}^{-1} {||q||}_{0,s}^2 \le \int _{s}\psi _{s}q^2 {\text {d}}s\le C_{{\text {bub}}}{||q||}_{0,s}^2, \end{aligned}$$
and
$$\begin{aligned}h_{E}^{-1/2}{||\psi _{s}q||}_{0,E} + h_{E}^{1/2} {||\nabla (\psi _{s}q)||}_{0,E} \le C_{{\text {bub}}}{||q||}_{0,s}. \end{aligned}$$
Here, with slight abuse of notation, the symbol q is also used to denote the constant prolongation of q in the direction normal to \(s\).
We shall first use the above two results to prove an inverse inequality for virtual element functions, made possible by the fact that functions in \(\mathcal {W}_h^{E}\) and \(V_h^E\) have polynomial Laplacians.
Lemma 10
(Inverse inequality) Suppose that Assumption 1 is satisfied. Let \(E\in {\mathcal {T}}_h\) and let \(w\in H^1(E)\) be such that \(\varDelta w\in \mathcal {P}_{p}(E)\). There exists a constant \(C_{{\text {inv}}}\), independent of \(w\), h and \(E\), such that
$$\begin{aligned}{||\varDelta w||}_{0,E} \le C_{{\text {inv}}}h_{E}^{-1} {|w|}_{1,E}. \end{aligned}$$
Proof
We first require an auxiliary polynomial inverse inequality \({||q||}_{0,E} \le C_{{\text {inv}}}h_{E}^{-1} {||q||}_{H^{-1}(E)}\), valid for all \(q \in \mathcal {P}_{p}(E)\). This may be proven by selecting \(v= q \psi _{E}\) in the definition of the dual norm, viz.
$$\begin{aligned} {||q||}_{H^{-1}(E)} := \sup _{0\ne v\in H^1_0(E)} \frac{ \int _Eq v{\text {d}}\varvec{x}}{{||\nabla v||}_{0,E}} \ge \frac{ \int _E\psi _{E}q^2 {\text {d}}\varvec{x}}{{||\nabla (\psi _{E}q)||}_{0,E}}, \end{aligned}$$
(4.1)
and using Lemma 8. Applying this to \(\varDelta w\in \mathcal {P}_{p}(E)\), we find that
$$\begin{aligned} {||\varDelta w||}_{0,E}&\le C_{{\text {inv}}}h_{E}^{-1} {||\varDelta w||}_{H^{-1}(E)}. \end{aligned}$$
Now, using (4.1), along with an integration by parts, we deduce
$$\begin{aligned} {||\varDelta w||}_{H^{-1}(E)}&= \sup _{0\ne v\in H^1_0(E)} \frac{ \int _E\varDelta wv{\text {d}}\varvec{x}}{{||\nabla v||}_{0,E}} = \sup _{0\ne v\in H^1_0(E) } \frac{ -\int _E\nabla w\cdot \nabla v{\text {d}}\varvec{x}}{{||\nabla v||}_{0,E}}. \end{aligned}$$
The result then follows by applying the Cauchy-Schwarz inequality. \(\square \)
The above inverse estimate will be used to prove an approximation theorem (Theorem 11 below) for the virtual element spaces considered in this work. The proof of Theorem 11 is inspired by [35, Prop. 4.2], where a related result is obtained in the much simpler setting of the original virtual element space of [7] for \(d=2\) only. As the construction in [35, Prop. 4.2] does not appear to generalize to \(d=3\), we use a different construction for the Clément-type interpolant below.
We begin by recalling some classical polynomial interpolation results on simplicial triangulations. Assumption 1 implies the existence of a globally shape-regular sub-triangulation \(\widehat{{\mathcal {T}}_h}\) of \({\mathcal {T}}_h\), cf. Remark 2. We use this to define \(v_{c}\) as the classical Clément interpolant [24] of \(v\) of degree \(p\) over the sub-triangulation \(\widehat{{\mathcal {T}}_h}\). Then, the following approximation estimates hold [24] for any \(v\in H^1(\varOmega )\):
$$\begin{aligned} {||v- v_{c}||}_{0,T}+ h{|v- v_{c}|}_{1,T} \le \hat{C}_{{\text {Clem}}}h {|v|}_{1,\widetilde{T}}, \end{aligned}$$
(4.2)
for all \(T\in \widehat{{\mathcal {T}}_h}\), with \(\hat{C}_{{\text {Clem}}}\) a positive constant depending only on the polynomial degree \(p\) and on the mesh regularity. Here, \(\widetilde{T}\) denotes the usual finite element patch relative to T.
Theorem 11
(Approximation using virtual element functions) Suppose that Assumption 1 is satisfied and let \(V_h\) denote the virtual element space (3.5). For \(v\in H^1(\varOmega )\), there exists a \({v_{{\text {I}}}}\in V_h\), such that, for all elements \(E\in {\mathcal {T}}_h\), we have
$$\begin{aligned}{||v- {v_{{\text {I}}}}||}_{0,E} +h_{E}{|v- {v_{{\text {I}}}}|}_{1,E} \le C_{{\text {Clem}}}h_{E} {|v|}_{1,\widetilde{E}}, \end{aligned}$$
\(C_{{\text {Clem}}}\) being a positive constant, depending only on the polynomial degree \(p\) and the mesh regularity.
Proof
We denote by \(v_{c}\) the Clément interpolant defined over a sub-triangulation \(\widehat{{\mathcal {T}}_h}\) and satisfying (4.2). It is assumed that all edges of the polygonal/polyhedral mesh \({\mathcal {T}}_h\) are also edges of the sub-triangulation \(\widehat{{\mathcal {T}}_h}\), cf. Remark 2.
Case
\(d=2\). We start by interpolating \(v_{c}\) into the enlarged virtual element space \(\mathcal {W}_h\). More specifically, we define \({w_{{\text {I}}}}\) elementwise as the solution of the problem
$$\begin{aligned} {\left\{ \begin{array}{ll} -\varDelta {w_{{\text {I}}}}= -\varDelta \varPi ^0_{p}v_{c}&{}\text { in } E, \\ {w_{{\text {I}}}}= v_{c}&{}\text { on } \partial E. \end{array}\right. } \end{aligned}$$
(4.3)
Then, since \(\varDelta \varPi ^0_{p}v_{c}\in \mathcal {P}_{p-2}(E) \subset \mathcal {P}_{p}(E)\) and \(v_{c}\) is a polynomial of degree \(p\) on each edge of \(E\), we may conclude that \({w_{{\text {I}}}}|_{E} \in \mathcal {W}_h^{E}\). Moreover, since \(v_{c}\) is continuous on \(\varOmega \), it follows that \({w_{{\text {I}}}}\in \mathcal {W}_h\).
Arguing as in [35, Proposition 4.2], we may show that
$$\begin{aligned} {|{w_{{\text {I}}}}- \varPi ^0_{p}v_{c}|}_{1,E} \le {|v_{c}- \varPi ^0_{p}v_{c}|}_{1,E}, \end{aligned}$$
(4.4)
and, therefore,
$$\begin{aligned} {|v_{c}- {w_{{\text {I}}}}|}_{1,E} \le 2{|v_{c}- \varPi ^0_{p}v_{c}|}_{1,E}. \end{aligned}$$
(4.5)
Now, \({w_{{\text {I}}}}\) allows us to construct an interpolant \({v_{{\text {I}}}}\in V_h\) using the definition of \(V_h^E\) [given in (3.3)] on each \(E\in {\mathcal {T}}_h\). By definition, the two interpolants \({v_{{\text {I}}}}\) and \({w_{{\text {I}}}}\) are equal on the mesh skeleton \({\mathcal {S}}_h\) and for all \(E\in {\mathcal {T}}_h\), \({\mathcal {M}}^{E}_{\varvec{\alpha }}({v_{{\text {I}}}})={\mathcal {M}}^{E}_{\varvec{\alpha }}({w_{{\text {I}}}})\) if \({|\varvec{\alpha }|}\le p-2\), while \({\mathcal {M}}^{E}_{\varvec{\alpha }}({v_{{\text {I}}}})={\mathcal {M}}^{E}_{\varvec{\alpha }}(\varPi ^{*}_{p}{w_{{\text {I}}}})\) if \(p-1\le {|\varvec{\alpha }|}\le p\). Consider, now, \({|{w_{{\text {I}}}}-{v_{{\text {I}}}}|}_{1,E}\) on each \(E\in {\mathcal {T}}_h\). Integration by parts yields
$$\begin{aligned} {|{w_{{\text {I}}}}- {v_{{\text {I}}}}|}_{1,E}^2 = -(\varDelta ({w_{{\text {I}}}}- {v_{{\text {I}}}}), {w_{{\text {I}}}}- {v_{{\text {I}}}})_{E}, \end{aligned}$$
(4.6)
as \({w_{{\text {I}}}}\) and \({v_{{\text {I}}}}\) coincide on \(\partial {E}\). Since \({w_{{\text {I}}}}- {v_{{\text {I}}}}\in \mathcal {W}_h^{E}\), we have \(\varDelta ({w_{{\text {I}}}}- {v_{{\text {I}}}}) \in \mathcal {P}_{p}(E)\). Let \(q_{p, p-1} \in \mathcal {P}_{p}(E) / \mathcal {P}_{p-2}(E)\) be defined by \(q_{p, p-1}=\varDelta ({w_{{\text {I}}}}- {v_{{\text {I}}}}) - \varPi ^0_{p-2} \varDelta ({w_{{\text {I}}}}- {v_{{\text {I}}}})\). Identity (4.6) can then be rewritten as
$$\begin{aligned} {|{w_{{\text {I}}}}- {v_{{\text {I}}}}|}_{1,E}^2 = -(q_{p, p-1}, {w_{{\text {I}}}}- {v_{{\text {I}}}})_{E} = -(q_{p, p-1}, {w_{{\text {I}}}}- \varPi ^{*}_{p} {w_{{\text {I}}}})_{E}, \end{aligned}$$
since \({v_{{\text {I}}}}\) and \({w_{{\text {I}}}}\) have the same moments of up to degree \(p-2\), while \({v_{{\text {I}}}}\) and \(\varPi ^{*}_{p}{w_{{\text {I}}}}\) share the same moments of degree \(p\) and \(p-1\). The Cauchy-Schwarz inequality then implies that
$$\begin{aligned} {|{w_{{\text {I}}}}- {v_{{\text {I}}}}|}_{1,E}^2 \le {||q_{p, p-1}||}_{0,E} {||{w_{{\text {I}}}}- \varPi ^{*}_{p} {w_{{\text {I}}}}||}_{0,E}. \end{aligned}$$
Further, from the stability of the \(L^2\) projection we get
$$\begin{aligned} {||q_{p,p-1}||}_{0,E} = {||({\text {I}}- \varPi ^0_{p-2}) \varDelta ({w_{{\text {I}}}}- {v_{{\text {I}}}})||}_{0,E} \le {||\varDelta ({w_{{\text {I}}}}- {v_{{\text {I}}}})||}_{0,E}, \end{aligned}$$
where \({\text {I}}\) denotes the identity operator on the space \(\mathcal {P}_{p}(E)\). Thus,
$$\begin{aligned} {|{w_{{\text {I}}}}- {v_{{\text {I}}}}|}_{1,E}^2&\le {||\varDelta ({w_{{\text {I}}}}- {v_{{\text {I}}}})||}_{0,E} {||{w_{{\text {I}}}}- \varPi ^{*}_{p} {w_{{\text {I}}}}||}_{0,E}\\&\le C_{{\text {inv}}}h_{E}^{-1} {|{w_{{\text {I}}}}- {v_{{\text {I}}}}|}_{1,E} {||{w_{{\text {I}}}}- \varPi ^{*}_{p} {w_{{\text {I}}}}||}_{0,E}, \end{aligned}$$
by Lemma 10. Further, adding and subtracting \(\varPi ^0_{p}{w_{{\text {I}}}}\) and using the stability of \(\varPi ^{*}_{p}\) and then using the Poincaré inequality (either on each 2-simplex of the shape-regular sub-triangulation \(E\) or directly on \(E\), cf. [40]), we obtain
$$\begin{aligned} {|{w_{{\text {I}}}}- {v_{{\text {I}}}}|}_{1,E}&\le C_{{\text {inv}}}(1+C_0^*) h_{E}^{-1} {||{w_{{\text {I}}}}- \varPi ^0_{p} {w_{{\text {I}}}}||}_{0,E}\\&\le C_{{\text {P}}}C_{{\text {inv}}}(1+C_0^*) {|{w_{{\text {I}}}}- \varPi ^0_{p} {w_{{\text {I}}}}|}_{1,E}, \end{aligned}$$
for some uniform constants \(C_0^*\) and \(C_{{\text {P}}}>0\) which depend on the shape regularity constant. Then, the triangle inequality, the stability of \(\varPi ^0_{p}\) with constant, say, \(C_0\), and (4.5) imply that
$$\begin{aligned} {|{w_{{\text {I}}}}- {v_{{\text {I}}}}|}_{1,E}&\le C_{{\text {P}}}C_{{\text {inv}}}(1+C_0^*) ({|{w_{{\text {I}}}}- \varPi ^0_{p}v_{c}|}_{1,E} + {|\varPi ^0_{p}v_{c}- \varPi ^0_{p} {w_{{\text {I}}}}|}_{1,E}) \\&= C_{{\text {P}}}C_{{\text {inv}}}(1+C_0^*) ({|{w_{{\text {I}}}}- \varPi ^0_{p}v_{c}|}_{1,E} + C_0{|v_{c}- {w_{{\text {I}}}}|}_{1,E}) \\&\le C_{{\text {P}}}C_{{\text {inv}}}(1+C_0^*)(1+2C_0) {|v_{c}- \varPi ^0_{p}v_{c}|}_{1,E}. \end{aligned}$$
Finally, the triangle inequality, the above bound, and (4.5), imply
$$\begin{aligned} {|v_{c}- {v_{{\text {I}}}}|}_{1,E} \le {|v_{c}- {w_{{\text {I}}}}|}_{1,E}+{|{w_{{\text {I}}}}- {v_{{\text {I}}}}|}_{1,E} \le C_1{|v_{c}- \varPi ^0_{p}v_{c}|}_{1,E}, \end{aligned}$$
(4.7)
with \(C_1:= (2+C_{{\text {P}}}C_{{\text {inv}}}(1+C_0^*)(1+2C_0))\). Since \({v_{{\text {I}}}}\) and \(v_{c}\) are equal on \(\partial {E}\), we may apply the Poincaré inequality to this to obtain a bound on \({||v_{c}- {v_{{\text {I}}}}||}_{0,E}\), with an extra power of \(h_E\).
The required bounds of \({|v- {v_{{\text {I}}}}|}_{r,E}\), \(r=0,1\), now follow by the triangle inequality, adding and subtracting \(v\) and \(\varPi ^0_{p}v\) to the right-hand side of (4.7), using once again the triangle inequality, and applying the bounds (4.2) and Theorem 7.
Case
\(d=3\). The proof in this case is based on using on each face \(s\in \partial {E}\) the construction just considered for \(d=2\) and then extending this inside \(E\).
Let \(\mathcal {W}_h^{s}\) and \(V_h^s\) be the interface spaces respectively defined by (3.1) and (3.3) applied to the interface. For each \(s\in \partial {E}\), we consider \({w_{{\text {I}}}^s}\in \mathcal {W}_h^{s}\) as the solution of the 2-dimensional boundary value problem (4.3) set on \(s\), with \(v_{c}\) representing the three-dimensional Clément interpolant of \(v\) with respect to the 3-dimensional sub-triangulaiton \(\widehat{{\mathcal {T}}_h}\).
Further, from \({w_{{\text {I}}}^s}\) we may use the 2-dimensional construction to obtain \({v_{{\text {I}}}^s}\in V_h^s\). This face interpolant satisfies (4.7) with \(s\) in place of \(E\), namely:
$$\begin{aligned} {|v_{c}- {v_{{\text {I}}}^s}|}_{1,s}\le C_1{|v_{c}- \varPi ^{0,s}_{p}v_{c}|}_{1,s}, \end{aligned}$$
(4.8)
with \(\varPi ^{0,s}_{p}v_{c}\) denoting the \(L^2\)-projection of the restriction of \(v_{c}\) to \(s\). Collecting the face-wise definitions we obtain a continuous interpolant \({v_{{\text {I}}}^{\partial {E}}}\) on \(\partial {E}\). With this, we first construct \({w_{{\text {I}}}}\) on \(E\) as the solution of the problem
$$\begin{aligned} {\left\{ \begin{array}{ll} -\varDelta {w_{{\text {I}}}}= -\varDelta \varPi ^0_{p}v_{c}\text { in } E, \\ {w_{{\text {I}}}}= {v_{{\text {I}}}^{\partial {E}}}\text { on } \partial E, \end{array}\right. } \end{aligned}$$
so that \({w_{{\text {I}}}}\in \mathcal {W}_h^{E}\) by definition, as in the case \(d= 2\) (cf. (3.1)).
In view of bounding \({|{w_{{\text {I}}}}- \varPi ^0_{p}v_{c}|}_{1,E}\), it is convenient to first split the trace \(({w_{{\text {I}}}}- \varPi ^0_{p}v_{c})|_{\partial {E}}=({v_{{\text {I}}}^{\partial {E}}}-v_{c}|_{\partial {E}})+(v_{c}-\varPi ^0_{p}v_{c})|_{\partial {E}}\). Recall that, for all \(s\in \partial {E}\), we have \(({v_{{\text {I}}}^s}-v_{c})|_{\partial s}=0\). Moreover, by Assumption 1, over s we may construct a shape-regular pyramid \(P_s\subset E\) with \(|P_s|\ge \rho |E|\). By the Trace Theorem applied to \(s\in \partial P_s\), there exists \({\varphi }_s\in H^1(P_s)\) with \({\varphi }_s |_{\partial P_s{\setminus } s} = 0\) and a constant \(C_{\mathrm{T}}>0\) such that
$$\begin{aligned} {|{\varphi }_s|}_{1,P_s}\le C_{\mathrm{T}}{||{v_{{\text {I}}}^s}-v_{c}||}_{1/2,s}. \end{aligned}$$
The constant \({C}_\mathrm{T}\) can be bounded uniformly over all \(s\) by a generalised scaling argument, cf. [21] and the references therein. Hence, defining \({\varphi }=\sum _{s\in \partial {E}}{\varphi }_s+v_{c}-\varPi ^0_{p}v_{c}\), where each \({\varphi }_s\) should be interpreted as its extension to zero on \(E\), we have by construction that \({\varphi }|_{\partial {E}}=({w_{{\text {I}}}}- \varPi ^0_{p}v_{c})|_{\partial {E}}\). Thus, as in the case \(d=2\), we have
$$\begin{aligned} {|{w_{{\text {I}}}}- \varPi ^0_{p}v_{c}|}_{1,E}&\le {|{\varphi }|}_{1,E}\nonumber \\&\le \sum _{s\in \partial {E}}{|{\varphi }^s|}_{1,P_s}+{|v_{c}-\varPi ^0_{p}v_{c}|}_{1,E}\nonumber \\&\le C_{\mathrm{T}}\sum _{s\in \partial {E}}{||{v_{{\text {I}}}^s}-v_{c}||}_{1/2,s}+{|v_{c}-\varPi ^0_{p}v_{c}|}_{1,E}. \end{aligned}$$
(4.9)
It just remains to bound the first term on the right-hand side. To this end, we use the Sobolev Interpolation Theorem and Poincaré inequality (facewise, cf. the case \(d=2\) above):
$$\begin{aligned} {||{v_{{\text {I}}}^s}- v_{c}||}_{1/2,s}^2&\le {||{v_{{\text {I}}}^s}- v_{c}||}_{0,s}^2+C_\mathrm{S}{||{v_{{\text {I}}}^s}- v_{c}||}_{0,s}{|{v_{{\text {I}}}^s}- v_{c}|}_{1,s}\nonumber \\&\le (1+C_\mathrm{S}h_E^{-1}){||{v_{{\text {I}}}^s}- v_{c}||}_{0,s}^2+C_\mathrm{S}h_E{|{v_{{\text {I}}}^s}- v_{c}|}_{1,s}\nonumber \\&\le (C_{{\text {P}}}^2 (h_E+C_\mathrm{S})+ C_\mathrm{S}) h_E{|{v_{{\text {I}}}^s}- v_{c}|}_{1,s}^2\nonumber \\&\le (C_{{\text {P}}}^2 (h_E+C_\mathrm{S})+ C_\mathrm{S}) C_1 h_E{|v_{c}- \varPi ^{0,s}_{p}v_{c}|}_{1,s}^2, \end{aligned}$$
(4.10)
for some constant \({C}_\mathrm{S}>0\) which, again, can be bounded uniformly over all s by a generalised scaling argument. To obtain the last bound above we used (4.8) applied to \(s\in \partial {E}\). The interface terms above are further bounded by applying Theorem 7, yielding
$$\begin{aligned} {|v_{c}- \varPi ^{0,s}_{p}v_{c}|}_{1,s} \le C_{{\text {proj}}}{|v_{c}|}_{1,s}\le C_{{\text {proj}}}h_{E}^{-1/2}{|v_{c}|}_{1,E}. \end{aligned}$$
Using this bound in (4.10) and the latter in (4.9), we finally obtain
$$\begin{aligned} {|{w_{{\text {I}}}}- \varPi ^0_{p}v_{c}|}_{1,E}&\le C_2 {|v_{c}|}_{1,E}, \end{aligned}$$
(4.11)
with \(C_2>0\) depending on the (uniformly bounded) number \(\nu _{E}\) of interfaces of \(E\) and on the constants \(C_{\mathrm{T}}\), \(C_{{\text {P}}}, C_{\mathrm{S}}, C_{{\text {inv}}}\), and \(C_{{\text {proj}}}\).
Now, given \({w_{{\text {I}}}}\), we can construct an interpolant \({v_{{\text {I}}}}\in V_h\) exactly as in the 2-dimensional case and following the same (dimension-independent) argument derive the bound (4.7). This latter bound, combined with (4.11), yields
$$\begin{aligned} {|v_{c}- {v_{{\text {I}}}}|}_{1,E} \le 2{|v_{c}-\varPi ^0_{p}v_{c}|}_{1,E}+{|{w_{{\text {I}}}}- \varPi ^0_{p}v_{c}|}_{1,E} \le C_3 {|v_{c}|}_{1,E}, \end{aligned}$$
(4.12)
for some \(C_3>0\) depending on \(C_1\) and \(C_2\).
From (4.12) we can derive the required bound in the \(L^2\)-norm by resorting to the scaled Poincaré-Friedrichs inequality [16] and recalling (4.12):
$$\begin{aligned} {||v_{c}- {v_{{\text {I}}}}||}_{0,E}&\le C_{{\text {P}}}\left( h_E{|v_{c}- {v_{{\text {I}}}}|}_{1,E}+h_E^{-1/2}{|\int _{\partial {E}} (v_{c}- {v_{{\text {I}}}})ds|} \right) \nonumber \\&\le C_{{\text {P}}}(C_3 h_E{|v_{c}|}_{1,E}+h_E^{1/2}\sum _{s\in \partial {E}}{||v_{c}- {v_{{\text {I}}}^s}||}_{0,s}) \end{aligned}$$
(4.13)
The interface terms on the right-hand side can be further bounded using the Poincaré inequality once more and (4.8):
$$\begin{aligned} {||v_{c}- {v_{{\text {I}}}^s}||}_{0,s}&\le C_{{\text {P}}}h_E{|v_{c}- {v_{{\text {I}}}^s}|}_{1,s} \le C_{{\text {P}}}C_1 h_E{|v_{c}- \varPi ^{0,s}_{p}v_{c}|}_{1,s}\nonumber \\&\le C_{{\text {P}}}C_1 C_{{\text {proj}}}h_E{|v_{c}|}_{1,s}\le C_{{\text {P}}}C_1 C_{{\text {proj}}}h_E^{1/2}{|v_{c}|}_{1,E}. \end{aligned}$$
Finally, combining this bound with (4.13) yields
$$\begin{aligned} {||v_{c}- {v_{{\text {I}}}}||}_{0,E}&\le C_4 h_E{|v_{c}|}_{1,E}, \end{aligned}$$
with \(C_4>0\) depending on \(C_{{\text {P}}}, C_1,\)
\(C_3\), \(C_{{\text {proj}}}\), and \(\nu _{E}\).
The statement of the theorem now follows, as in the case \(d=2\). \(\square \)
Remark 12
For \(d=3\), the proof of the above VEM approximation result makes use of both the Trace Theorem and Sobolev Interpolation Theorem applied to each mesh interface. This was necessitated by the hierarchical construction of the local virtual element spaces with respect to spatial dimension. The associated constants are uniformly bounded but depend on the polygonal shape of the mesh interfaces, and as such are not easily accessible in general. However, if the mesh interfaces are triangular or the method is constructed on the sub-triangulation of each mesh interface, the proof does only depend on easily computable quantities.