Dirac particles’ tunnelling from 5-dimensional rotating black strings
influenced by the generalized uncertainty principle

Deyou Chen 111E-mail<EMAIL_ADDRESS>
Institute of Theoretical Physics, China West Normal University,

Nanchong 637009, China

Abstract: The standard Hawking formula predicts the complete evaporation of
black holes. Taking into account effects of quantum gravity, we investigate
fermions’ tunnelling from a 5-dimensional rotating black string. The
temperature is determined not only by the string, but also affected by the
quantum number of the emitted fermion and the effect of the extra spatial
dimension. The quantum correction slows down the increase of the temperature,
which naturally leads to the remnant in the evaporation.

##  1 Introduction

The semi-classical tunnelling method is an effective way to describe the
Hawking radiation [2]. Using this method, the tunnelling behavior of massless
particles across the horizon was veritably described in [3]. In the research,
the varied background spacetime was taken into account. The tunnelling rate
was related to the change of the Bekenstein-Hawking entropy and the
temperature was higher than the standard Hawking temperature. In the former
researches, the standard temperatures were derived [4, 5, 6, 7, 8, 9], which
imply the complete evaporation of black holes. Thus the varied background
spacetime accelerates the black holes’ evaporation. This result was also
demonstrated in other complicated spacetimes [10, 11, 12, 13, 14]. Extended
this work to massive particles, the tunnelling radiation of general spacetimes
was investigated in [15, 16]. The same result was derived by the relation
between the phase velocity and the group velocity.

In [17], the standard Hawking temperatures were recovered by fermions
tunnelling across the horizons. In the derivation, the action of the emitted
particle was derived by the Hamilton-Jacobi equation [18]. This derivation is
based on the complex path analysis [19]. In this method, we don’t need the
consideration of that the particle moves along the radial direction [20, 21,
22, 23]. This is a difference from the work of Parikh and Wilczek [3].

The tunnelling radiation beyond the semi-classical approximation was discussed
in [24, 25, 26]. Their ansatz is also based on the Hamilton-Jacobi method. The
key point is to expand the action in a powers of $\hbar$. Using the expansion,
one can get the quantum corrections over the semiclassical value. The
corrected temperature is lower than the standard Hawking temperature. The
higher order correction entropies were derived by the first law of black hole
thermodynamics.

Taking into account effects of quantum gravity, the semi-classical tunnelling
method was reviewed in the recent work [27, 28]. In [27], the tunnelling of
massless particles through quantum horizon of a Schwarzschild black hole was
investigated by the influence of the generalized uncertainty principle (GUP).
Through the modified commutation relation between the radial coordinate and
the conjugate momentum and the deformed Hamiltonian equation, the radiation
spectrum with the quantum correction was derived. The thermodynamic quantities
were discussed. In the fermionic fields, with the consideration of effects of
quantum gravity, the generalized Dirac equation in curved spacetime was
derived by the modified fundamental commutation relations [29], which is [28]

$\displaystyle\left[i\gamma^{0}\partial_{0}+i\gamma^{i}\partial_{i}\left(1-\beta
m^{2}\right)+i\gamma^{i}\beta\hbar^{2}\left(\partial_{j}\partial^{j}\right)\partial_{i}+\frac{m}{\hbar}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\right.$
$\displaystyle\left.+i\gamma^{\mu}\Omega_{\mu}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\right]\psi=0.$ (1)

This derivation is based on the existence of a minimum measurable length. The
length can be realized in a model of GUP

$\displaystyle\Delta x\Delta p\geq\frac{\hbar}{2}\left[1+\beta(\Delta
p)^{2}+\beta<p>^{2}\right],$ (2)

where $\beta=\beta_{0}\frac{l^{2}_{p}}{\hbar^{2}}$ is a small value,
$\beta_{0}<10^{34}$ is a dimensionless parameter and $l_{p}$ is the Planck
length. Eq. (2) was derived by the modified Heisenberg algebra
$\left[x_{i},p_{j}\right]=i\hbar\delta_{ij}\left[1+\beta p^{2}\right]$, where
$x_{i}$ and $p_{i}$ are position and momentum operators defined respectively
as [29, 30]

$\displaystyle x_{i}$ $\displaystyle=$ $\displaystyle x_{0i},$ $\displaystyle
p_{i}$ $\displaystyle=$ $\displaystyle p_{0i}(1+\beta p_{0}^{2}),$ (3)

$p_{0}^{2}=\sum p_{0j}p_{0j}$, $x_{0i}$ and $p_{0j}$ satisfy the canonical
commutation relations $\left[x_{0i},p_{0j}\right]=i\hbar\delta_{ij}$. Thus the
minimal position uncertainty is gotten as

$\displaystyle\Delta x$ $\displaystyle=$
$\displaystyle\hbar\sqrt{\beta}\sqrt{1+\beta<p>^{2}},$ (4)

which means that the minimum measurable length is $\Delta
x_{0}=\hbar\sqrt{\beta}$ [29]. To let $\Delta x_{0}$ have a physical meaning,
the condition $\beta>0$ must be satisfied. It was showed in [29]. Based on the
GUP, the black hole’s remnant was first researched by Adler et al. [31].
Incorporate eq. (3) into the Dirac equation in curved spacetime, the modified
Dirac equation was derived[28]. Using this modified equation, fermions’
tunnelling from the Schwarzschild spacetime was investigated. The temperature
was showed to be related to the quantum number of the emitted fermion. An
interested result is that the quantum correction slows down the increase of
the temperature. It is natural to lead to the remnant.

In this paper, taking into account effects of quantum gravity, we investigate
fermions’ tunnelling from a 5-dimensional rotating black string. The key point
in this paper is to construct a tetrad and five gamma matrices. The result
shows that in the frame of quantum gravity, the temperature is affected not
only by the quantum number of the emitted fermion, but also by the effect of
the extra compact dimension. The quantum correction slows down the increase of
the temperature. The remnant is naturally observed in the evaporation.

In the next section, we perform the dragging coordinate transformation on the
metric and construct five gamma matrices, then investigate the fermion’s
tunnelling from the 5-dimensional rotating string. The remnant is observed.
Section 3 is devoted to our conclusion.

##  2 Tunnelling radiation with the influence of the generalized uncertainty
principle

The Kerr metric describes a rotating black hole solution of the Einstein
equations in four dimensions. When we add an extra compact spatial dimension
to it, the metric becomes

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta}{\rho^{2}}\left(dt-a\sin^{2}\theta
d\varphi\right)^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\left[adt-(r^{2}+a^{2})d\varphi\right]^{2}$
(5)
$\displaystyle+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+g_{zz}dz^{2},$

where $\Delta=r^{2}-2Mr+a^{2}=(r-r_{+})(r-r_{-})$,
$\rho^{2}=r^{2}+a^{2}\cos^{2}\theta$, $g_{zz}$ is usually set to $1$. The
above metric describes a rotating uniform black string.
$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$ are the locations of the event (inner)
horizons. $M$ and $a$ are the mass and angular momentum unit mass of the
string, respectively. A fermion’s motion satisfies the generalized Dirac
equation (1). To investigate the tunnelling behavior of the fermion, it can
directly choose a tetrad and construct gamma matrices from the metric (5). The
metric (5) describes a rotating spacetime. The energy and mass near the
horizons are dragged by the rotating spacetime. It is not convenient to
discuss the fermion’s tunnelling behavior. For the convenience of constructing
the tetrad and gamma matrices, we perform the dragging coordinate
transformation $d\phi=d\varphi-\Omega dt$, where

$\Omega=\frac{\left(r^{2}+a^{2}-\Delta\right)a}{\left(r^{2}+a^{2}\right)^{2}-\Delta
a^{2}\sin^{2}\theta},$ (6)

on the metric (5). Then the metric (5) takes on the form as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-F(r)dt^{2}+\frac{1}{G(r)}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}+g_{zz}dz^{2}$
(7) $\displaystyle=$
$\displaystyle-\frac{\Delta\rho^{2}}{(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}{\theta}}dt^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+g_{zz}dz^{2}$
$\displaystyle+\rho^{2}d\theta^{2}+\frac{\sin^{2}{\theta}}{\rho^{2}}\left[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}{\theta}\right]d\phi^{2}.$

Now the tetrad is directly constructed from the above metric. It is

$\displaystyle
e_{\mu}^{a}=diag(\sqrt{F},1/\sqrt{G},\sqrt{g_{\theta\theta}},\sqrt{g_{\phi\phi}},\sqrt{g_{zz}}).$
(8)

Then gamma matrices are easily constructed as follows

$\displaystyle\gamma^{t}=\frac{1}{\sqrt{F}}\left(\begin{array}[]{cc}0&I\\\
-I&0\end{array}\right),$
$\displaystyle\gamma^{\theta}=\sqrt{g^{\theta\theta}}\left(\begin{array}[]{cc}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right),$ (13)
$\displaystyle\gamma^{r}=\sqrt{G}\left(\begin{array}[]{cc}0&\sigma^{3}\\\
\sigma^{3}&0\end{array}\right),$
$\displaystyle\gamma^{\phi}=\sqrt{g^{\phi\phi}}\left(\begin{array}[]{cc}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),$ (18)
$\displaystyle\gamma^{z}=\sqrt{g^{zz}}\left(\begin{array}[]{cc}-I&0\\\
0&I\end{array}\right).$ (21)

When measure the quantum property of a spin-1/2 fermion, we can get two
values. They correspond to two states with spin up and spin down. The wave
functions of two states of a fermion in the metric (7) spacetime take on the
form as

$\displaystyle\psi_{\left(\uparrow\right)}=\left(\begin{array}[]{c}A\\\ 0\\\
B\\\
0\end{array}\right)\exp\left(\frac{i}{\hbar}I_{\uparrow}\left(t,r,\theta,\phi,z\right)\right),$
(26)

$\displaystyle\psi_{\left(\downarrow\right)}=\left(\begin{array}[]{c}0\\\ C\\\
0\\\
D\end{array}\right)\exp\left(\frac{i}{\hbar}I_{\downarrow}\left(t,r,\theta,\phi,z\right)\right),$
(31)

where $A,B,C,D$ are functions of $(t,r,\theta,\phi,z)$, and $I$ is the action
of the fermion, $\uparrow$ and $\downarrow$ denote the spin up and spin down,
respectively. In this paper, we only investigate the state with spin up. The
analysis of the state with spin down is parallel. To use the WKB
approximation, we insert the wave function (26) and the gamma matrices into
the generalized Dirac equation (1). Dividing by the exponential term and
considering the leading terms yield four equations. They are

$\displaystyle-\frac{B}{\sqrt{F}}\partial_{t}I-B\sqrt{G}(1-\beta
m^{2})\partial_{r}I+A\sqrt{g^{zz}}(1-\beta m^{2})\partial_{z}I$
$\displaystyle-Am(1-\beta m^{2}-\beta
Q)+B\beta\sqrt{G}Q\partial_{r}I-A\beta\sqrt{g^{zz}}Q\partial_{z}I=0,$ (32)

$\displaystyle\frac{A}{\sqrt{F}}\partial_{t}I-A\sqrt{G}(1-\beta
m^{2})\partial_{r}I-B\sqrt{g^{zz}}(1-\beta m^{2})\partial_{z}I$
$\displaystyle-Bm(1-\beta m^{2}-\beta
Q)+A\beta\sqrt{G}Q\partial_{r}I+B\beta\sqrt{g^{zz}}Q\partial_{z}I=0,$ (33)

$\displaystyle-B\left(i\sqrt{g^{\theta\theta}}\partial_{\theta}I+\sqrt{g^{\phi\phi}}\partial_{\phi}I\right)(1-\beta
m^{2}-\beta Q)=0,$ (34)

$\displaystyle-A\left(i\sqrt{g^{\theta\theta}}\partial_{\theta}I+\sqrt{g^{\phi\phi}}\partial_{\phi}I\right)(1-\beta
m^{2}-\beta Q)=0,$ (35)

where
$Q=g^{rr}\left({\partial_{r}I}\right)^{2}+g^{\theta\theta}\left({\partial_{\theta}I}\right)^{2}+g^{\phi\phi}\left({\partial_{\phi}I}\right)^{2}+g^{zz}\left({\partial_{z}I}\right)^{2}$.
It is difficult to get the expression of the action from the above equations.
Considering the property of the spacetime, we carry out separation of
variables as

$\displaystyle I=-(\omega-j\Omega)t+W(r)+\Theta(\theta,\phi)+Jz,$ (36)

where $\omega$ is the energy of the emitted fermion, $j$ is the angular
momentum and $J$ is a conserved momentum corresponding to the compact
dimension. Eqs. (34) and (35) are irrelevant to $A,B$. Inserting Eq. (36) into
them yields

$\displaystyle
i\sqrt{g^{\theta\theta}}\partial_{\theta}\Theta+\sqrt{g^{\phi\phi}}\partial_{\phi}\Theta=0,$
(37)

which implies that $\Theta$ is a complex function other than the constant
solution. In the former research, it was found that the contribution of
$\Theta$ could be canceled in the derivation of the tunnelling rate. Using Eq.
(37), an important relation is easily gotten as

$\displaystyle
g^{\theta\theta}(\partial_{\theta}\Theta)^{2}+g^{\phi\phi}(\partial_{\phi}\Theta)^{2}=0.$
(38)

Now our interest is the first two equations. Inserting Eq. (36) into Eqs. (32)
and (33), canceling $A$ and $B$ and neglecting the higher order terms of
$\beta$, we get

$\displaystyle A(\partial_{r}W)^{4}+B(\partial_{r}W)^{2}+C=0,$ (39)

where

$\displaystyle A$ $\displaystyle=$ $\displaystyle 2\beta G^{2}F,$
$\displaystyle B$ $\displaystyle=$ $\displaystyle-[1-4\beta
g^{zz}\left({\partial_{z}I}\right)^{2}]GF,$ $\displaystyle C$ $\displaystyle=$
$\displaystyle[1-2\beta m^{2}-2\beta
g^{zz}\left({\partial_{z}I}\right)^{2}](m^{2}-g^{zz}\left({\partial_{z}I}\right)^{2})F+\left({\partial_{t}I}\right)^{2}.$
(40)

Solving the above equation at the event horizon yields the imaginary part of
the radial action. Based on the invariance under canonical transformations, we
adopt the method developed in [33]. The tunnelling rate is

$\displaystyle\Gamma$ $\displaystyle\propto$ $\displaystyle
exp[-\frac{1}{\hbar}Im\oint p_{r}dr]=exp\left[-\frac{1}{\hbar}Im\left(\int
p_{r}^{out}dr-\int p_{r}^{in}dr\right)\right]$ (41) $\displaystyle=$
$\displaystyle exp\left[\mp\frac{2}{\hbar}Im\int p_{r}^{out,in}dr\right].$

In the above equation, $\oint p_{r}dr$ is invariant under canonical
transformations. Here let $p_{r}=\partial_{r}W$. Thus the solutions of $Im\int
p_{r}^{out,in}dr$ are determined by Eq. (39), which is

$\displaystyle Im\oint p_{r}dr$ $\displaystyle=$ $\displaystyle 2ImW^{out}$
(42) $\displaystyle=$ $\displaystyle 2Im\int
dr\sqrt{\frac{(E-j\Omega)^{2}+(1-2\beta m^{2}-2\beta
g^{zz}J^{2})(m^{2}-g^{zz}J^{2})F}{GF(1-4\beta g^{zz}J^{2})}}$
$\displaystyle\times\left[1+\beta\left(\frac{(E-j\Omega)^{2}}{F}+m^{2}-g^{zz}J^{2}\right)\right]$
$\displaystyle=$ $\displaystyle
2\pi\frac{(\omega-j\Omega_{+})(r_{+}^{2}+a^{2})}{r_{+}-r_{-}}\left[1+\beta\Xi(J,\theta,r_{+},j)\right],$

where $g^{zz}=1$, $\Omega_{+}=\frac{a}{r_{+}^{2}+a^{2}}$ is the angular
velocity at the event horizon. $\Xi(J,\theta,r_{+},j)$ is a complicated
function of $J,\theta,r_{+},j$, therefore, we don’t write down here. It should
be that $\Xi(J,\theta,r_{+},j)>0$. If adopt Eq. (42) to calculate the
tunnelling rate, we will derive two times Hawking temperature, which was
showed in [32]. This is not in consistence with the standard temperature. With
careful observations, Akhmedova et. al. found that the contribution coming
from the temporal part of the action was ignored [33]. When they took into
account the temporal contribution, the factor of two in the temperature was
resolved.

To find the temporal contribution, we use the Kruskal coordinates $(T,R)$. The
region exterior to the string $(r>r_{+})$ is described by

$\displaystyle T$ $\displaystyle=$ $\displaystyle
e^{\kappa_{+}r_{*}}sinh(\kappa_{+}t),$ $\displaystyle R$ $\displaystyle=$
$\displaystyle e^{\kappa_{+}r_{*}}cosh(\kappa_{+}t),$ (43)

where
$r_{*}=r+\frac{1}{2\kappa_{+}}ln\frac{r-r_{+}}{r_{+}}-\frac{1}{2\kappa_{-}}ln\frac{r-r_{-}}{r_{-}}$
is the tortoise coordinate, and
$\kappa_{\pm}=\frac{r_{+}-r_{-}}{2(r_{\pm}^{2}+a^{2})}$ denote the surface
gravity at the outer (inner) horizons. The description of the interior region
is given by

$\displaystyle T$ $\displaystyle=$ $\displaystyle
e^{\kappa_{+}r_{*}}cosh(\kappa_{+}t),$ $\displaystyle R$ $\displaystyle=$
$\displaystyle e^{\kappa_{+}r_{*}}sinh(\kappa_{+}t).$ (44)

To connect these two patches across the horizon, we need to rotate the time
$t$ as $t\rightarrow t-i\kappa_{+}\frac{\pi}{2}$. As pointed in [33], this
rotation would lead to an additional imaginary contribution coming from the
temporal part, namely, $Im(E\Delta t^{out,in})=\frac{1}{2}\pi E\kappa_{+}$,
where $E=\omega-j\Omega_{+}$. Thus the total temporal contribution is
$Im(E\Delta t)=\pi E\kappa_{+}$. Therefore, the tunnelling rate is

$\displaystyle\Gamma$ $\displaystyle\propto$ $\displaystyle
exp\left[-\frac{1}{\hbar}\left(Im(E\Delta t)+Im\oint p_{r}dr\right)\right]$
(45) $\displaystyle=$
$\displaystyle-4\pi\frac{(\omega-j\Omega_{+})(r_{+}^{2}+a^{2})}{\hbar(r_{+}-r_{-})}\left[1+\frac{1}{2}\beta\Xi(J,\theta,r_{+},j)\right].$

This is the Boltzman factor expression and implies the temperature

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{\hbar(r_{+}-r_{-})}{4\pi(r_{+}^{2}+a^{2})\left[1+\frac{1}{2}\beta\Xi(J,\theta,r_{+},j)\right]}$
(46) $\displaystyle=$ $\displaystyle
T_{0}\left[1-\frac{1}{2}\beta\Xi(J,\theta,r_{+},j)\right],$

where $T_{0}=\frac{\hbar(r_{+}-r_{-})}{4\pi(r_{+}^{2}+a^{2})}$ is the standard
Hawking temperature of the Kerr string and shares the same expression of the
temperature of the 4-dimensional Kerr black hole. It shows that the corrected
temperature is determined by the mass, angular momentum and extra dimension of
the string, but also affected by the quantum number (energy, mass and angular
momentum) of the fermion. Therefore, the property of the emitted fermion
affects the temperature when the effects of quantum gravity are taken into
account.

When $a=0$, the metric (5) is reduced to the Schwarzschild string metric. Then
the imaginary part of the radial action (42) is reduced to

$\displaystyle Im\oint p_{r}dr$ $\displaystyle=$ $\displaystyle 2\pi\omega
r_{+}\left[1+\beta\left(2\omega^{2}+3m^{2}/2+J^{2}/2\right)\right].$ (47)

Adopting the same process, we get the temperature of the Schwarzschild string
as

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{\hbar}{4\pi
r_{+}\left[1+\beta\left(\omega^{2}+3m^{2}/4+J^{2}/4\right)\right]}$ (48)
$\displaystyle=$ $\displaystyle\frac{\hbar}{8\pi
M}\left[1-\beta\left(\omega^{2}+3m^{2}/4+J^{2}/4\right)\right].$

It shows that the effect of the extra dimension and the quantum number
(energy, mass and angular momentum) of the fermion affect the temperature of
the Schwarzschild string. It is obviously that the quantum correction slows
down the crease of the temperature. Finally, the string can not evaporate
completely and there is a blanched state. At the this state, the remnant is
left. The effect of the extra dimension plays an role of impediment during the
evaporation. When $J=0$, Eq. (48) describes the temperature of the
4-dimensional Schwarzschild black hole. The remnant was derived as $\geq
M_{p}/{\beta_{0}}$, where $M_{p}$ is the Planck mass and $\beta_{0}$ is a
dimensionless parameter accounting for quantum gravity effects [28].

##  3 Conclusion

In this paper, we investigated the fermion’s tunnelling from the 5-dimensional
Kerr string spacetime. To incorporate the influence of quantum gravity, we
adopted the generalized Dirac equation derived in [28]. The corrected
temperature is not only determined by the mass, angular momentum and extra
dimension, but also affected by the quantum number of the emitted fermion. The
quantum correction slows down the increase of the temperature. Finally, the
balance state appears. At this state, the string can not evaporate completely
and the remnant is left. This can be seen as the direct consequence of the
generalized uncertainty principle.

Acknowledgements

This work is supported by the National Natural Science Foundation of China
with Grant No. 11205125.

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# The role of stabilization in the virtual element method: a survey

Lorenzo Mascotto  Department of Mathematics and Applications, University of
Milano-Bicocca, 20125 Milan, Italy (lorenzo.mascotto@unimib.it); IMATI-CNR,
Pavia, Italy; Faculty of Mathematics, University of Vienna, 1090 Vienna,
Austria<EMAIL_ADDRESS>
###### Abstract

The virtual element method was introduced 10 years ago and it has generated a
large number of theoretical results and applications ever since. Here, we
overview the main mathematical results concerning the stabilization term of
the method as an introduction for newcomers in the field. In particular, we
summarize the proofs of some results for two dimensional “nodal” conforming
and nonconforming virtual element spaces to pinpoint the essential tools used
in the stability analysis. We discuss their extensions to several other
virtual elements. Finally, we show several ways to prove interpolation
estimates, including a recent one that is based on employing the stability
bounds.

AMS subject classification: 65N12; 65N15.

Keywords: virtual element method; stability bound; interpolation estimate.

##  1 Introduction

The virtual element method was introduced ten years ago [9] as a
generalization of the finite element method to polytopic meshes. Local virtual
element spaces consist of solutions to local problems with polynomial data.
Therefore, virtual element functions are not known in closed form; in the
spirit of the mimetic finite differences [48, 15], only the evaluation of
their degrees of freedom is required in the design of the method.

Consequently, the bilinear forms appearing in the variational formulation of
given partial differential equations are not computable and have to be
approximated based on two main ingredients: projections from local virtual
element spaces onto polynomial spaces; bilinear forms that stabilize the
scheme.

This entails that the error analysis for virtual elements has the form of a
Strang-type result, where several variational crimes have to be taken into
account. In particular, one has to cope with certain stability bounds and
interpolation estimates in virtual element spaces.

The first work on the virtual element method contains the following statement
concerning the stability estimates [9, Section 4.6]:

“ _In general, the choice of the bilinear form $S^{K}(\cdot,\cdot)$ [the local
virtual element stabilization] would depend on the problem and on the degrees
of freedom. From $(4.20)$ it is clear that $S^{K}(\cdot,\cdot)$ must scale
like $a^{K}(\cdot,\cdot)$ [the “grad-grad bilinear form] on the kernel of
$\Pi^{\nabla}_{p}$ [an $H^{1}$ polynomial projector]. Choosing then the
canonical basis $\varphi_{1},\dots,\varphi_{N_{K}}$ as_
$\chi_{i}(\varphi_{j})=\delta_{i,j},\quad
i,j=1,\dots,N_{K},\qquad\text{[$\chi_{i}$ is the $i$-th local degree of
freedom]}$ _the local stiffness matrix is given by_
$a^{K}_{h}(\varphi_{i},\varphi_{j})=a^{K}(\Pi^{\nabla}_{p}\varphi_{i},\Pi^{\nabla}_{p}\varphi_{j})+S^{K}((I-\Pi^{\nabla}_{p})\varphi_{i},(I-\Pi^{\nabla}_{p})\varphi_{j}).$
_In our case it is easy to check that, on a “reasonable” polygon,
$a^{K}(\varphi_{i},\varphi_{j})\approx 1$. Note that this holds true for all
$i=1,2,\dots,N_{K}$ [$N_{K}$ is the dimension of the local discrete space]
since we defined the local degrees of freedom suitably. […] However, several
types of misbehaviour can occur for awkwardly-shaped polygons, in particular
if two or more vertices tend to coalesce, although, in our numerical
experiments, the method appears to be quite robust in this respect._”

So, some facts were made clear since the very inception of the method:

  * •

the stabilization is not required to have approximation properties: it only
has to scale as the corresponding “continuous” bilinear form;

  * •

a reasonable choice for the stabilization is the “dofi-dofi” one given by
$S^{K}(\varphi_{i},\varphi_{j})=\delta_{i,j}$ for all $i$, $j=1,\dots,N_{K}$;

  * •

the “dofi-dofi” stabilization needs to be carefully tuned/changed in presence
of “ _awkwardly-shaped polygons_ ”.

Similar considerations are contained in other works tracing back to the early
years of the virtual elements literature, say, the period 2013-2016: there,
one can find heuristic motivations but no proofs of the stability bounds,
i.e., bounds of the form $\alpha^{*}a^{K}(v_{h},v_{h})\leq
S^{K}(v_{h},v_{h})\leq\alpha^{*}a^{K}(v_{h},v_{h})$ for suitable discrete
functions $v_{h}$ and positive constants $\alpha_{*}<\alpha^{*}$.

In 2017, the first paper [16] on the theoretical aspects of the stabilization
in virtual elements was published: stability properties for two dimensional
nodal conforming virtual elements were investigated on rather general
geometries. Ever since, several related contributions have been proposed;
amongst them we mention the other three pioneering works [39, 28, 29]. Most of
the literature on this topic is concerned with nodal conforming virtual
elements; fewer works can be found on other types of virtual elements such as
face, edge, Stokes, immersed-like virtual elements.

In light of this, the virtual element literature on the theoretical aspects of
the stabilization might be classified into three periods:

  * •

the early years (2013-2016), when the properties of the stabilization were
introduced and motivated heuristically;

  * •

the pioneering years (2017-2018), when the first papers [16, 28, 29, 39] on
the theoretical aspects of the stabilization were published;

  * •

the consolidation years (2019-2023), when several other works on the topic
were written, the pioneering analysis was generalized, and other types of
virtual elements were considered.

We deemed useful to collect and review all contributions that we are aware of
about the theoretical aspects of the stabilization. We hope that this work
could represent a starting guide in this area. More details and the topics we
are going to cover can be found at the end of this section.

Disclaimer. As we are interested here only in reviewing the literature on
theoretical aspects of the virtual element stabilization, we do not review
contributions on more “practical” issues as those in [54, 55, 14, 49, 45, 52].

#### Notation and assumptions.

Given $D\subset\mathbb{R}^{d}$, $d=1,2,3$, a Lipschitz domain with measure
$|D|$, we introduce the Sobolev space $H^{s}(D)$, $s\geq 0$, and endow it with
the usual norm ${\left\|{\cdot}\right\|}_{s,D}$, seminorm
${\left|{\cdot}\right|}_{s,D}$, and bilinear form $(\cdot,\cdot)_{s,D}$. If
$s=0$, we let the Sobolev space $H^{0}(D)$ be the Lebesgue space $L^{2}(D)$.
Negative order Sobolev spaces are defined by duality.

We set the spaces $\mathbb{P}_{p}(K)$ and $\mathbb{P}_{p}(e)$ of polynomials
of maximum degree $p$ over a polygon $K$ and an interval $e$. We use the
notation $\mathbb{P}_{-1}(K)=\\{0\\}$. For the space $\mathbb{P}_{p}(K)$,
given $\mathbf{x}_{K}=(x_{K},y_{K})$ and $h_{K}$ denote the centroid and
diameter of $K$, it is convenient to introduce the basis
$\\{m_{\boldsymbol{\alpha}}\\}$ of scaled and centred monomials

$m_{\boldsymbol{\alpha}}(\mathbf{x})\\!:=\\!\left(\frac{\mathbf{x}-\mathbf{x}_{K}}{h_{K}}\right)^{\boldsymbol{\alpha}}\\!\\!=\\!\left(\frac{x-x_{K}}{h_{K}}\right)^{\alpha_{1}}\\!\left(\frac{y-y_{K}}{h_{K}}\right)^{\alpha_{2}},\;\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2})\in\mathbb{R}^{2},\;|\boldsymbol{\alpha}|=\alpha_{1}+\alpha_{2}=0,\dots,p.$
(1)

Analogously, we can define the basis $\\{m_{\alpha}^{e}\\}$,
$\alpha=1,\dots,p$, of shifted and scaled monomials for $\mathbb{P}_{p}(e)$.

Given two quantities $a$ and $b$, we write $a\lesssim b$ meaning that there
exists $c$ depending on the shape of $K$, but not on its size, such that
$a\leq c\ b$.

#### Outline.

We prove the stability bounds for two dimensional nodal conforming virtual
elements in Section 2: here, we focus on two paradigmatic stabilizations;
derive corresponding stability bounds; underline what the main tools in the
proof are; review the relevant literature. We discuss the generalization to
nonconforming “nodal” elements and arbitrarily regular “elliptic-type”
elements in Section 3. “Nonelliptic-type” elements, such as face, edge,
Stokes-like, and immersed virtual elements, are overviewed in Section 4.
Comments on “$p$-version” spaces, and the role of the stabilization in
relability and efficiency bounds for residual error estimators are given in
Section 5. In Section 6, we show that the stability bound easily implies
interpolation estimates. We draw some conclusions in Section 7.

##  2 Basic results: stability in nodal conforming virtual elements

In this section, we review the role of the stabilization in two dimensional
nodal conforming virtual elements. After designing local virtual elements with
a set of unisolvent degrees of freedom, we describe computable polynomial
projectors and discrete bilinear forms in Section 2.1. We introduce two
paradigmatic stabilizations and provide an easy proof of stability bounds in
Section 2.2. In Section 2.3, we review the essential literature concerning the
theory behind the stabilization in nodal conforming virtual elements.

###  2.1 Nodal conforming virtual elements

Following [9], given a polygonal element $K$ and a positive integer number
$p$, we define the nodal conforming virtual element

$V_{h}(K):=\left\\{v_{h}\in H^{1}(K)\ \middle|\ \Delta
v_{h}\in\mathbb{P}_{p-2}(K),\;v_{h}{}_{|e}\in\mathbb{P}_{p}(e)\quad\forall
e\in\mathcal{E}^{K}\right\\}.$ (2)

We endow the space $V_{h}(K)$ with the following set of unisolvent degrees of
freedom: given $v_{h}$ in $V_{h}(K)$,

  * •

the point values of $v_{h}$ at the vertices of $K$;

  * •

on each edge $e$ of $K$, the point values of $v_{h}$ at the $p-1$ internal
Gauß-Lobatto nodes;

  * •

given the scaled monomial basis $\\{m_{\boldsymbol{\alpha}}\\}$ of
$\mathbb{P}_{p-2}(K)$ as in (1), the scaled moments

$\frac{1}{|K|}\int_{K}m_{\boldsymbol{\alpha}}\ v_{h}.$

The second set of degrees of freedom can be replaced by suitably scaled edge
moments or uniformly spaced nodes. We collect the above degrees of freedom in
the set $\\{\operatorname{dof}_{j}\\}_{j=1}^{N_{K}}$, $N_{K}$ being the
dimension of $V_{h}(K)$.

The degrees of freedom of $V_{h}(K)$ allow for the computation of several
polynomial projections [9]. In what follows, we need the operator
$\Pi^{\nabla}_{p}:H^{1}(K)\to\mathbb{P}_{p}(K)$ defined as

$\begin{cases}(\nabla q_{p},\nabla(v-\Pi^{\nabla}_{p}v))_{0,K}\qquad\forall
q_{p}\in\mathbb{P}_{p}(K)\\\ \int_{\partial
K}(v-\Pi^{\nabla}_{p}v)=0,\end{cases}$ (3)

and the operator $\Pi^{0}_{p-2}:L^{2}(K)\to\mathbb{P}_{p-2}(K)$ defined as

$(q_{p-2},v-\Pi^{0}_{p-2}v)_{0,K}\qquad\qquad\forall
q_{p-2}\in\mathbb{P}_{p-2}(K).$ (4)

Given the degrees of freedom of a function $v_{h}$ in $V_{h}(K)$, we can
compute $\Pi^{\nabla}_{p}v_{h}$ and $\Pi^{0}_{p-2}v_{h}$; see [9].

Other options to fix the constant part of $\Pi^{\nabla}_{p}$, see the second
condition in (3), can be found in the literature. For instance, it is
alternatively possible to fix the average in the bulk or the arithmetic
average of the values at the vertices of the polygon.

The standard discretization of the bilinear form
$a^{K}(\cdot,\cdot):=(\nabla\cdot,\nabla\cdot)_{0,K}$ in nodal conforming
virtual elements is given by

$a^{K}_{h}(u_{h},v_{h}):=a^{K}(\Pi^{\nabla}_{p}u_{h},\Pi^{\nabla}_{p}v_{h})+S^{K}((I-\Pi^{\nabla}_{p})u_{h},(I-\Pi^{\nabla}_{p})v_{h}).$
(5)

The bilinear form $S^{K}(\cdot,\cdot):V_{h}(K)\times V_{h}(K)$ is required to
be computable via the degrees of freedom and satisfies the following stability
bounds: there exist $0<\alpha_{*}\leq\alpha^{*}$ independent of $h_{K}$ such
that

$\alpha_{*}{\left|{v_{h}}\right|}_{1,K}^{2}\leq
S^{K}(v_{h},v_{h})\leq\alpha_{*}{\left|{v_{h}}\right|}_{1,K}^{2}\qquad\qquad\forall
v_{h}\in V_{h}(K)\cap\ker(\Pi^{\nabla}_{p}).$ (6)

###  2.2 Two stabilizations for nodal conforming virtual elements

We introduce two stabilizations that are common in the literature of nodal
conforming virtual elements. The first one is known as the “dofi-dofi”
stabilization [9] and is given by

$S^{K}(u_{h},v_{h}):=\sum_{j=1}^{N_{K}}\operatorname{dof}_{j}(u_{h})\operatorname{dof}_{j}(v_{h})\qquad\qquad\forall
u_{h},v_{h}\in V_{h}(K).$ (7)

The second one, which we shall refer to as “projected” stabilization, is [13]

$S^{K}(u_{h},v_{h}):=h_{K}^{-1}(u_{h},v_{h})_{0,\partial
K}+h_{K}^{-2}(\Pi^{0}_{p-2}u_{h},\Pi^{0}_{p-2}v_{h})_{0,K}\qquad\qquad\forall
u_{h},v_{h}\in V_{h}(K).$ (8)

Both stabilizations are computable using the degrees of freedom of $V_{h}(K)$.

######  Lemma 2.1.

The bilinear forms $S^{K}(\cdot,\cdot)$ in (7) and (8) satisfy (6).

###### Proof.

We prove the assertion for the “projected” stabilization (8) only; the details
for the “dofi-dofi” stabilization (7) are similar but slightly more involved.
The assertion for the stabilization in (7) follows combining the bounds for
the stabilization in (8) and the techniques, e.g., in [16, 29, 39].

The lower bound. We recall an inverse estimate for functions with polynomial
Laplacian; see, e.g., [32, Lemma 10] or [13, Theorem 2]:

${\left\|{\Delta v_{h}}\right\|}_{0,K}\lesssim
h_{K}^{-1}{\left|{v_{h}}\right|}_{1,K}\qquad\qquad\forall v_{h}\in
H^{1}(K),\;\Delta v_{h}\in\mathbb{P}_{p-2}(K).$ (9)

Furthermore, we have the polynomial inverse estimate

${\left|{q_{p}}\right|}_{\frac{1}{2},\partial K}\lesssim
h_{K}^{-\frac{1}{2}}{\left\|{q_{p}}\right\|}_{0,\partial K}\qquad\qquad\forall
q_{p}\in\mathcal{C}^{0}(\partial K),\quad
q_{p}{}_{|e}\in\mathbb{P}_{p}(e),\quad\forall e\in\mathcal{E}^{K}.$ (10)

Integrating by parts, recalling that $\Delta v_{h}$ belongs to
$\mathbb{P}_{p-2}(K)$, applying the Cauchy-Schwarz inequality, invoking the
definition of negative Sobolev norms, and using the inverse estimates (9) and
(10), the lower bound in (6) follows:

$\begin{split}{\left|{v_{h}}\right|}^{2}_{1,K}&=-\int_{K}\Delta v_{h}\
v_{h}+\int_{\partial K}(\mathbf{n}_{K}\cdot\nabla v_{h})v_{h}=-\int_{K}\Delta
v_{h}\ \Pi^{0}_{p-2}v_{h}+\int_{\partial K}(\mathbf{n}_{K}\cdot\nabla
v_{h})v_{h}\\\ &\leq{\left\|{\Delta
v_{h}}\right\|}_{0,K}{\left\|{\Pi^{0}_{p-2}v_{h}}\right\|}_{0,K}+{\left\|{\mathbf{n}_{K}\cdot\nabla
v_{h}}\right\|}_{-\frac{1}{2},\partial
K}{\left|{v_{h}}\right|}_{\frac{1}{2},\partial K}\\\ &\lesssim{\left\|{\Delta
v_{h}}\right\|}_{0,K}{\left\|{\Pi^{0}_{p-2}v_{h}}\right\|}_{0,K}+(h_{K}{\left\|{\Delta
v_{h}}\right\|}_{0,K}+{\left|{v_{h}}\right|}_{1,K}){\left|{v_{h}}\right|}_{\frac{1}{2},\partial
K}\\\
&\lesssim{\left|{v_{h}}\right|}_{1,K}\left(h_{K}^{-1}{\left\|{\Pi^{0}_{p-2}v_{h}}\right\|}_{0,K}+h_{K}^{-\frac{1}{2}}{\left\|{v_{h}}\right\|}_{0,\partial
K}\right).\end{split}$

The upper bound. Using the stability of $\Pi^{0}_{p-2}$ in the $L^{2}$ norm,
the trace inequality, and the Poincaré inequality (recall we are assuming
$v_{h}$ belongs to $\ker(\Pi^{\nabla}_{p})$), the upper bound in (6) follows:

$S^{K}(v_{h},v_{h}):=h_{K}^{-1}{\left\|{v_{h}}\right\|}^{2}_{0,\partial
K}+h_{K}^{-2}{\left\|{\Pi^{0}_{p-2}v_{h}}\right\|}^{2}_{0,K}\leq
h_{K}^{-1}{\left\|{v_{h}}\right\|}^{2}_{0,\partial
K}+h_{K}^{-2}{\left\|{v_{h}}\right\|}^{2}_{0,K}\lesssim{\left|{v_{h}}\right|}^{2}_{1,K}.$

∎

From the proof of Lemma 2.1, we can see that

  * •

the lower bound in (6) follows from (polynomial and virtual) inverse
estimates, but no zero average condition (based on the fact that $v_{h}$
belongs also to $\ker(\Pi^{\nabla}_{p})$) is used;

  * •

the upper bound in (6) is proven based only on “direct estimates”, such as the
Poincaré inequality and the trace inequality, and therefore is valid for
$H^{1}$ functions with zero average.

For these reasons, Lemma 2.1 immediately generalizes as follows.

######  Corollary 2.2.

The bilinear forms $S^{K}(\cdot,\cdot)$ in (7) and (8) satisfy the following
stability bounds: there exist $0<\alpha_{*}\leq\alpha^{*}$ such that

$\alpha_{*}{\left|{v_{h}}\right|}_{1,K}^{2}\leq S^{K}(v_{h},v_{h})\quad\forall
v_{h}\in V_{h}(K),\qquad
S^{K}(v,v)\leq\alpha^{*}{\left|{v}\right|}_{1,K}^{2}\quad\forall v\in
H^{1}(K),\;(v,1)_{0,D}=0,$

where $D$ is any subset with nonzero measure of either $K$ or $\partial K$.

The stability bounds in Lemma 2.1 and Corollary 2.2 can be easily extended to
the case of the so-called enhanced virtual element spaces introduced in [1].
More precisely, define

$\begin{split}V_{h}(K):=\Big{\\{}v_{h}\in H^{1}(K)&\ \Big{|}\ \Delta
v_{h}\in\mathbb{P}_{p}(K);\quad v_{h}{}_{|e}\in\mathbb{P}_{p}(e);\quad
v_{h}{}_{|\partial K}\in\mathcal{C}^{0}(\partial K);\\\
&\quad\int_{K}(v_{h}-\Pi^{\nabla}_{p}v_{h})m_{\boldsymbol{\alpha}}=0\;\forall|\boldsymbol{\alpha}|=p-1,p\Big{\\}}.\end{split}$
(11)

This space can be endowed with the same degrees of freedom of the standard
space virtual element space in (2). Such degrees of freedom allow us to
compute the projectors $\Pi^{\nabla}_{p+2}$ and $\Pi^{0}_{p}$; see (3) and
(4). It is immediate to check that the stabilization

$S^{K}(u_{h},v_{h}):=h_{K}^{-1}(u_{h},v_{h})_{0,\partial
K}+h_{K}^{-2}(\Pi^{0}_{p}u_{h},\Pi^{0}_{p}v_{h})_{0,K}\qquad\qquad\forall
u_{h},v_{h}\in V_{h}(K)$

satisfies Lemma 2.1 and Corollary 2.2.

###  2.3 Early essential literature and later contributions

The first work containing mathematical proofs on the stabilization is [16].
This and other three works constitute the early backbone of the stability
analysis of the virtual element method. In particular, we pinpoint and rapidly
describe these four works, which were published in 2017-2018:

  * •

[16] contains the first analysis of the stabilization in two dimensional nodal
conforming virtual elements; an arbitrary number of edges is allowed;
stability bounds are there derived for different stabilizations (including the
“dofi-dofi” one);

  * •

[28, 29] contain the stability analysis for two and three dimensional nodal
conforming virtual elements; arbitrary numbers of edges and faces are allowed;
stability bounds are derived for different stabilizations (including the
“dofi-dofi” one);

  * •

[39] contains the stability analysis for two dimensional nodal conforming
virtual elements; more standard geometries are employed.

It is our opinion that the presentation in [39] is the simplest one and is
therefore recommended to virtual elements novices; the presentation in [16]
and [28, 29] is more technical and applies to more general geometries, whence
these contributions are recommended to more expert readers.

After 2018, other works were devoted to additional mathematical aspects of the
stability in nodal conforming virtual elements. Amongst others, we mention
nodal conforming virtual elements on curved domains [24]; robustness with
respect to anisotropic elements [33, 34]; virtual elements stabilized by means
of higher-order polynomial energy projection terms [25, 26, 37, 47]; lack of
robustness with respect to the degree of accuracy of the method [13]; nodal
serendipity virtual elements [21]; presence of stabilization terms in residual
error estimators [19, 32]. We shall elaborate more on the two last topics in
Section 5 below.

The literature stability analysis for other types of elements is more limited.
The reason for this is that nodal conforming virtual element functions have a
“second order elliptic structure”. In other virtual elements, local spaces
consist of functions solving local problems (with polynomial data) that are in
general more technical to handle.

##  3 Stability in other “elliptic-type” virtual elements

In this section, we investigate the role of the stabilization for other
“elliptic-type” virtual elements that still have an “elliptic structure”. In
particular, we focus on “second order” nonconforming virtual elements in
Section 3.1; investigate their stability properties in Section 3.2; review
conforming and nonconforming arbitrarily regular virtual element spaces in
Section 3.3.

###  3.1 “Nodal” nonconforming virtual elements

Following [8], given a polygonal element $K$, $\mathcal{E}^{K}$ its set of
edges, and a positive integer number $p$, we define the “nodal” nonconforming
virtual element

$V_{h}(K):=\left\\{v_{h}\in H^{1}(K)\ \middle|\ \Delta
v_{h}\in\mathbb{P}_{p-2}(K),\;\mathbf{n}_{K}\cdot\nabla
v_{h}{}_{|e}\in\mathbb{P}_{p-1}(e)\quad\forall e\in\mathcal{E}^{K}\right\\}.$
(12)

We endow the space $V_{h}(K)$ with the following set of unisolvent degrees of
freedom: given $v_{h}$ in $V_{h}(K)$,

  * •

on each edge $e$ of $K$, given the scaled monomial basis
$\\{m_{\alpha}^{e}\\}$ of $\mathbb{P}_{p-1}(e)$ discussed in Section 1, the
scaled moments

$\frac{1}{|e|}\int_{e}m_{\alpha}^{e}\ v_{h};$

  * •

given the scaled monomial basis $\\{m_{\boldsymbol{\alpha}}\\}$ of
$\mathbb{P}_{p-2}(K)$ as in (1), the scaled moments

$\frac{1}{|K|}\int_{K}m_{\boldsymbol{\alpha}}\ v_{h}.$

We collect the above degrees of freedom in the set
$\\{\operatorname{dof}_{j}\\}_{j=1}^{N_{K}}$, being $N_{K}$ the dimension of
$V_{h}(K)$.

The degrees of freedom of $V_{h}(K)$ allow for the computation of several
polynomial projections [8]. In what follows, we need the operators
$\Pi^{\nabla}_{p}:H^{1}(K)\to\mathbb{P}_{p}(K)$ as in (3),
$\Pi^{0}_{p-2}:L^{2}(K)\to\mathbb{P}_{p-2}(K)$ as in (4), and
$\Pi^{0,e}_{p-1}:L^{2}(e)\to\mathbb{P}_{p-1}(e)$ defined as

$(q_{p-1}^{e},v-\Pi^{0,e}_{p-1}v)_{0,e}\qquad\qquad\forall
q_{p-1}^{e}\in\mathbb{P}_{p-1}(e).$

Given the degrees of freedom of a function $v_{h}$ in $V_{h}(K)$, we can
compute $\Pi^{\nabla}_{p}v_{h}$, $\Pi^{0}_{p-2}v_{h}$, and
$\Pi^{0,e}_{p-1}v_{h}$ for all edges $e$; see [8]. The standard discretization
of the bilinear form $a^{K}(\cdot,\cdot):=(\nabla\cdot,\nabla\cdot)_{0,K}$ in
“nodal” nonconforming virtual elements is as in (5).

Also in the “nodal” nonconforming virtual element setting, the stabilization
$S^{K}(\cdot,\cdot):V_{h}(K)\times V_{h}(K)$ is a bilinear form that is
computable via the degrees of freedom, and is such that there exist
$0<\alpha_{*}\leq\alpha^{*}$ independent of $h_{K}$ for which the stability
bounds in (6) are valid.

###  3.2 Stability bounds in “nodal” nonconforming virtual elements

We introduce two stabilizations for “nodal” nonconforming virtual elements.
The first one is the “dofi-dofi” stabilization as in (7). The second one,
which we shall refer to as “projected” stabilization, is

$S^{K}(u_{h},v_{h}):=h_{K}^{-1}\sum_{e\in\mathcal{E}^{K}}(\Pi^{0,e}_{p-1}u_{h},\Pi^{0,e}_{p-1}v_{h})_{0,e}+h_{K}^{-2}(\Pi^{0}_{p-2}u_{h},\Pi^{0}_{p-2}v_{h})_{0,K}\quad\forall
u_{h},v_{h}\in V_{h}(K).$ (13)

Both stabilization are computable using the degrees of freedom of $V_{h}(K)$.

######  Lemma 3.1.

The bilinear forms $S^{K}(\cdot,\cdot)$ in (7) and (13) satisfy (6).

###### Proof.

We prove only the assertion for the “projected” stabilization (13); to this
aim, we follow the guidelines in [50, Theorem 3.2]. The details for the “dofi-
dofi” stabilization (7) are similar but slightly more involved.

The following polynomial inverse inequality holds true [50, Theorem 3.2]:

${\left\|{q_{p}}\right\|}_{0,\partial K}\lesssim
h_{K}^{-\frac{1}{2}}{\left\|{q_{p}}\right\|}_{-\frac{1}{2},\partial
K}\qquad\qquad\forall q_{p}\in\mathcal{C}^{0}(\partial K),\quad
q_{p}|_{|e}\in\mathbb{P}_{p}(e)\quad\forall e\in\mathcal{E}^{K}.$ (14)

The lower bound. Integrating by parts, recalling the properties of functions
in “nodal” nonconforming virtual element spaces in (12), applying the Cauchy-
Schwarz inequality twice, using the polynomial inverse inequality (14),
invoking the Neumann trace inequality [53, Theorem A.33], and recalling the
virtual inverse estimate (9), we can write

$\begin{split}{\left|{v_{h}}\right|}^{2}_{1,K}&=-\int_{K}\Delta v_{h}\
v_{h}+\sum_{e\in\mathcal{E}^{K}}\int_{e}(\mathbf{n}_{K}\cdot\nabla
v_{h})v_{h}=-\int_{K}\Delta v_{h}\
\Pi^{0}_{p-2}v_{h}+\sum_{e\in\mathcal{E}^{K}}\int_{e}(\mathbf{n}_{K}\cdot\nabla
v_{h})\Pi^{0,e}_{p-1}v_{h}\\\ &\leq{\left\|{\Delta
v_{h}}\right\|}_{0,K}{\left\|{\Pi^{0}_{p-2}v_{h}}\right\|}_{0,K}+\sum_{e\in\mathcal{E}^{K}}{\left\|{\mathbf{n}_{K}\cdot\nabla
v_{h}}\right\|}_{0,e}{\left\|{\Pi^{0,e}_{p-1}v_{h}}\right\|}_{0,e}\\\
&\leq{\left\|{\Delta
v_{h}}\right\|}_{0,K}{\left\|{\Pi^{0}_{p-2}v_{h}}\right\|}_{0,K}+{\left\|{\mathbf{n}_{K}\cdot\nabla
v_{h}}\right\|}_{0,\partial
K}\Big{(}\sum_{e\in\mathcal{E}^{K}}{\left\|{\Pi^{0,e}_{p-1}v_{h}}\right\|}_{0,e}^{2}\Big{)}^{\frac{1}{2}}\\\
&\lesssim{\left\|{\Delta
v_{h}}\right\|}_{0,K}{\left\|{\Pi^{0}_{p-2}v_{h}}\right\|}_{0,K}+{\left\|{\mathbf{n}_{K}\cdot\nabla
v_{h}}\right\|}_{-\frac{1}{2},\partial
K}h_{K}^{-\frac{1}{2}}\Big{(}\sum_{e\in\mathcal{E}^{K}}{\left\|{\Pi^{0,e}_{p-1}v_{h}}\right\|}_{0,e}^{2}\Big{)}^{\frac{1}{2}}\\\
&\lesssim{\left\|{\Delta
v_{h}}\right\|}_{0,K}{\left\|{\Pi^{0}_{p-2}v_{h}}\right\|}_{0,K}+(h_{K}{\left\|{\Delta
v_{h}}\right\|}_{0,K}+{\left|{v_{h}}\right|}_{1,K})h_{K}^{-\frac{1}{2}}\Big{(}\sum_{e\in\mathcal{E}^{K}}{\left\|{\Pi^{0,e}_{p-1}v_{h}}\right\|}_{0,e}^{2}\Big{)}^{\frac{1}{2}}\\\
&\lesssim{\left|{v_{h}}\right|}_{1,K}\left(h_{K}^{-1}{\left\|{\Pi^{0}_{p-2}v_{h}}\right\|}_{0,K}+h_{K}^{-\frac{1}{2}}\Big{(}\sum_{e\in\mathcal{E}^{K}}{\left\|{\Pi^{0,e}_{p-1}v_{h}}\right\|}_{0,e}^{2}\Big{)}^{\frac{1}{2}}\right).\end{split}$

The upper bound. Using the stability of $\Pi^{0}_{p-2}$ and $\Pi^{0,e}_{p-1}$
in the $L^{2}(K)$ and $L^{2}(e)$ (for all edges $e$ of $K$) norms,
respectively, the trace inequality, and the Poincaré inequality, the upper
bound in (6) follows:

$\begin{split}S^{K}(v_{h},v_{h})&:=h_{K}^{-1}\sum_{e\in\mathcal{E}^{K}}{\left\|{\Pi^{0,e}_{p-1}v_{h}}\right\|}^{2}_{0,e}+h_{K}^{-2}{\left\|{\Pi^{0}_{p-2}v_{h}}\right\|}^{2}_{0,K}\leq
h_{K}^{-1}\sum_{e\in\mathcal{E}^{K}}{\left\|{v_{h}}\right\|}^{2}_{0,e}+h_{K}^{-2}{\left\|{v_{h}}\right\|}^{2}_{0,K}\\\
&=h_{K}^{-1}{\left\|{v_{h}}\right\|}^{2}_{0,\partial
K}+h_{K}^{-2}{\left\|{v_{h}}\right\|}^{2}_{0,K}\lesssim{\left|{v_{h}}\right|}^{2}_{1,K}.\end{split}$

∎

As in the case of nodal conforming virtual elements, Lemma 3.1 can be
generalized in the sense of Corollary 2.2.

Moreover, as in Section 2.2, it is possible to define an enhanced version of
the space in (12):

$\begin{split}V_{h}(K):=\Big{\\{}v_{h}\in H^{1}(K)&\ \Big{|}\ \Delta
v_{h}\in\mathbb{P}_{p}(K);\quad\mathbf{n}_{K}\cdot\nabla
v_{h}{}_{|e}\in\mathbb{P}_{p-1}(e)\quad\forall e\in\mathcal{E}^{K};\\\
&\quad\int_{K}(v_{h}-\Pi^{\nabla}_{p}v_{h})m_{\boldsymbol{\alpha}}=0\quad\forall|\boldsymbol{\alpha}|=p-1,p\Big{\\}}.\end{split}$

We can endow this space with the same degrees of freedom as for the space in
(12), which allow for the computation of higher order polynomial projectors
$\Pi^{\nabla}_{p+2}$ and $\Pi_{p}^{0}$; see (3) and (4).

It can be checked that the following stabilization satisfies Lemma 3.1:

$S^{K}(u_{h},v_{h}):=h_{K}^{-1}\sum_{e\in\mathcal{E}^{K}}(\Pi^{0,e}_{p-1}u_{h},\Pi^{0,e}_{p-1}v_{h})_{0,e}+h_{K}^{-2}(\Pi^{0}_{p}u_{h},\Pi^{0}_{p}v_{h})_{0,K}\qquad\qquad\forall
u_{h},v_{h}\in V_{h}(K).$

Other theoretical results on the stabilization for “nodal” nonconforming
virtual elements can be found in [27].

###  3.3 Arbitrarily regular virtual element spaces

Arbitrarily regular virtual element spaces were one of the first
generalization of the nodal conforming virtual element method detailed in
Section 2.1 and trace back to almost the inception of the method; see [18,
31]. Over the years, several conforming and nonconforming variants have been
proposed; see, e.g., [4, 5, 56].

Differently from standard $\mathcal{C}^{0}$ elements, arbitrarily regular
virtual elements are designed so as global $\mathcal{C}^{k}$ spaces, $k>0$,
can be constructed. This is accomplished by an enrichment of interface and
bulk degrees of freedom. In particular, local virtual element spaces are the
set of solutions to polyharmonic problems with polynomial data.

Thence, the “elliptic structure” of the spaces can be still employed while
deriving the stability bounds. For this reason, the analysis is quite similar
to that for standard “nodal” conforming and nonconforming virtual elements. We
refer to [40] and [38] for the stability bounds in nonconforming and
conforming arbitrarily regular virtual elements. It is quite interesting the
fact that both references are rather recent (2020 and 2022, respectively).
This is probably motivated by the fact that (_i_) stability bounds are derived
in any dimension; (_ii_) even though the structure of the arbitrarily regular
virtual element spaces is still “elliptic”, the interaction of high order
derivatives and the presence of multiple boundary conditions render the
analysis of the stability bounds quite involved.

We report here, for the 2D case, a stability result for $\mathcal{C}^{1}$
virtual elements. Let $p\geq 3$ 111The case $p=2$ requires a slightly
different but similar treatment, and is therefore omitted. and consider the
space  $V_{h}(K):=\left\\{v_{h}\in H^{2}(K)\ \middle|\
\Delta^{2}v_{h}\in\mathbb{P}_{p-4}(K),\;v_{h}{}_{|e}\in\mathbb{P}_{p}(e),\;\mathbf{n}_{K}\cdot\nabla
v_{h}\in\mathbb{P}_{p-1}(e)\quad\forall e\in\mathcal{E}^{K}\right\\}.$ We
endow this space with the following set of degrees of freedom [18, 46]: given
$v_{h}$ in $V_{h}(K)$,  • the point values of $v_{h}$ at the vertices of $K$;
• the point values of $h_{K}\partial_{1}v_{h}$ and $h_{K}\partial_{2}v_{h}$ at
the vertices of $K$, where $\partial_{1}$ and $\partial_{2}$ are the
derivatives along the edges matching at such vertices; • for all edges $e$ in
$\mathcal{E}^{K}$, given $\\{m_{\alpha}^{e}\\}$ the scaled and shifted
monomial basis of $\mathbb{P}_{p-4}(e)$, the moments
$\frac{1}{h_{e}}\int_{e}m_{\alpha}^{e}\ v_{h};$ • for all edges $e$ in
$\mathcal{E}^{K}$, given $\\{m_{\alpha}^{e}\\}$ the scaled and shifted
monomial basis of $\mathbb{P}_{p-3}(e)$, the moments $\int_{e}m_{\alpha}^{e}\
\mathbf{n}_{K}\cdot\nabla v_{h};$ • given $\\{m_{\boldsymbol{\alpha}}\\}$ the
scaled and shifted monomial basis of $\mathbb{P}_{p-4}(e)$, the moments
$\frac{1}{|K|}\int_{K}m_{\boldsymbol{\alpha}}\ v_{h}.$ Based on such degrees
of freedom, the following stabilization is computable:
$S^{K}(u_{h},v_{h}):=h_{K}^{-4}(\Pi^{0}_{p-4}u_{h},\Pi^{0}_{p-4}v_{h})_{0,K}+h_{K}^{-3}(u_{h},v_{h})_{0,\partial
K}+h_{K}^{-1}(\mathbf{n}_{K}\cdot\nabla u_{h},\mathbf{n}_{K}\cdot\nabla
v_{h})_{0,\partial K}.$ (15) Proceeding as in Section 2.2 and [38] yields the
following result.

######  Lemma 3.2.

The stabilization in (15) satisfies the two following bounds: there exist
$0<\alpha_{*}\leq\alpha^{*}$ such that
$\begin{split}&\alpha_{*}{\left|{v_{h}}\right|}_{2,K}^{2}\leq
S^{K}(v_{h},v_{h})\quad\forall v_{h}\in V_{h}(K),\\\
&S^{K}(v,v)\leq\alpha^{*}{\left|{v}\right|}_{2,K}^{2}\qquad\forall v\in
H^{1}(K)\quad\text{ such that }\quad(v,1)_{0,D}=0,\;(\nabla
v,(1,1))_{0,D}=0,\end{split}$ where $D$ is any subset with nonzero measure of
either $K$ or $\partial K$.

##  4 Stability in “nonelliptic” virtual elements

In this section, we review the literature on the stabilization for some
“nonelliptic” virtual elements: face (Section 4.1); edge (Section 4.2);
Stokes-like (Section 4.3); immersed (Section 4.4) virtual elements.

###  4.1 Face virtual elements

Face virtual elements were designed in [30, 11, 10] and generalize the
Raviart-Thomas elements to polytopic meshes. Both in two and three dimensions,
local face spaces consist of functions solving local “div-rot” problems and
with polynomial normal components on faces. The degrees of freedom consist of
bubble-like moments in the element and moments of normal components over
faces.

The first stability analysis (on 2D curved elements) can be found in [42].
Lowest order two and three dimensional face spaces were analyzed in [20],
general order two and three dimensional (standard and serendipity) face spaces
in [22], and three dimensional face spaces on curved polyhedra in [43].
Related results are discussed in [57].

Compared to the elliptic case, the stability analysis is here complicated by
the “div-rot” structure of the spaces. Notably, the analysis needs the design
of certain Helmholtz-like decompositions, as well as employing more
sophisticated results from the theory of vector potential and div-rot systems;
see, e.g., [41, 2].

###  4.2 Edge virtual elements

Edge virtual elements were first designed in [10] and generalize the Nédélec
element to polytopic elements. In three dimensions, local edge spaces consist
of functions solving local “div-curlcurl” problems inside the element; local
“div-rot” problems on faces; polynomial tangential traces over edges. The
degrees of freedom consist of bubble-like moments in the element and on faces,
and moments of tangential components over edges.

The first stability analysis (lowest order, in two and three dimensions) can
be found in [20]; the general order (standard and serendipity) case was later
tackled in [22].

Compared to the the case of face elements, the analysis of edge elements in
three dimensions is further complicated by the fact that virtual element
functions are solutions to different types of problems inside the element
(“div-curlcurl” problems) and on faces (“div-rot” problems), which require a
different treatment to establish suitable stability estimates.

###  4.3 Stokes-like virtual elements

Lowest order Stokes-like virtual elements appeared in [3] and were later
generalized to the general order case in [17]. Local spaces consist of
functions solving local Stokes-like with polynomial data; in particular, the
divergence of Stokes-like virtual element functions is polynomial. This allows
for the design of methods that are automatically divergence free. In two
dimensions, the degrees of freedom consist of bubble-like moments in the
element and point values on the boundary; in three dimensions, of element and
face moments, and point values on the edges.

Stokes-like virtual elements are very successful and were the topic of a
plethora of articles. However, to the best of our knowledge, the first and
only paper dealing with the stability analysis of Stokes-like virtual element
spaces is [23]. The stability bounds are essentially derived based on the
stable inf-sup structure of the local Stokes problems, and polynomial and
(novel) virtual element inverse estimates.

###  4.4 Immersed-like virtual elements

More recently, immersed-like virtual elements have been introduced in two and
three dimensions for elliptic and Maxwell problems; see [35, 36]. Such spaces
are particularly effective when handling problems with nonsmooth coefficients.
The irregular behaviour of the exact solution is captured by local immersed-
like virtual element functions in a Trefftz fashion, as they are solutions to
local problems involving the same nonsmooth coefficients as in the continuous
problem.

##  5 Dependence on the degree of accuracy and error estimators

Two situations where the presence of the stabilization might be troublesome
are the $p$\- and $hp$-versions of the method, and when proving the
reliability and efficiency of error estimators. We elaborate on these two
topics in Sections 5.1 and 5.2, respectively.

###  5.1 Stability bounds depending on the degree of accuracy

The $p$\- and $hp$-versions of a Galerkin method aim at achieving convergence
by increasing the dimension of the local approximation spaces (while keeping
the mesh fixed), and by mesh refinement and $p$-version at once, respectively.

Stability bounds with explicit dependence on the degree of accuracy $p$ were
investigated in several works. Amongst them, we recall the following
contributions: $p$-explicit stability bounds were first derived in [13] for
the “projected” stabilization in (8); this result was extended [6] to the
“dofi-dofi” stabilization in (7); the $p$-version nonconforming virtual
elements was addressed in [50].

In all cases, at least one of the constants $\alpha_{*}$ and $\alpha^{*}$ in
(6) depend on $p$. This is not surprising as several inverse estimates are
employed to derive the lower bound in (6); see the lower bounds in Lemmas 2.1
and 3.1. However, the suboptimality with respect to the degree of accuracy is
rather mild in practice; see [13, Section 4.1].

###  5.2 The stability in residual error estimators

In finite elements, an error estimator $\eta$ given by the combination of
local error estimators $\eta_{K}$ is a quantity that can be computed by means
of the discrete solution $u_{h}$ to the method and has to be designed so as to
be comparable to the error of the method.

For the Poisson problem, standard reliability and efficiency estimates read

${\left|{u-u_{h}}\right|}_{1,\Omega}\lesssim\eta(u_{h})+\operatorname{HOT}_{\Omega}(f),\qquad\qquad\eta_{K}(u_{h})\lesssim{\left|{u-u_{h}}\right|}_{1,\omega_{K}}+\operatorname{HOT}_{\omega_{K}}(f),$
(16)

where $\omega_{K}$ is the patch of elements around $K$ and
$\operatorname{HOT}_{D}(f)$ is a high-order oscillation term on a domain $D$
involving the right-hand side $f$.

Residual error estimators in virtual elements [19, 32] satisfy analogous
bounds that differ from (16) in two aspects: (_i_) the discrete solution
$u_{h}$ is not available in closed form, so it must be replaced by a
polynomial projection; (_ii_) the stabilization appears in the error estimator
and error estimates.

So, for a computable virtual element error estimator $\widetilde{\eta}$ given
by the combination of local virtual element error estimators
$\widetilde{\eta}_{K}$, the virtual element counterpart of (16) has the
following form:

$\begin{split}&{\left|{u-u_{h}}\right|}_{1,\Omega}\lesssim\widetilde{\eta}(\Pi^{\nabla}_{p}u_{h})+\sum_{K\in\mathcal{T}_{h}}S^{K}((I-\Pi^{\nabla}_{p})u_{h},(I-\Pi^{\nabla}_{p})u_{h})^{\frac{1}{2}}+\operatorname{HOT}_{\Omega}(f),\\\
&\widetilde{\eta}_{K}(\Pi^{\nabla}_{p}u_{h})\lesssim{\left|{u-u_{h}}\right|}_{1,\omega_{K}}+\\!\\!\\!\\!\sum_{K\in\mathcal{T}_{h},\
K\subset\omega_{K}}\\!\\!\\!\\!S^{K}((I-\Pi^{\nabla}_{p})u_{h},(I-\Pi^{\nabla}_{p})u_{h})^{\frac{1}{2}}+\operatorname{HOT}_{\omega_{K}}(f).\end{split}$

The presence of the extra stabilization term in the bounds above might be
inauspicious. On the one hand, such term is not robust with respect to the
degree of accuracy as discussed in Section 5.1. On the other hand, adaptive
mesh refinements may lead to polygonal meshes with several hanging nodes and
possibly very small facets; in that case, one should be extremely careful in
designing stabilization that are robust with respect to badly-shaped elements.

A partial breakthrough facing these issues is contained in [12], where the
stabilization term is removed in the reliability and efficiency estimates.
However, this has been proved under restrictive assumptions: meshes must
consist of elements with triangular shape; the 3D version is not covered;
lowest order elements are employed. So, general reliability and efficiency
estimates that are robust with respect to the stabilization are currently not
available, albeit the subject of current study.

##  6 The stability bounds may imply interpolation estimates

Interpolation estimates by means of functions in the virtual element space are
an important ingredient in the error analysis of the method. In this section,
we first report standard ways of proving interpolation estimates in conforming
and nonconforming virtual elements; see Sections 6.1 and 6.2, respectively.
Next, in Section 6.3, we show a more recent strategy to derive interpolation
estimates based on the stability bounds covering the conforming and
nonconforming cases at once.

###  6.1 Standard proof for interpolation estimates: conforming elements

The first proof of interpolation estimates for nodal conforming virtual
elements traces back to 2015; see [51]. Several variants have been proposed
ever since, most of them extending the ideas therein.

Here, we only provide details on the two dimensional nodal conforming virtual
elements; see [51, Proposition 4.2]. Given an element $K$ split into a shape-
regular triangulation $\mathcal{T}(K)$, $v$ in $H^{1}(K)$, $q_{p}$ in
$\mathbb{P}_{p}(K)$, and $v_{C}$ the Clément quasi-interpolant of $v$ over
$\mathcal{T}(K)$, we define $v_{I}$ over $K$ as the weak solution to

$\begin{cases}-\Delta v_{I}=-\Delta q_{p}&\text{in }K\\\ v_{I}=v_{C}&\text{on
}\partial K.\end{cases}$

A minimum energy argument entails

${\left|{v-v_{I}}\right|}_{1,K}\leq
2{\left|{v-q_{p}}\right|}_{1,K}+{\left|{v-v_{C}}\right|}_{1,K},$

whence interpolation estimates follow from polynomial quasi-interpolation and
approximation estimates.

Thus, deriving interpolation estimates for conforming virtual elements
strongly hinges upon using the structure of the local virtual element spaces,
i.e., the stability of the continuous formulation. When considering other
types of virtual elements, this may result in interpolation estimates that are
much harder to derive. Prototypical examples of this fact are interpolation
estimates in Stokes-like [17]; Hellinger-Reissner-like [7, 44]; face and edge
[20, 22] virtual elements.

###  6.2 Standard proof for interpolation estimates: nonconforming elements

The first proof of interpolation estimates for “nodal” nonconforming virtual
elements traces back to 2018; see [50]. Several variants have been proposed
ever since, most of them extending the ideas therein.

Here, we only provide details on the two dimensional “nodal” nonconforming
virtual elements; see [50, Proposition 3.1] and the later work [46, Corollary
4.1].

Given $K$ in $\mathcal{T}_{h}$ and $v$ in $H^{1}(K)$, we define $v_{I}$ in the
“nodal” nonconforming space $V_{h}(K)$ defined in (12) as the degrees of
freedom interpolant of $v$. Notably, $v_{I}$ is uniquely identified by

$\int_{K}(v-v_{I})q_{p-2}^{K}=0\quad\forall
q_{p-2}^{K}\in\mathbb{P}_{p-2}(K),\qquad\int_{e}(v-v_{I})q_{p-1}^{e}=0\quad\forall
q_{p-1}^{e}\in\mathbb{P}_{p-1}(e)\quad\forall e\in\mathcal{E}^{K}.$

For any $q_{p}^{K}$ in $\mathbb{P}_{p}(K)$, recalling that $\Delta v_{I}$
belongs to $\mathbb{P}_{p-2}(K)$ and $\mathbf{n}_{K}\cdot\nabla v_{I}{}_{|e}$
belongs to $\mathbb{P}_{p-1}(e)$ for all the edges $e$ of $K$, we readily
deduce

$\begin{split}{\left|{v-v_{I}}\right|}_{1,K}&=-\int_{K}\Delta(v-v_{I})\
(v-v_{I})+\sum_{e\in\mathcal{E}^{K}}\int_{e}\mathbf{n}_{K}\cdot\nabla(v-v_{I})\
(v-v_{I})\\\ &=-\int_{K}\Delta(v-q_{p}^{K})\
(v-v_{I})+\sum_{e\in\mathcal{E}^{K}}\int_{e}\mathbf{n}_{K}\cdot\nabla(v-q_{p}^{K})\
(v-v_{I})\\\
&=\int_{K}\nabla(v-q_{p}^{K})\cdot\nabla(v-v_{h})\leq{\left|{v-q_{p}^{K}}\right|}_{1,K}{\left|{v-v_{I}}\right|}_{1,K},\end{split}$

whence we deduce

$\inf_{v_{h}\in
V_{h}(K)}{\left|{v-v_{h}}\right|}_{1,K}\leq{\left|{v-v_{I}}\right|}_{1,K}\leq\inf_{q_{p}\in\mathbb{P}_{p}(K)}{\left|{v-q_{p}^{K}}\right|}_{1,K}.$

Extending this approach to other types of virtual elements may be not
immediate.

###  6.3 A more recent proof for interpolation estimates

There is a recent alternative approach originally employed in [23] to derive
interpolation estimates, which applies to conforming and nonconforming virtual
elements at once, and is based on having stability bounds at hand.

Again, to simplify the presentation we stick to the two dimensional nodal
conforming virtual element case and consider the stabilization in (7), even
though the following steps might be generalized to several other elements and
dimensions, as well as to other stabilizations. Let $K$ be a given element,
$v$ be in $H^{s}(K)$, $s>1$, and $q_{p}$ be any function in
$\mathbb{P}_{p}(K)$ with the same average of $v$ over $K$. We denote the
degrees of freedom interpolant of any sufficiently smooth function by adding a
subscript $\cdot_{I}$ to such function. Observe that $q_{p}=(q_{p})_{I}$.
Recalling Corollary 2.2 and using that $S^{K}(v_{I},v_{I})=S^{K}(v,v)$ by
definition (7), we readily get

$\begin{split}{\left|{v-v_{I}}\right|}_{1,K}&\leq{\left|{v-q_{p}}\right|}_{1,K}+{\left|{v_{I}-q_{p}}\right|}_{1,K}\leq{\left|{v-q_{p}}\right|}_{1,K}+\alpha_{*}^{-1}S^{K}(v_{I}-q_{p},v_{I}-q_{p})^{\frac{1}{2}}\\\
&={\left|{v-q_{p}}\right|}_{1,K}+\alpha_{*}^{-1}S^{K}((v-q_{p})_{I},(v-q_{p})_{I})^{\frac{1}{2}}={\left|{v-q_{p}}\right|}_{1,K}+\alpha_{*}^{-1}S^{K}(v-q_{p},v-q_{p})^{\frac{1}{2}}\\\
&\leq\left(1+\frac{\alpha^{*}}{\alpha_{*}}\right){\left|{v-q_{p}}\right|}_{1,K}.\end{split}$

Thus, upon having at disposal stability bounds of the form (6), we proved that
the best interpolation error is bounded by the best polynomial error up to the
stability constants. The above estimates readily extend to nonconforming nodal
virtual elements.

##  7 Conclusions and outlook

The first paper on the virtual element method was published ten years ago. The
analysis of the stabilization of the method is more recent and is still a
current and lively area of research. Several works on the stability in nodal
conforming virtual elements have been published; fewer on other types of
virtual elements. In this contribution, we reviewed the literature about the
role and theoretical analysis of the stabilization in the virtual element
method. We presented paradigmatic proofs for “nodal” conforming and
nonconforming virtual elements; we only mentioned the main results for other
types of elements. We further underlined that the stability bounds imply
interpolation estimates.

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