TITLE: Is this writing incorrect?
QUESTION [1 upvotes]: If we want to find $\frac{d}{dx}\cos x^2 $ then is this writing incorrect $\frac{d}{dx} \cos x^2= \frac{d}{dx^2}\cos x^2 \times \frac{d}{dx} x^2 $

REPLY [1 votes]: No. This expression is not wrong. $cos(x^2)$ is just a composite function.
Derivative of such functions can be taken by using chain rule.


TITLE: Explicit Formula for $\zeta(s)$
QUESTION [2 upvotes]: In the explicit expression for $$\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2})  $$
$ x^\rho$ denotes $x^{\mathrm{Re} \rho}$. I wanted to know if there is some formula for $$\sum_{\rho} \frac{x^{\rho}}{\rho}$$ and whether that divisor also denotes the real part of $\rho$ or not.

REPLY [1 votes]: No, there is no particularly good formula for
$$
\sum_\rho \frac{x^\rho}{\rho},
$$
except of course for the explicit formula itself. But this is enough for many applications, such as proving an explicit prime number theorem with error-term (along with the zero-free region for the zeta function).
In this answer, I give a quick sketch of how one gets the explicit formula. In short, one uses the residue theorem on the integral
$$ \int_{(2)} -\frac{\zeta'(s)}{\zeta(s)} \frac{X^s}{s} ds,$$
and there are poles at zeroes of the zeta function. These poles have residues $x^\rho/\rho$. This does not mean the real part. And contrary to your claim, the imaginary parts are very important, both for the actual value and for convergence.


TITLE: How can I approach this linear algebra problem?
QUESTION [0 upvotes]: Part A) I can only say there is necessarily n real eigenvalues producing n pairwise orthogonal eigenvectors because A is symmetric, but what else is there to say about the eigenvalues utilizing the fact that A has orthonormal columns?

REPLY [0 votes]: If $A$ has orthonormal columns, then $A^T A = I$.
$A$ is also symmetric, we we have $A^2 = I$.
Now you can check that the eigenvalues must satisfy $\lambda^2 = 1$.


TITLE: Bijective continuous map from compact space to NON HAUSDORFF SPACE prove homeomorphism
QUESTION [1 upvotes]: Let $X,Y$ be topological spaces with $X$ compact and let $f:X \to Y$ be continuous and bijective. Prove that $f$ is a homeomorphism.

First of all, the image of a compact space under a continuous map is compact. Thus by the bijectivity of $f$, $Y$ is also compact. Now I am not sure, how to proceed. Can anyone give me a hint?

REPLY [5 votes]: This isn't true.  For instance, take any finite set with more than one element and let $X$ be that set with the discrete topology and $Y$ be that set with the indiscrete topology.  Then $X$ is compact and the identity map is a continuous bijection $X\to Y$, but it is not a homeomorphism.


TITLE: Remembering exact sine cosine and tangent values?
QUESTION [4 upvotes]: There exists a common trick to remember exact sine cosine and tangent values. The trick is relatively long, so instead of reposting it, please refer to my answer on this page.
Although I have used this trick for a while, I've never understood why it works. I understand the tangent values (sine/cosine according to right angle ratios) and the why the cosine values are in "the opposite order" of sine values (due to the cosine function being a $90^\circ$ phase shift on the  sine function) but why does doing the above trick provide the correct values for sine (and cosine in opposite order)?

REPLY [5 votes]: This is just a briefer restatement of the same rule, which I find easier to remember. I don't think there is a particular reason why it works. (If there were, there would be some nice generalization, which there does not seem to be.)
$\theta \hskip2cm 0^\circ \hskip 1cm 30^\circ \hskip 1cm 45^\circ \hskip1cm 60^\circ \hskip1cm 90^\circ $
$\sin(\theta) \hskip1cm {\sqrt{0} \over 2}\hskip1cm {\sqrt{1}\over 2}\hskip1cm {\sqrt{2}\over 2}\hskip1cm {\sqrt{3}\over 2}\hskip1cm {\sqrt{4}\over 2}$
$\cos(\theta) \hskip1cm {\sqrt{4} \over 2}\hskip1cm {\sqrt{3}\over 2}\hskip1cm {\sqrt{2}\over 2}\hskip1cm {\sqrt{1}\over 2}\hskip1cm {\sqrt{0}\over 2}$
Edit: You might say that these values correspond to looking at triangles where the Pythagorean theorem becomes one of the following:
$$4=0+4=1+3=2+2=3+1=4+0$$
and that these triangles happen to have very nice angles.


TITLE: What is the difference between a topological space which has a $\sigma$-locally finite base to a paracompact topological space
QUESTION [2 upvotes]: I am reading about metrization theorems and can't figure out What is the difference between a topological space which has a $\sigma$-locally finite base to a paracompact topological space.
A topological space is called paracompact, if every cover has an open refinement which is locally finite.
A topological space has a $\sigma$-locally finite base if it has a base which is a countable union of locally finite families.
Any help?
Thank you!

REPLY [2 votes]: The following is either quoted or paraphrased from "Counterexamples in Topology" by Lynn Arthur Steen and J. Arthur Seebach, Jr. I will skip some of the later proofs.
"A topological space is metrizable iff it is regular (T4) and has a $\sigma$-locally finite base, that is, a base which is the countable union of locally finite families. Although this requirement is very close to paracompactness, and though every metric space is paracompact, there exist regular paracompact spaces which are not metrizable." (p.37)
Thus, we need to find a space that is regular but not second countable, meaning that it does not contain a countable basis. This, in turn means the space is still paracompact, but not metrizable, and thus the space is not $\sigma$-locally finite.
The "Right Half-Open Interval Topology" on the set X of all real numbers is defined by open sets of the form $[a,b)$ with $a,b \in \Bbb R \subset X$.
The space is not second countable: Define $S:=\lbrace\left[x_i,y_i \right)| i \in \Bbb Z^+ \rbrace$. If it is a countable basis, we can find an $a \in X$ with $a \neq x_i$ for any $i \in \Bbb Z^+$. However, $[a,b)$ cannot be formed by any union of any elements of $S$. Thus, the space is not metrizable. (75-76)
The space is completely normal, which implies it is normal, which implies it is regular. Because it is Lindelöf, which means every open cover has a countable subcover, we conclude that is is paracompact. (75-76)
In sum, paracompact but does not have a $\sigma$-locally finite base, since no metric can be defined that generates the desired topology.


TITLE: Can a symmetric matrix always be represented as the sum of a positive-definite and negative-definite matrix?
QUESTION [8 upvotes]: I was wondering if it is possible to decompose any symmetric matrix into a positive definite and a negative definite component. I can't seem to think of a counterexample if the statement is false.

REPLY [2 votes]: This answer is years late but I like this proof.
The set of positive definite matrices is open in the set of symmetric matrices and must thus contain $\frac{n(n+1)}{2}$ matrices which are linearly independent. Take that as a basis for the vector space of symmetric matrices. Then every element can be written as a linear combination of positive definite matrices, and we can simply group the basis elements by the positive coefficients and negative coefficients, giving us that a symmetric matrix M can be written as the sum of a positive definite matrix and the negation of a positive definite matrix.


TITLE: Combinatorics Pigeonhole problem
QUESTION [4 upvotes]: Hello to all! So i have to do this problem: In the course of an year of 365 days Peter solves combinatorics problems. Each day he solves at least 1 problem, but no more than 500 for the year. Prove that for the year there exists an interval of consecutive days in which he had solved exactly 229 problems. PS: I think that the pigeonhole principle can be used here , but i just can't show a meaningful and descriptive way of proving it.Big thanks to anyone who can help !

REPLY [7 votes]: Let $a_1$ be the number of combinatorics problems solved on the first day, $a_2$ be the total number of combinatorics problems solved on the first and second days, and so on. The sequence of numbers $a_1,a_2,...,a_{365}$ is an increasing sequence since each term of the sequence is larger than the one that precedes it and at least one combinatorics problem is solved each day. Since he solves no more than $500$ combinatorics problems for the year we know that $1\leq a_1\leq a_2\leq \cdots \leq a_{365}\leq 500$. The sequence $a_1+229$, $a_2+229$, and so on is also an increasing sequence. So $230\leq a_1+229\leq a_2+229\leq \cdots \leq a_{365}+229 \leq 729$. Each of the $730$ numbers, that is $a_1,a_2,...,a_1+229+a_2+229,...$ is an integer between $1$ and $729$. It follows that two of them are equal since no two of the numbers $a_1,a_2,...$ are equal and no two of the numbers $a_1+229,a_2+229,...$ are equal. There must exist an $i$ and a $j$ such that $a_i=a_j+229$. Thus on days $j+1$, $j+2$,..., $i$ there exists consecutive days when the student solves $229$ combinatorics problems.

REPLY [1 votes]: Lets consider partial sums. $A(x)$ is a sum of problems after $x$ days. Now what is the remainder of $A(x)$ divided by $229$? $A(x)$ changes as days go by, there are only $229$ possible remainders and $365$ days - so there must be two days $x$ and $y$, where $A(x)$ and $A(y)$ have the same remainder. $A(x)$ and $A(y)$ represent sums of problems of two overlapping periods. $A(x)$ is the larger period (you can assume this). $A(x)-A(y)$ is divisible by 229 and is positive. $A(x)-A(y)$ also represents a sum of problems of a certain period (since the two periods are overlapping, just take the larger period and disregard the part of the larger period that is the smaller period.
In any $55$ days he must have solved at least $55$ problems, therefore in $310$ days he must have solved at most $500-55=445$ problems which is less than $458$. There are only $229$ remainders, but $310$ days therefore in the $310$ days, there must be two days $x$ and $y$ such that $A(x)-A(y)$ is divisible by $229$, we also know that it is smaller than $458$, therefore it has to be equal to $229$.


TITLE: Integral with simple fractions: $\int_\ \frac{\cos x }{\sin x \sqrt{1+\cos^2x}} \, dx$
QUESTION [1 upvotes]: I have a problem with this integral
$$\int_\ \frac{\cos x }{\sin x  \sqrt{1+\cos^2x}} \, dx$$
Using substitution $u = \sin x $ we get
$$\int_\ \frac{1 }{\ u  \sqrt{2-u^2}} \, du$$
I think the next step is to use simple fractions, but I don't know how to do it.

REPLY [2 votes]: Hint
Multiply both numerator and denominator by $u$ and then use the substitution $2-u^2=t^2.$ To get
$$\int \frac{u}{u^2\sqrt{2-u^2}} \, du=\int \frac{1}{t^2-2} \, dt.$$
Now you can use partial fractions.


TITLE: Find $\frac{dy}{dx}$ when, $y \arcsin x - x \arctan y = 1$
QUESTION [1 upvotes]: Yesterday was my class test and I found this question.
Find $\displaystyle \frac{dy}{dx}$ when, $\displaystyle y \arcsin x - x \arctan y = 1$
I have read the question for arctanx as $1/1 + x^2$. But that is applied only, when the question is
Find difference is along x (dy/dx) and the function has the x as the variable.
How would I solve the above equation?

REPLY [2 votes]: Use the total differentiation of an implicit function $F =\operatorname F(x,y) = 0$. You end with
$\displaystyle \frac {\operatorname dy}{\operatorname dx} = - \dfrac{ \frac {\operatorname dF}{\operatorname dx} }{ \frac { \operatorname dF }{\operatorname dy}}$
and you wil arrive to what lab bhattarcharjee gives.


TITLE: Water heating up as it flows through a perfectly insulated pipe
QUESTION [0 upvotes]: I'm trying to solve the following problem:

We want to know what the outflow temperature of the water, $\rm T_{water, out}$ is as a function of time.
Knowing:

Water is flowing through a thick pipe in a turbulent way
The initial temperature of the thick pipe is known, and uniform, $\rm T_{0,Ext}$.
We know the internal radius of the pipe, $\rm r_{Int}$, and the external radius, $\rm r_{Ext}$.
We know the inflow temperature into the pipe, $\rm T_{water,in}$. This value remains constant through time.
The pipe's external wall is perfectly insulated (i.e. no heat can flow through it and the initial amount of heat held in the pipe is the only heat available to heat the water).

Current Approach :
I feel like the above should be a textbook example, but I can't find the solution to that problem anywhere and my maths are not good enough to be able to solve that problem from scratch. I've tried using papers describing similar problems but I just can't re-arrange the equations to fit my perfectly insulated external wall assumption.
My current approach has been to equate the heat gained by the water (1) to the heat flow transferred from the pipe internal wall to the fluid (2), to the heat transmitted through the pipe itself (3). But with a time dependent $T_{Ext}$ who will gradually cool I seem to always have an additional unknown I can't solve for as soon as I move away from $t_{0}$.
(1) $\rm Q = density_{water} × Flow\ Rate_{water} × Specific\ Heat_{water}(T_{water, out}-T_{water, in})$
(2) $\rm Q = 2× \pi × r_{Int}× Length_{pipe} × \left[ \frac {Thermal\ Conductivity_{water}× \nu} {2× r_{Int}}\right]×\left(T_{r_{Int}}-\frac {T_{water, out}-T_{water, in}}{2}\right)$
with Nu the Nusslet Number.
(3) $\rm Q = 2× \pi× Length_{pipe}× \frac {Thermal\ Conductivity_{pipe}}{(r_{Ext}/r_{Int})}×(T_{Ext}-T_{r_{Int}})$
Any help would be much appreciated.
Cheers

REPLY [0 votes]: The easiest way to initially proceed is to assume everything has come to steady-state conditions and there are no thermal gradients anywhere i.e., the system is equilibrated.
Then you equate the total friction losses (work performed on the parcels of water traversing the pipe length) to the increase in internal energy of the water flowing through the pipe at a certain rate and obtain the temperature rise in the exiting water as a function of flow rate.
This yields the steady-state solution. Then you add in the time-dependent terms one at a time, solve, and check the result against the steady-state solution.


TITLE: Is it true that $\overline{A \cap \overline{B}} = \overline{A \cap {B}}$?
QUESTION [0 upvotes]: I was trying to understand Willard's proof (the book is called Topology) about local compacteness that any subset of a Hausdorff space is the intersection of an open set and a closed set.
Then I came across this post: Locally compact subspace is an intersection of an open and closed set.
But I can't prove
$$(1) \quad \overline{W_x \cap \overline{M}} = \overline{W_x \cap {M}}$$
(wich is very similar to what Willard does $\overline{W_x \cap {M}} \subseteq X \implies  \overline{W_x} \cap M \subseteq X$).
How does one prove $(1)$?  Thank you for the help.

REPLY [4 votes]: No. Let $A=(0,1)\cup \{2\}$,  and $B=(2,3)$ then $\overline{A\cap \overline{B}}=\{2\}\ne\emptyset =\overline{A\cap B}$,


TITLE: Is $t=0$ a stationary point?
QUESTION [1 upvotes]: I have a parametric equation
$$
t^2 \textbf{i} + (t^3 +1) \textbf{j} =0
$$
So to find stationary point is to differentiate so I did and got
$$
2t \textbf{i} + 3t^2\textbf{j}
$$
Then equate to zero. Now my question is I know for sure one of the stationary points is $-2/3$ but is $0$ a stationary point too? Why or why not?
That's because when I plotted the graph out using $-4 < t < 4$, the stationary points don't make sense at all (it's not the max or min)... Can someone please try it out and see if it's something I did wrong? Thank you!!!

REPLY [1 votes]: A stationnary point depend only on the parametrisation of your graph. You could have the same graph, but with a different parametrisation, and different stationnary point.
Imagine your equation describe the movement of a mobile in respect to time. You can make the same graph, but going at different speeds or even stopping.
These stopping points are called "stationnary points"


TITLE: Help to prove fact about quintic and galois theory?
QUESTION [0 upvotes]: Consider $f = 2x^5 - 10x +5 \in Q[x]$. Let $L/Q$ be a splitting field of f. Show that $Gal(L/Q)$ injects as a subgroup of $S_5$.

REPLY [2 votes]: Hint: Show that every automorphism must permute the roots of $f$, and that an automorphism is completely determined by where it sends the roots.


text:
\begin{document}

\maketitle

\begin{abstract}
 Proper orthogonal decomposition (POD) stabilized methods for the Navier-Stokes equations are considered and analyzed.
 We consider two cases, the case in which the snapshots are based on a non inf-sup stable method and the case in which the
 snapshots are based on an inf-sup stable method. For both cases we construct approximations to the velocity and the pressure.
 For the first case, we analyze a method in which the snapshots are based on a stabilized scheme with equal order polynomials for the velocity and the pressure with Local Projection Stabilization (LPS) for the gradient of the velocity and the pressure. For the POD method we add the same kind of LPS stabilization for the gradient of the velocity  and the pressure than the direct method, together with grad-div stabilization.
  In the second case, the snapshots are based on an inf-sup stable Galerkin method with grad-div stabilization and for the POD model we apply also grad-div stabilization. In this case, since the snapshots are discretely divergence-free, the pressure can be removed from the formulation of the POD approximation to the velocity. To approximate the pressure, needed in many engineering applications, we use a supremizer pressure recovery method.
  Error bounds with constants independent on inverse powers of the viscosity parameter are proved for both methods. Numerical experiments show the accuracy and performance of the schemes.
\end{abstract}

\noindent{\bf AMS subject classifications.} 35Q30,  65M12, 65M15, 65M20, 65M60, 65M70,\\ 76B75. \\
\noindent{\bf Keywords.} Navier-Stokes equations, proper orthogonal decomposition, fully discrete schemes, non inf-sup stable elements, inf-sup stable elements, grad-div stabilization.

\section{Introduction}

Reduced order models (ROM) are a fairly extensive technique applied in many different fields to reduce the computational cost of direct
numerical simulations while keeping enough accurate numerical approximations. In particular, ROM have been extensively applied in recent years to model incompressible flows \cite{Rozza15,IliescuJohn14,RebholzIliescu17,Quarteroni16,RozzaStabile18,Wang12}. Proper Orthogonal Decomposition (POD) method provides the elements (modes) of the reduced basis from a given database (snapshots) which are computed by means of a direct or full order method (FOM).

In this paper, we study the numerical approximation of incompressible flows with stabilized POD-ROM.
We consider the Navier-Stokes equations
\begin{align}
\label{NS} \partial_t\bu -\nu \Delta \bu + (\bu\cdot\nabla)\bu + \nabla p &= \bff &&\text{in }\ (0,T]\times\Omega,\nonumber\\
\nabla \cdot \bu &=0&&\text{in }\ (0,T]\times\Omega,
\end{align}
in a bounded domain $\Omega \subset {\mathbb R}^d$, $d \in \{2,3\}$ with initial condition $\bu(0)=\bu^0$. In~\eqref{NS},
$\bu$ is the velocity field, $p$ the kinematic pressure, $\nu>0$ the kinematic viscosity coefficient,
 and $\bff$ represents the accelerations due to external body forces acting
on the fluid. The Navier-Stokes equations \eqref{NS} must be complemented with boundary conditions. For simplicity,
we only consider homogeneous
Dirichlet boundary conditions $\bu = \boldsymbol 0$ on $\partial \Omega$.

In practice, for computing the snapshots one can use an inf-sup stable or a non inf-sup stable method. In this paper, we analyze two methods, one starting from a non inf-sup stable method and the other starting from an inf-sup stable method. For both cases we construct approximations to the velocity and the pressure since in many engineering problems not only approximations to the velocity are required, but also to the pressure (e.g., to compute forces on bodies in the flow and for incompressible shear flows \cite{NPM05}, as the mixing layer or the wake flow \cite{Tadmor11}, where neglecting the pressure may lead to large amplitude errors).

In \cite{samuele_pod} a POD stabilized method for the Navier-Stokes equations is introduced and analyzed. The method uses snapshots for the
velocity that do not satisfy a weakly discrete divergence-free condition since are based on equal order (say order $l$) velocity and pressure approximations. A LPS-type stabilization term for the pressure is introduced in the POD model. This term is inspired in the term-by-term LPS stabilization introduced in \cite{Cha_maca}, see also \cite{ahmed_etal}, \cite{nos_lps}. The stabilization term is based on a fluctuation operator of a locally stable projection onto the space of polynomials of degree $l-1$. The method in \cite{samuele_pod} allows to compute both approximations to the velocity and the pressure. Apart from LPS stabilization
for the pressure no other stabilization terms are considered in this method. A penalty term is added to the variational formulation to avoid the zero mean condition for the pressure. This technique introduces an extra parameter $\sigma$ that is assumed to be small (of order $10^{-6}$). In the error analysis of the
method there is a factor behaving as $\sigma$. To  handle a divergence error term appearing in the error analysis in \cite{samuele_pod} a saturation property
is applied. If we denote by $\alpha$ the saturation constant between the space of divergence POD velocity approximations and the space of POD pressure
approximations then the bounds in \cite{samuele_pod} are multiplied by $\alpha^2/\sigma$. The experimental value of $\alpha$ in the numerical examples
in \cite{samuele_pod} has size around $10^{-2}$ so that $\alpha^2/\sigma\approx 10^2$. The error bounds in \cite{samuele_pod} are not independent
on inverse powers of the viscosity parameter.

In the first part of the present paper we consider the same method of \cite{samuele_pod} with some differences. First of all, we do not
add a penalty term for the pressure. Since both, the parameter $\sigma$ of the penalty term and its inverse $\sigma^{-1}$,  appear in the error analysis of \cite{samuele_pod} multiplying the
constants in the error bounds it is a problem to choose the optimal value for $\sigma$, while keeping an optimal order for the method.  Secondly, apart from term-by-term LPS stabilization for the pressure  we add LPS
stabilization for the gradient of the velocity and also grad-div stabilization to the POD model. To compute the snapshots, we use a stabilized FOM. More precisely, the snapshots for the model are based on equal order velocity and pressure approximations of a stabilized finite element method that uses term-by-term LPS stabilization for the gradient of the velocity and the pressure. In \cite{nos_lps} error bounds for this method are proved in which the constants do not depend on inverse powers of the viscosity. An error bound for the $L^2$ error of the velocity of order $l+1/2$ is obtained (where $l$ is the local degree of the velocity and pressure approximations). Starting from an optimal order FOM allows to improve the accuracy of the model we present. As in \cite{samuele_pod}, the method we consider in this paper provides both approximations to the velocity and the pressure. In this paper, to handle a divergence error term appearing in the error analysis, we use some properties of the POD basis functions inherited from the direct model (the stabilized method), instead of a saturation assumption.   Then, we are able to handle all the terms in the error analysis using the same kind of stabilization in the POD model as in the FOM method. Moreover, we prove error bounds in which the constants do not depend on inverse powers of the viscosity parameter. To this end, the added grad-div term is essential.

To our knowledge this is the first time this kind of viscosity independent bounds are obtained for POD models. In \cite{pod_da_nos}  a POD data assimilation scheme for the Navier-Stokes equations is analyzed. The method
in \cite{pod_da_nos}  uses also grad-div stabilization. This method only provides approximations for the velocity. Some of the constants in the error bounds in \cite{pod_da_nos}  are also viscosity independent. Indeed, we follow some of the ideas in  \cite{pod_da_nos} for the error analysis of the present paper. However, in \cite{pod_da_nos} the snapshots are based on a non stabilized standard Galerkin method and, as a consequence, the error coming from the snapshots depends on inverse powers of the viscosity. Although a stabilized method could be used for the snapshots in \cite{pod_da_nos} to avoid this dependency, it could not be an equal order velocity-pressure method (as in the present paper) since the discrete divergence-free condition of the snapshots is strongly used in the error analysis of \cite{pod_da_nos}. To solve this problem,
we also introduce and analyze a second method in which the snapshots are computed with an inf-sup stable Galerkin method with grad-div stabilization for which error bounds with constants independent on the viscosity have been proved in \cite{NS_grad_div}. For the POD model we also add grad-div stabilization. In a first step, we compute only a POD approximation to the velocity, since the pressure can be removed from the formulation using the discrete divergence-free condition of the POD basis functions. In a second step, following \cite{schneier} we apply a supremizer \cite{Rozza15,RozzaVeroy07} pressure recovery method to get a POD pressure approximation. We get bounds both for the velocity and the pressure in which the constants do not depend on inverse powers of the viscosity.

The outline of the paper is as follows. In Section \ref{sec:PN} we introduce some preliminaries and notations. Section \ref{sec:POD} is devoted to recall the POD method and get some a priori bounds for the $L^2$-orthogonal projection of the FOM velocity onto the POD velocity space. In Section \ref{sec:LPS-ROM} we introduce and analyze the stabilized POD method based on a non inf-sup stable FOM. In Section \ref{sec:grad-div-ROM} we introduce and analyze the grad-div stabilized POD method based on an inf-sup stable FOM. In Section \ref{sec:num} some numerical experiments show the accuracy and performance of the methods. Finally, Section \ref{sec:Concl} presents the main conclusions.

\section{Preliminaries and notation}\label{sec:PN}
The following Sobolev embeddings \cite{Adams} will be used in the analysis: For
$q \in [1, \infty)$, there exists a constant $C=C(\Omega, q)$ such
that

\begin{equation}\label{sob1}
\|v\|_{L^{q'}} \le C \| v\|_{W^{s,q}}, \,\,\quad
\frac{1}{q'}
\ge \frac{1}{q}-\frac{s}{d}>0,\quad q<\infty, \quad v \in
W^{s,q}(\Omega)^{d}.
\end{equation}
The following inequality can be found in  \cite[Remark 3.35]{John}
\begin{eqnarray}\label{diver_vol}
\|\nabla \cdot \bv\|_0\le \|\nabla   \bv \|_0,\quad \bv\in H_0^1(\Omega)^d.
\end{eqnarray}
Let us denote by $Q=L_0^2(\Omega)=\left\{q\in L^2(\Omega)\mid (q,1)=0\right\}$.
Let $\mathcal{T}_{h}=(\tau_j^h,\phi_{j}^{h})_{j \in J_{h}}$, $h>0$ be a family of partitions of $\overline\Omega$, where $h$ denotes the maximum diameter of the elements $\tau_j^h\in \mathcal{T}_{h}$, and $\phi_j^h$ are the mappings from the reference simplex $\tau_0$ onto $\tau_j^h$.
We shall assume that the partitions are shape-regular and quasi-uniform.
We define the following finite element spaces
\begin{eqnarray*}
Y_h^l&=& \left\{v_h\in C^0(\overline\Omega)\mid {v_h}_{\mid_K}\in {\Bbb P}_l(K),\quad \forall K\in \mathcal T_h\right\}, \ l\ge 1,\nonumber\\
{\boldsymbol Y}_h^l&=&(Y_h^l)^d,\quad {\boldsymbol X}_h^l={\boldsymbol Y}_h^l\cap H_0^1(\Omega)^d, \nonumber\\
Q_h^l&=&Y_h^l\cap L_0^2(\Omega).
\end{eqnarray*}
\begin{eqnarray}\label{eq:V}
{\boldsymbol V}_{h,l}={\boldsymbol X}_h^l\cap \left\{ {\boldsymbol \chi}_{h} \in H_0^1(\Omega)^d \mid
(q_{h}, \nabla\cdot{\boldsymbol\chi}_{h}) =0  \quad\forall q_{h} \in Q_{h}^{l-1}
\right\},\quad l\geq 2.
\end{eqnarray}If the family of
meshes is quasi-uniform then  the following inverse
inequality holds for each $\bv_{h} \in Y_{h}^{l}$, see e.g., \cite[Theorem 3.2.6]{Cia78},
\begin{equation}
\label{inv} \| \bv_{h} \|_{W^{m,p}(K)} \leq c_{\mathrm{inv}}
h_K^{n-m-d\left(\frac{1}{q}-\frac{1}{p}\right)}
\|\bv_{h}\|_{W^{n,q}(K)},
\end{equation}
where $0\leq n \leq m \leq 1$, $1\leq q \leq p \leq \infty$, and $h_K$

is the diameter of~$K \in \mathcal T_h$.
Let ${\boldsymbol V}=\left\{ {\boldsymbol \chi} \in H_0^1(\Omega)^d \mid
\nabla\cdot{\boldsymbol\chi} =0\right\}$. We consider a modified Stokes projection that was introduced in \cite{grad-div1} and that we denote by $\bs_h^m:{\boldsymbol V}\rightarrow {\boldsymbol V}_{h,l}$
satisfying
\begin{eqnarray}\label{stokespro_mod_def}
(\nabla \bs_h^m,\nabla \bvar_h)=(\nabla \bu,\nabla \bvar_h),\quad \forall \, \,
\bvar_{h} \in \bV_{h,l},
\end{eqnarray}
and the following error bound, see \cite{grad-div1}:
\begin{equation}
\|\bu-\bs_h^m\|_0+h\|\bu-\bs_h^m\|_1\le C\|\bu\|_j h^j,\qquad
1\le j\le l+1.
\label{stokespro_mod}
\end{equation}
From \cite{chenSiam}, we also have
\begin{align}
\|\nabla \bs_h^m\|_{L^\infty}\le C\|\nabla \bu\|_{L^\infty} \label{cotainfty1},
\end{align}
where $C$ does not depend on $\nu$ and~\cite[Lemma~3.8]{bosco_titi_yo}
\begin{align}
\label{cota_sh_inf_mu}
\|\bs_h^m\|_{L^\infty}  & \le C(\|\bu\|_{d-2}\|\bu\|_2)^{1/2},
\\
\label{la_cota_mu}
\|\nabla\bs_h^m\|_{L^{2d/(d-1)}} & \le
 C\bigl(\|\bu\|_1\|\bu\|_2\bigr)^{1/2},
\end{align}
where the constant~$C$ is independent of~$\nu$.

As mentioned in the introduction our aim is to get error bounds with constants independent on inverse powers of $\nu$. To this end, instead of starting with the snapshots of
a Galerkin method, we start from the snapshots of a stabilized method. We consider the method analyzed in \cite{nos_lps}:
the approximation of the solution of (\ref{NS}) with the implicit Euler method in time and a LPS method with LPS stabilization of the gradient of the velocity and the pressure.

Assuming $l \geq 2$, the method reads: given  {${\bu}_h^0$ an approximation to $\bu^0$ in ${\boldsymbol X}_h^l$}, find $(\bu_h^{n+1},p_h^{n+1})\in {\boldsymbol X}_h^l\times Q_h^l$ such that {for $n\ge 0$}
\begin{eqnarray}\label{eq:gal_est}
\lefteqn{\hspace*{-16em}\left(\frac{\bu_h^{n+1}-\bu_h^n}{\Delta t},\bv_h\right)+\nu(\nabla \bu_h^{n+1},\nabla \bv_h)+b(\bu_h^{n+1},\bu_h^{n+1},\bv_h)-(p_h^{n+1},\nabla \cdot \bv_h)}
\nonumber\\
+
S_h(\bu_h^{n+1},\bv_h)&=&({\boldsymbol f}^{n+1},\bv_h) \quad \forall \bv_h\in {\boldsymbol X}_h^l,\\
(\nabla \cdot \bu_h^{n+1},q_h)+s_{\rm pres}(p_h^{n+1},q_h)&=&0 \quad \forall q_h\in Q_h^l.\nonumber
\end{eqnarray}
In \eqref{eq:gal_est} $(\bu_h^n,p_h^{n})$ is the stabilized approximation at time $t_n=n\Delta t$,  $\Delta t$
is the time step and $b_h(\cdot,\cdot,\cdot)$  is defined in the following way
$$
b_{h}(\bu_{h},\bv_{h},\bw_{h}) =((\bu_{h}\cdot \nabla ) \bv_{h}, \bw_{h})+ \frac{1}{2}( \nabla \cdot (\bu_{h})\bv_{h},\bw_{h}),
\quad \, \forall \, \bu_{h}, \bv_{h}, \bw_{h} \in {\boldsymbol X}_h^l.
$$
It is straightforward to verify that $b_h$ enjoys the skew-symmetry property
\begin{equation}\label{skew}
b_h(\bu,\bv,\bw)=-b_h(\bu,\bw,\bv) \qquad \forall \, \bu, \bv, \bw\in H_0^1(\Omega)^d.
\end{equation}
Also,
\begin{eqnarray}
S_h(\bu_h^{n+1},\bv_h)&=& \sum_{K\in \mathcal{T}_h}
\tau_{\nu,K}\left(\sigma^*_h (\nabla \bu_h^{n+1}), \sigma_h^*(\nabla \bv_h)\right)_K,\nonumber\\
s_{\rm pres}(p_h^{n+1},q_h)&=&\sum_{K\in \mathcal T_h}\tau_{p,K}(\sigma^*_h(\nabla p_h^{n+1}),\sigma^*_h(\nabla q_h))_K,\label{eq:spres}
\end{eqnarray}
where $\tau_{\nu,K}$ and $\tau_{p,K}$ are the gradient of the velocity and pressure stabilization parameters, respectively.
In addition,   $\sigma^*_h=Id-\sigma_h^{l-1}$, where $\sigma_h^{j}$ is a locally stable projection or interpolation operator from $L^2(\Omega)^d$ on ${\boldsymbol Y}_h^{j}.$
It will be assumed that
\begin{equation}\label{eq:tau_p2}
\alpha_1h_K\le \tau_{p,K}\le \alpha_2 h_K,
\end{equation}
and
\begin{equation}\label{eq:mu_2}
c_1h_K\le \tau_{\nu,K}\le c_2 h_K,
\end{equation}
with nonnegative constants $\alpha_1, \alpha_2, c_1, c_2$.
The following notation will be used
\begin{equation}\label{eq:norm_tau_p}
(f,g)_{\tau}=\sum_{K\in {\mathcal T}_h}\tau_{K}(f,g)_K, \quad
\|f\|_{\tau} = (f,f)_{\tau}^{1/2},
\end{equation}
where $\tau$ denotes either $\tau_p$ or $\tau_{\nu}$. Following \cite{ahmed_etal,Cha_maca}, we consider an approximation $\hat\bu_h^{n}\in {\boldsymbol X}_h^l$ of $\bu^n$ satisfying
\begin{equation}\label{eq:approxIMAJNA}
(\bu^n-\hat\bu_h^{n},\bv_h)=0,\quad\forall\bv_h\in{\boldsymbol Y}_h^{l-1}.
\end{equation}

Assume $\nu\le h$ and
\begin{eqnarray*}\label{eq:cond_dif}
C \|\hat\bu_h^{n+1}\|_{L^\infty}^2\left( \max_{K\in\mathcal T_h} \tau_{p,K}\right)\left(\max_{K\in\mathcal T_h} \tau_{\nu,K}^{-1}\right)\le \frac {1}{16},
\end{eqnarray*}
and let the parameter $\varepsilon$  be chosen
sufficiently small so that
\begin{equation}
\label{eq:epsilon}
C \varepsilon h \|\hat\bu_h^{n+1}\|_{L^\infty}^2\left(\max_{K\in\mathcal T_h} \tau_{\nu,K}^{-1}\right)\le \frac {1}{16}.
\end{equation}
Let
$$
M_u=1+C\|\bu\|_{L^\infty(H^3)}\left(1+\|\bu\|_{L^\infty(H^3)}\right)
$$
and
\begin{eqnarray*}\label{eq:tilde_Kb}
K_{u,p}=\left(\left(1+\varepsilon^{-1}+\|\bu\|_{L^\infty(H^3)}^2\right)\|\bu\|_{L^\infty(H^{l+1})}^2+\|\partial_t\bu\|_{L^\infty(H^{l+1})}^2
+\|p\|_{L^\infty(H^{l+1})}^2\right),
\end{eqnarray*}
$\varepsilon$ being the value in~(\ref{eq:epsilon}). Then, the following bound is proved in \cite[Theorem 6.1]{nos_lps}: for $T=M\Delta t$, $n\le M$
 \begin{eqnarray}\label{eq:err_after_gron_22b}
&&\|\bu^n-\bu_h^{n}\|_0^2+h^2\|\nabla(\bu^n-\bu_h^{n})\|_0^2
+\Delta t\sum_{j=1}^n\|\sigma_h^*(\nabla  (\bu^{j}-\bu_h^j))\|_{\tau_\nu}^2
\qquad
  \\
&&\quad+\Delta t\sum_{j=1}^n\|\sigma_h^*(\nabla  (p^j-p_h^{j}))\|_{\tau_p}^2\le C e^{2T M_u}\left(  T  K_{u,p} h^{2l+1}+ (\Delta t)^2\int_{t_0}^{t_n}\|\partial_{tt}\bu\|_{0}^2
~dt\right).\nonumber
\end{eqnarray}
From \eqref{eq:err_after_gron_22b} we can write
\begin{eqnarray}\label{eq:cota_est}
&&\|\bu^n-\bu_h^n\|_0+ h\|\bu^n-\bu_h^n\|_1+\left(\Delta t\sum_{j=1}^n S_{h}( \bu^j-\bu_h^{j},\bu^j-\bu_h^{j})\right)^{1/2}
\nonumber\\
&&\qquad+\left(\Delta t\sum_{j=1}^n s_{\rm pres}( p^j-p_h^{j},p^j-p_h^{j})\right)^{1/2}\nonumber\\
&&\ \quad\le C(\bu,p,l+1) (h^{l+1/2}+\Delta t), \quad 1\le n\le M,
\end{eqnarray}
where the constant $C(\bu,p,l+1)$ does not depend on inverse powers of $\nu$. From \cite[Remark 6.2]{nos_lps} and \cite{sim_pre_nos}
the following bound for the $L^2$ error of the pressure can be obtained for $1\le n\le M$
\begin{eqnarray}\label{eq:cota_est_pre}
\|p^{n}-p_h^{n}\|_0+ h\|p^{n}-p_h^{n}\|_1\le C(\bu,p,l+1) h^{-1/2}(h^{l+1/2}+\Delta t).
\end{eqnarray}
\section{Proper orthogonal decomposition}\label{sec:POD}
We will consider a proper orthogonal decomposition (POD) method.
Let us fix $T>0$ and $M>0$ and take $\Delta t=T/M$ and let us consider the following spaces
$$
{\cal \bV}=<\bu_h^1,\ldots,\bu_h^M>,\quad {\cal W}=<p_h^{1},\ldots,p_h^{M}>.
$$
Let $d_v$ be the dimension of the space $\cal \bV$ and let $d_p$ be the dimension of the space ${\cal W}$.

Let $K_v$, $K_p$ be the correlation matrices corresponding to the snapshots $K_v=((k_{i,j}^v))\in {\mathbb R}^{M\times M}$,
$K_p=((k_{i,j}^p))\in {\mathbb R}^{M\times M}$ where
$$
k_{i,j}^v=\frac{1}{M}(\bu_h^i,\bu_h^j),\quad k_{i,j}^p=\frac{1}{M}(p_h^{i},p_h^{j}),
$$
and $(\cdot,\cdot)$ is the inner product in $L^2(\Omega)^d$. Following \cite{kunisch} we denote by
$ \lambda_1\ge  \lambda_2,\ldots\ge \lambda_{d_v}>0$ the positive eigenvalues of $K_v$ and by
$\bv_1,\ldots,\bv_{d_v}\in {\mathbb R}^{M}$ the associated eigenvectors. Analogously, we denote by
$ \gamma_1\ge  \gamma_2,\ldots\ge \gamma_{d_p}>0$ the positive eigenvalues of $K_p$ and by
$\bw_1,\ldots,\bw_{d_p}\in {\mathbb R}^{M}$ the associated eigenvectors. Then, the (orthonormal) POD bases are given by
\begin{eqnarray}\label{lachi}
\bvar_k=\frac{1}{\sqrt{M}}\frac{1}{\sqrt{\lambda_k}}\sum_{j=1}^M v_k^j \bu_h(\cdot,t_j),\quad
\psi_k=\frac{1}{\sqrt{M}}\frac{1}{\sqrt{\gamma_k}}\sum_{j=1}^M w_k^j p_h(\cdot,t_{j}),
\end{eqnarray}
where $v_k^j$ is the $j$-th component of the eigenvector $\bv_k$ (respectively $w_k^j$ is the $j$-th component of the eigenvector $\bw_k$) and the following error formulas hold, see \cite[Proposition~1]{kunisch}
\begin{eqnarray}\label{eq:cota_pod_0}
\frac{1}{M}\sum_{j=1}^M\left\|\bu_h^j-\sum_{k=1}^r(\bu_h^j,\bvar_k)\bvar_k\right\|_{0}^2&=&\sum_{k=r+1}^{d_v}\lambda_k,\\
\frac{1}{M}\sum_{j=1}^M\left\|p_h^{j}-\sum_{k=1}^r(p_h^{j},\psi_k)\psi_k\right\|_{0}^2&=&\sum_{k=r+1}^{d_p}\gamma_k,\label{eq:cota_pod_0_pre}
\end{eqnarray}
where we have used the notation $\bu_h^j=\bu_h(\cdot,t_j)$, $p_h^{j}=p_h(\cdot,t_{j})$.

Denoting by $S$ the stiffness matrix for the POD basis: $S^v=((s_{i,j}^v))\in {\mathbb R}^{d_v\times d_v}$, $S^p=((s_{i,j}^p))\in {\mathbb R}^{d_p\times d_p}$  with
$s_{i,j}^v=(\nabla \bvar_i,\nabla \bvar_j)$
and $s_{i,j}^p=(\nabla \psi_i,\nabla \psi_j)$,
then for any $\bv \in {\cal \bV}$, $w\in {\cal W}$ the following inverse inequalities hold, see \cite[Lemma 2]{kunisch}
\begin{equation}\label{eq:inv_S}
||\nabla \bv ||_0\le \sqrt{\|S^v\|_2}\|\bv\|_0,\quad
||\nabla w ||_0\le \sqrt{\|S^p\|_2}\|w\|_0,
\end{equation}
where $\|\cdot\|_2$ denotes the spectral norm of the matrix.

From \eqref{eq:inv_S}, applying \eqref{eq:cota_pod_0} we get
\begin{eqnarray}\label{inv_1}
&&\frac{1}{M}\sum_{j=1}^M\left\|\nabla \bu_h^j-\sum_{k=1}^r(\bu_h^j,\bvar_k)\nabla \bvar_k\right\|_0^2\nonumber\\
&&\quad\le \frac{\|S^v\|_2}{M}\sum_{j=1}^M\left\|\bu_h^j-\sum_{k=1}^r(\bu_h^j,\bvar_k)\bvar_k\right\|_0^2\le \|S^v\|_2\sum_{k=r+1}^{d_v} \lambda_k.
\end{eqnarray}
Instead of \eqref{inv_1} we can also apply the following result that is taken from \cite[Lemma 3.2]{iliescu}
\begin{eqnarray}\label{inv_2}
\frac{1}{M}\sum_{j=1}^M\left\|\nabla\bu_h^j-\sum_{k=1}^r(\bu_h^j,\bvar_k)\nabla\bvar_k\right\|_0^2=
\sum_{k=r+1}^{d_v}\lambda_k \|\nabla\bvar_k\|_0^2.
\end{eqnarray}
Analogously, we obtain
\begin{eqnarray}\label{inv_1_p}
\frac{1}{M}\sum_{j=1}^M\left\|\nabla p_h^{j}-\sum_{k=1}^r(p_h^{j},\psi_k)\nabla\psi_k\right\|_0^2\leq
\|S^p\|_2\sum_{k=r+1}^{d_p}\gamma_k,
\end{eqnarray}
and
\begin{eqnarray}\label{inv_2_p}
\frac{1}{M}\sum_{j=1}^M\left\|\nabla p_h^{j}-\sum_{k=1}^r(p_h^{j},\psi_k)\nabla\psi_k\right\|_0^2=
\sum_{k=r+1}^{d_p}\gamma_k \|\nabla\psi_k\|_0^2.
\end{eqnarray}
In the sequel we will denote by
$$
{\cal \bV}^r=<\bvar_1,\bvar_2,\ldots,\bvar_r>,\quad{\cal W}^r=<\psi_1,\psi_2,\ldots,\psi_r>,
$$
and by $P_r^v$, $P_r^p$ the $L^2$-orthogonal projection onto ${\cal \bV}^r$ and ${\cal W}^r$, respectively.

\subsection{A priori bounds for the orthogonal projection onto ${\cal \bV}^r$}
In this section we will prove some a priori bounds for the orthogonal projection $P_r^v \bu_h^j$, $j=0,\cdots,M$ that are
obtained from a priori bounds for the Galerkin approximation $\bu_h^j$, $j=0,\cdots,M$. Then, we start getting a priori
bounds for the stabilized approximation $\bu_h^n$. We follow the same arguments we introduced in \cite{pod_da_nos}.
We start with the $L^\infty$ norm, using \eqref{inv}, \eqref{cota_sh_inf_mu}, \eqref{eq:cota_est} and  \eqref{stokespro_mod}
we get
\begin{eqnarray}\label{eq:uh_infty}
\|\bu_h^j\|_{L^\infty}&\le& \|\bu_h^j-\bs_h^m(\cdot,t_j)\|_{L^\infty}+\|\bs_h^m(\cdot,t_j)\|_{L^\infty}
\nonumber\\
&\le& C h^{-d/2}\|\bu_h^j-\bs_h^m(\cdot,t_j)\|_0+C(\|\bu^j\|_{d-2}\|\bu^j\|_2)^{1/2}\nonumber\\
&\le& C h^{-d/2}C(\bu,p,2)(h^{3/2}+\Delta t)+C(\|\bu^j\|_{d-2}\|\bu^j\|_2)^{1/2}\nonumber\\
&\le& C_{\bu,{\rm inf}}:=C\left( C(\bu,p,2)+(\|\bu^j\|_{d-2}\|\bu^j\|_2)^{1/2}\right),
\end{eqnarray}
whenever we assume the following condition holds for the time step
\begin{eqnarray}\label{eq:Delta_t0}
\Delta t \le C h^{d/2}.
\end{eqnarray}

Now we bound the $L^\infty$ norm of the gradient, using \eqref{inv}, \eqref{cotainfty1}, \eqref{eq:cota_est} and  \eqref{stokespro_mod}
we get
\begin{eqnarray}\label{eq:grad_uh_infty}
\|\nabla\bu_h^j\|_{L^\infty}&\le& \|\nabla\bu_h^j-\nabla\bs_h^m(\cdot,t_j)\|_{L^\infty}+\|\nabla\bs_h^m(\cdot,t_j)\|_{L^\infty}
\nonumber\\
&\le& C h^{-d/2}\|\bu_h^j-\bs_h^m(\cdot,t_j)\|_1+C\|\nabla \bu^j\|_{L^\infty}\nonumber\\
&\le& C h^{-d/2}C(\bu,p,3)h^{-1}\left(h^{5/2}+\Delta t \right)+C\|\nabla \bu^j\|_{L^\infty}\nonumber\\
&\le& C_{\bu,1,{\rm inf}}:= C\left(C(\bu,p,3)+\|\nabla  \bu\|_{L^\infty(L^\infty)}\right),
\end{eqnarray}
whenever we assume the following condition holds for the time step
\begin{eqnarray}\label{eq:Delta_t}
\Delta t \le C h^{(d+2)/2}.
\end{eqnarray}
Since condition \eqref{eq:Delta_t} implies \eqref{eq:Delta_t0} in the sequel we will assume the stronger condition \eqref{eq:Delta_t} holds.

Finally, we bound the $L^{2d/(d-1)}$ norm.  Using \eqref{inv}, \eqref{la_cota_mu}, \eqref{eq:cota_est} and  \eqref{stokespro_mod}
and assuming again condition \eqref{eq:Delta_t} holds (indeed the weaker condition $\Delta t \le C h^{3/2}$ would be enough) we get
\begin{eqnarray}\label{eq:uh_2d_dmenosuno}
\|\nabla \bu_h^j\|_{L^{2d/(d-1)}}&\le& \|\nabla(\bu_h^j-\bs_h^m(\cdot,t_j))\|_{L^{2d/(d-1)}}+\|\nabla \bs_h^m(\cdot,t_j)\|_{L^{2d/(d-1)}}
\nonumber\\
&\le& C h^{-1/2}\|\bu_h^j-\bs_h^m(\cdot,t_j)\|_1+C\bigl(\|\bu\|_1\|\bu\|_2\bigr)^{1/2}\nonumber\\
&\le& C h^{-1/2}C(\bu,p,2)h^{-1}(h^{3/2}+\Delta t)+C\bigl(\|\bu\|_1\|\bu\|_2\bigr)^{1/2}\nonumber\\
&\le& C_{\bu,{\rm ld}}:=C\left( C(\bu,p,2)+\bigl(\|\bu\|_1\|\bu\|_2\bigr)^{1/2}\right).
\end{eqnarray}
Now, we prove a priori bounds in the same norms for $P_r^v \bu_h^j$. We first observe that
$$
P_r^v \bu_h^j=(P_r^v \bu_h^j-\bu_h^j)+\bu_h^j.
$$
Since we have already proved error bounds for the second term on the right-hand side above we only need to bound the first one. To this end
we observe that it is easy to get
\begin{equation}\label{estasi}
\bu_h^j-P_r^v \bu_h^j=\sqrt{M}\sum_{k=r+1}^{d_v} \sqrt{\lambda_k} v_k^j\bvar_k.
\end{equation}
And then
\begin{eqnarray}\label{cota_bos1}
\|\bu_h^j-P_r^v \bu_h^j\|_0&=&\sqrt{M}\left(\sum_{k=r+1}^{d_v}\lambda_k|v_k^j|^2\right)^{1/2}\le \sqrt{M}\sqrt{\lambda_{r+1}}
\left(\sum_{k=r+1}^{d_v}|v_k^j|^2\right)^{1/2}\nonumber\\
&\le& \sqrt{M}\sqrt{\lambda_{r+1}},
\end{eqnarray}
where in the last inequality we have used that
\begin{eqnarray}\label{menor1}
\left(\sum_{k=r+1}^{d_v}|v_k^j|^2\right)^{1/2}\le 1
\end{eqnarray}
 since the matrix with columns
the vectors $\bv_k$ can be enlarged to an $M\times M$ orthogonal matrix.
Applying inverse inequality \eqref{inv}, \eqref{eq:uh_infty} and \eqref{cota_bos1} we get
\begin{eqnarray}\label{eq:cotaPlinf2}
\|P_r^v \bu_h^j\|_{L^\infty}&\le& \|\bu_h^j\|_{L^\infty}+C h^{-d/2}\|\bu_h^j-P_r^v \bu_h^j\|_0\nonumber\\
&\le& C_{\rm inf}:=
C_{\bu,{\rm inf}}+C h^{-d/2}\sqrt{M}\sqrt{\lambda_{r+1}}.
\end{eqnarray}
Now, we observe that from \eqref{inv_1} we get
$$
\|\nabla(\bu_h^j-P_r^v \bu_h^j)\|_0\le \sqrt{M}\|S^v\|_2^{1/2}\left(\sum_{k=r+1}^{d_v} \lambda_k\right)^{1/2}.
$$
Applying this inequality together with inverse inequality \eqref{inv} and \eqref{eq:grad_uh_infty} we obtain
\begin{eqnarray}\label{eq:cotanablaPlinf}
\|\nabla P_r^v \bu_h^j\|_{L^\infty}\le C_{1,\rm inf}:=C_{\bu,1,{\rm inf}}+C h^{-d/2}\sqrt{M}\|S^v\|_2^{1/2}\left(\sum_{k=r+1}^{d_v} \lambda_k\right)^{1/2}.
\end{eqnarray}
 Finally, arguing in the same way but applying \eqref{eq:uh_2d_dmenosuno} instead of \eqref{eq:grad_uh_infty} we can write
 \begin{eqnarray}\label{eq:cotanablaPld}
\|\nabla P_r^v\bu^j\|_{L^{2d/(d-1)}}&\le& C_{\rm ld}:=C_{\bu,{\rm ld}}+Ch^{-1/2}\sqrt{M}\|S^v\|_2^{1/2}\left(\sum_{k=r+1}^{d_v} \lambda_k\right)^{1/2}.
\end{eqnarray}

\section{A POD stabilized method from a non inf-sup stable FOM}\label{sec:LPS-ROM}
For a given initial condition $\bu_r^0$  we consider the following POD stabilized method in which, for simplicity, as a time integrator we apply the implicit Euler method. For $n\ge 1$, find $(\bu_r^n,p_r^{n})\in {\cal \bV}^r\times{\cal W}^r$ such that
\begin{eqnarray}\label{eq:pod_method1}
&&\left(\frac{\bu_r^{n}-\bu_r^{n-1}}{\Delta t},\bvar\right)+\nu(\nabla \bu_r^n,\nabla\bvar)+b_h(\bu_r^n,\bu_r^n,\bvar)
-(p_r^{n},\nabla \cdot \bvar)
\nonumber\\
&&\quad\quad\quad\ \ +S_h(\bu_r^n,\bvar)+\mu(\nabla \cdot\bu_r^n,\nabla \cdot\bvar)=(\bff^{n},\bvar),\quad \forall \bvar\in {\cal \bV}^r,\\\nonumber
&&(\nabla \cdot \bu_r^n,\psi)+s_{\rm pres}(p_r^{n},\psi)=0,\quad \quad\quad\quad \ \forall \psi\in {\cal W}^r,
\end{eqnarray}
where the gradient of the velocity and pressure stabilization terms are defined in \eqref{eq:spres}, and $\mu$ is the positive grad-div stabilization parameter.

To get the error bounds of the method we will compare $\bu_r^n$ with $P_r^v \bu^n_h$ and $p_r^n$ with $P_r^p p^n_h$.
Let us denote by
$$
\bbeta_h^n=P_r^v\bu^n_h-\bu_h^{n},\quad \xi_h^n=P_r^p p^{n}_h-p_h^n.
$$
It is easy to get
\begin{eqnarray}\label{eq:prov_prop}
&&\left(\frac{P_r^v \bu^{n}_h-P_r^v \bu^{n-1}_h}{\Delta t},\bvar\right)+\nu(\nabla  P_r^v \bu^n_h,\nabla\bvar)+b_h(P_r^v \bu^n_h,P_r^v \bu^n_h,\bvar)
\nonumber\\
&&\quad -(P_r^p p^{n}_h,\nabla \cdot \bvar)+S_h(P_r^v \bu_h^n,\bvar)+\mu(\nabla \cdot P_r^v \bu^n_h,\nabla \cdot\bvar)=(\bff^{n},\bvar)
 \nonumber\\
&&\quad +\nu(\nabla\bbeta_h^n,\nabla\bvar)+S_h(\bbeta_h^n,\bvar)-(\xi_h^n,\nabla \cdot \bvar)+\mu(\nabla \cdot\bbeta_h^n,\nabla \cdot \bvar)\nonumber\\
&&\quad+b_h(P_r^v\bu^n_h,P_r^v \bu^n_h,\bvar)-b_h(\bu^{n}_h,\bu^{n}_h,\bvar),\quad \forall \bvar\in {\cal \bV}^r,\\\nonumber
&&\quad(\nabla \cdot P_r^v \bu^n,\psi)+s_{\rm pres}(P_r p^{n},\psi)=(\nabla \cdot \bbeta_h^n,\psi)+s_{\rm pres}(\xi_h^n,\psi),\quad  \forall \psi\in {\cal W}^r.
\end{eqnarray}
Let us denote by
$$
\be_r^n=\bu_r^n-P_r^v \bu^n_h,\quad z_r^n=p_r^n-P_r^p p^n_h.
$$
Subtracting \eqref{eq:prov_prop} from \eqref{eq:pod_method1} and taking $\bvar=\be_r^n$ and $\psi=z_r^{n}$ we get
\begin{eqnarray}\label{eq:error}
&&\frac{1}{2\Delta t}\left(\|\be_r^n\|_0^2-\|\be_r^{n-1}\|_0^2\right)+\nu\|\nabla \be_r^n\|_0^2+S_h(\be_r^n,\be_r^n)
+\mu\|\nabla \cdot \be_r^n\|_0^2+s_{\rm pres}(z_r^{n},z_r^{n})\nonumber\\
&&\quad\le \left(b_h(P_r^v  \bu^n_h,P_r^v \bu^n_h,\be_r^n)-b_h(\bu_r^n,\bu_r^n,\be_r^n)\right)-\nu(\nabla\bbeta_h^n,\nabla\be_r^n)
-S_h(\bbeta_h^n,\be_r^n)\nonumber\\
&&\quad+(\xi_h^n,\nabla \cdot \be_r^n)-\mu(\nabla \cdot\bbeta_h^n,\nabla \cdot \be_r^n)-\left(b_h(P_r^v\bu^n_h,P_r^v \bu^n_h,\be_r^n)-b_h(\bu^{n}_h,\bu^{n}_h,\be_r^n)\right)\nonumber\\
&&\quad -(\nabla \cdot \bbeta_h^n,z_r^n)-s_{\rm pres}(\xi_h^n,z_r^n)\nonumber\\
&&\quad\le I+II+III+IV+V+VI+VII+VIII.
\end{eqnarray}
We will bound the terms on the right-hand side of \eqref{eq:error}.
We first observe that using the skew-symmetric property \eqref{skew}, \eqref{eq:cotaPlinf2} and \eqref{eq:cotanablaPlinf} we get
\begin{eqnarray}\label{eq:cota_er_1}
|I|\le |b_h(\be_r^n,P_r^v \bu_h^n,\be_r^n)|&\le& \|\nabla P_r^v\bu_h^n\|_{L^\infty}\|\be_r^n\|_0^2+\frac{1}{2}\|\nabla \cdot \be_r^n\|_0
\|P_r^v\bu_h^n\|_{L^\infty}\|\be_r^n\|_0\nonumber\\
&\le& C_{1,{\rm inf}}\|\be_r^n\|_0^2+\frac{C_{\rm inf}}{2}\|\nabla \cdot \be_r^n\|_0\|\be_r^n\|_0\nonumber\\
&\le& \left(C_{1,{\rm inf}}+\frac{C_{\rm inf}^2}{4\mu}\right)\|\be_r^n\|_0^2+\frac{\mu}{4}\|\nabla \cdot \be_r^n\|_0^2.
\end{eqnarray}
For the second term we obtain
\begin{eqnarray}\label{eq:cota_er_2}
|II|\le \frac{\nu}{2}\|\nabla \be_r^n\|_0^2+\frac{\nu}{2}\|\nabla\bbeta_h^n\|_0^2.
\end{eqnarray}
For the third
\begin{eqnarray}\label{eq:cota_er_3}
|III|\le \frac{1}{2}S_h(\be_r^n,\be_r^n)+\frac{1}{2}S_h(\bbeta_h^n,\bbeta_h^n).
\end{eqnarray}
For the forth
\begin{eqnarray}\label{eq:cota_er_4}
|IV|\le \frac{\mu}{8}\|\nabla \cdot \be_r^n\|_0^2+\frac{2}{\mu}\|\xi_h^n\|_0^2.
\end{eqnarray}
For the fifth term we get
\begin{eqnarray}\label{eq:cota_er_5}
|V|\le\frac{\mu}{8}\|\nabla \cdot \be_r^n\|_0^2+2\mu \|\nabla \cdot \bbeta_h^n\|_0^2.
\end{eqnarray}
To bound the sixth term we apply Sobolev embeddings \eqref{sob1}, \eqref{eq:cotaPlinf2},  \eqref{diver_vol}, \eqref{eq:cotanablaPld},
\eqref{eq:uh_infty} and \eqref{eq:uh_2d_dmenosuno}
\begin{eqnarray}\label{eq:cota_er_6}
|VI|&\le& |b_h(P_r^v\bu_h^n,\bbeta_h^n,\be_r^n)|+|b_h(\bbeta_h^n,\bu_h^n,\be_r^n)|
\nonumber\\&\le&
\|P_r^v\bu_h^n\|_{L^\infty}\|\nabla \bbeta_h^n\|_0\|\be_r^n\|_0+\frac{1}{2}\|\nabla \cdot P_r^v\bu_h^n\|_{L^{2d/(d-1)}}
\| \bbeta_h^n\|_{L^{2d}}\|\be_r^n\|_0\nonumber\\
&&\quad +\|\bbeta_h^n\|_{L^{2d}} \|\nabla  \bu_h^n\|_{L^{2d/(d-1)}}\|\be_r^n\|_0+\frac{1}{2}
\|\nabla \cdot\bbeta_h^n\|_0\|\bu_h^n\|_{L^\infty}\|\be_r^n\|_0\nonumber\\
&\le& C_{\rm inf}\|\nabla \bbeta_h^n\|_0\|\be_r^n\|_0+\frac{1}{2}C C_{\rm ld}\|\nabla \bbeta_h^n\|_0\|\be_r^n\|_0
\nonumber\\
&&\quad +C\|\nabla \bbeta_h^n\|_0C_{\bu,\rm ld}\|\be_r^n\|_0+\frac{1}{2}\|\nabla \bbeta_h^n\|_0C_{\bu,\rm inf}\|\be_r^n\|_0\nonumber\\
&\le& C\|\nabla \bbeta_h^n\|_0^2+\frac{1}{2}\|\be_r^n\|_0^2.
\end{eqnarray}
To bound the seventh term we will use the properties of the direct method used to compute the snapshots. From \eqref{estasi}, \eqref{lachi}
and \eqref{eq:pod_method1} we can write
\begin{eqnarray}\label{eq:cota_er_7}
|VII|&\le& \left|\sum_{k={r+1}}^{d_v}\sqrt{M}\sqrt{\lambda_k}v_k^j\left(\nabla \cdot \bvar_k,z_r^n\right)\right|
=\left|\sum_{k={r+1}}^{d_v}v_k^j\sum_{s=1}^Mv_k^s \left(\nabla \cdot \bu_h^s,z_r^n\right)\right|\nonumber\\
&\le&\left|\sum_{k={r+1}}^{d_v}v_k^j\sum_{s=1}^Mv_k^s s_{\rm pres}(p_h^s,z_r^n)\right|\nonumber\\
&\le&\left|
\sum_{k={r+1}}^{d_v}v_k^j\sum_{s=1}^Mv_k^s s_{\rm pres}(p_h^s,p_h^s)^{1/2}s_{\rm pres}(z_r^n,z_r^n)^{1/2}\right|\nonumber\\
&\le&\frac{1}{4}s_{\rm pres}(z_r^n,z_r^n)+\left(\sum_{k={r+1}}^{d_v}v_k^j\sum_{s=1}^Mv_k^s s_{\rm pres}(p_h^s,p_h^s)^{1/2}\right)^2\nonumber\\
&\le&\frac{1}{4}s_{\rm pres}(z_r^n,z_r^n)+\left(\sum_{k={r+1}}^{d_v}|v_k^j|^2\right)\sum_{k={r+1}}^{d_v}\left(\sum_{s=1}^Mv_k^s s_{\rm pres}(p_h^s,p_h^s)^{1/2}\right)^2\nonumber\\
&\le&\frac{1}{4}s_{\rm pres}(z_r^n,z_r^n)+\sum_{k={r+1}}^{d_v}\left(\sum_{s=1}^M|v_k^s|^2\right)\left(\sum_{s=1}^Ms_{\rm pres}(p_h^s,p_h^s)\right)
\nonumber\\
&\le&\frac{1}{4}s_{\rm pres}(z_r^n,z_r^n)+(d_v-r)\sum_{k=1}^M s_{\rm pres}(p_h^k,p_h^k),
\end{eqnarray}
where in the last inequality we have used \eqref{menor1}.

For the last term on the right-hand side of \eqref{eq:error} we get
\begin{eqnarray}\label{eq:cota_er_8}
|VIII|&\le&\frac{1}{4}s_{\rm pres}(z_r^n,z_r^n)+s_{\rm pres}(\xi_h^n,\xi_h^n).
\end{eqnarray}
Inserting \eqref{eq:cota_er_1}, \eqref{eq:cota_er_2}, \eqref{eq:cota_er_3}, \eqref{eq:cota_er_4}, \eqref{eq:cota_er_5}, \eqref{eq:cota_er_6}, \eqref{eq:cota_er_7} and \eqref{eq:cota_er_8} into \eqref{eq:error} we obtain
\begin{eqnarray*}\label{eq:error2}
&&\frac{1}{\Delta t}\left(\|\be_r^n\|_0^2-\|\be_r^{n-1}\|_0^2\right)+\nu\|\nabla \be_r^n\|_0^2+S_h(\be_r^n,\be_r^n)
+\mu\|\nabla \cdot \be_r^n\|_0^2+s_{\rm pres}(z_r^{n},z_r^{n})\nonumber\\
&&\le \left(2C_{1,\rm inf}+\frac{C_{\rm inf}^2}{2\mu}+1\right)\|\be_r^n\|_0^2
+C(\nu+\mu+1)\|\nabla \bbeta_h^n\|_0^2+S_h(\bbeta_h^n,\bbeta_h^n)
\nonumber\\
&&\quad+\frac{4}{\mu}\|\xi_h^n\|_0^2+2(d_v-r)(\Delta t)^{-1}\sum_{k=1}^M \Delta t s_{\rm pres}(p_h^k,p_h^k)+2s_{\rm pres}(\xi_h^n,\xi_h^n).
\end{eqnarray*}
Summing over times we reach
\begin{eqnarray}\label{eq:error2}
&&\|\be_r^n\|_0^2+\nu\sum_{j=1}^n\Delta t \|\nabla \be_r^j\|_0^2+\sum_{j=1}^n\Delta t S_h(\be_r^j,\be_r^j)
+\mu\sum_{j=1}^n\Delta t\|\nabla \cdot \be_r^j\|_0^2\nonumber\\
&&\quad+\sum_{j=1}^n\Delta ts_{\rm pres}(z_r^{j},z_r^{j})\nonumber\\
&&\le \|\be_r^0\|_0^2+\sum_{j=1}^n\Delta t \left(2C_{1,\rm inf}+\frac{C_{\rm inf}^2}{2\mu}+1\right)\|\be_r^j\|_0^2
+\btau_n,
\end{eqnarray}
where
\begin{eqnarray}\label{eq:tau}
\btau_n&=&\sum_{j=1}^n\Delta tS_h(\bbeta_h^j,\bbeta_h^j)+C(\nu+\mu+1)\sum_{j=1}^n\Delta t\|\nabla \bbeta_h^j\|_0^2\nonumber\\
&&\quad+\sum_{j=1}^n\Delta t\left(\frac{4}{\mu}\|\xi_h^j\|_0^2
+2s_{\rm pres}(\xi_h^j,\xi_h^j)\right)\nonumber\\
&&\quad+2(d_v-r)T(\Delta t)^{-1}\sum_{k=1}^M \Delta t s_{\rm pres}(p_h^k,p_h^k)\nonumber\\
&=&\btau_1+\btau_2+\btau_3+\btau_4.
\end{eqnarray}
Denoting by
$$
C_u=2C_{1,\rm inf}+\frac{C_{\rm inf}^2}{2\mu}+1,
$$
assuming $\Delta t C_u\le 1/2$ and applying Gronwall's Lemma \cite[Lemma 5.1]{Hey-RanIV} we get
\begin{eqnarray}\label{eq:error3}
&&\|\be_r^n\|_0^2+\nu\sum_{j=1}^n\Delta t \|\nabla \be_r^j\|_0^2+\sum_{j=1}^n\Delta t S_h(\be_r^j,\be_r^j)+\mu\sum_{j=1}^n\Delta t\|\nabla \cdot \be_r^j\|_0^2\nonumber\\
&&\quad+\sum_{j=1}^n\Delta ts_{\rm pres}(z_r^{j},z_r^{j})\le e^{2T C_u}\left(\|\be_r^0\|_0^2+\btau_n\right).
\end{eqnarray}
To conclude we will get an error bound for $\btau_n$. For the first term in \eqref{eq:tau}, applying the definition of $S_h$, the $L^2$-stability of $\sigma_h^*$, \eqref{eq:mu_2} and \eqref{inv_1} we get
\begin{eqnarray}\label{eq:trun1}
\btau_1\le C h \sum_{j=1}^n\Delta t \|\nabla\bbeta_h^j\|_0^2\le C h\|S^v\|_2\sum_{k=r+1}^{d_v} \lambda_k.
\end{eqnarray}
For the second term in \eqref{eq:tau}, applying again \eqref{inv_1} we obtain
\begin{eqnarray}\label{eq:trun2}
\btau_2\le  C(\nu+\mu+1)\|S^v\|_2\sum_{k=r+1}^{d_v} \lambda_k.
\end{eqnarray}
For the third term, we again apply the $L^2$-stability of $\sigma_h^*$, \eqref{eq:tau_p2}, \eqref{eq:cota_pod_0_pre} and \eqref{inv_1_p}  to prove
\begin{eqnarray}\label{eq:trun3}
\btau_3\le C (\mu^{-1}+h\|S^p\|_2)  \sum_{k=r+1}^{d_p} \gamma_k.
\end{eqnarray}
Finally, for the last term, applying \eqref{eq:cota_est} we get
\begin{eqnarray}\label{eq:trun4}
\btau_4\le C(\bu,p,l+1)^2(d_v-r)(\Delta t)^{-1} (h^{2l+1}+(\Delta t)^2).
\end{eqnarray}
Inserting \eqref{eq:trun1}, \eqref{eq:trun2}, \eqref{eq:trun3} and \eqref{eq:trun4} into \eqref{eq:error3} we reach
\begin{eqnarray}\label{eq:error4}
&&\|\be_r^n\|_0^2+\nu\sum_{j=1}^n\Delta t \|\nabla \be_r^j\|_0^2+\sum_{j=1}^n\Delta t S_h(\be_r^j,\be_r^j)+\mu\sum_{j=1}^n\Delta t\|\nabla \cdot \be_r^j\|_0^2\\
&&\quad+\sum_{j=1}^n\Delta ts_{\rm pres}(z_r^{j},z_r^{j})\le e^{2T C_u}\left(\|\be_r^0\|_0^2+
C(\nu+\mu+1+h)\|S^v\|_2\sum_{k=r+1}^{d_v} \lambda_k\right.\nonumber\\
&&\quad \left. +C (\mu^{-1}+h\|S^p\|_2)  \sum_{k=r+1}^{d_p} \gamma_k+C(\bu,p,l+1)^2(d_v-r)(\Delta t)^{-1} (h^{2l+1}+(\Delta t)^2)  \right)\nonumber.
\end{eqnarray}
\begin{remark}
Let us observe that, contrary to other error bounds in the literature, apart from the last term on the right-hand side of  \eqref{eq:error4},
all the error bound is written in terms of the eigenvalues. The last term appears due to the fact that  non inf-sup stable elements
have been used both in the formulation of the direct and the POD methods. The reason for being able to write most of bound in terms of the eigenvalues is that we have compared the POD velocity approximation with $P_r^v \bu_h^n$ instead of $P_r^v \bu^n$, as it is standard. On the other hand, it is interesting to observe that the constants in the error bound \eqref{eq:error4} are independent on inverse powers of the viscosity.
\end{remark}
\begin{Theorem}
Let $\bu$ be the velocity in the Navier-Stokes equations \eqref{NS}, let $\bu_r$ be the LPS POD
stabilized approximation defined in \eqref{eq:pod_method1}, and assume that the solution $(\bu,p)$ of \eqref{NS} is regular enough. Then, the
following bound holds
\begin{eqnarray}\label{eq:cota_finalLPS}
\sum_{j=1}^n\Delta t \|\bu_r^j-\bu^j\|_0^2&\le& 3e^{2T C_u}\left(\|\be_r^0\|_0^2+
C(\nu+\mu+1+h)\|S^v\|_2\sum_{k=r+1}^{d_v} \lambda_k\right.\nonumber\\
&&\quad \left. +C (\mu^{-1}+h\|S^p\|_2)  \sum_{k=r+1}^{d_p} \gamma_k\right.\\
&&\quad\left.+C(\bu,p,l+1)^2(d_v-r)(\Delta t)^{-1} (h^{2l+1}+(\Delta t)^2)  \right)\nonumber\\
&&\quad +3TC(\bu,p,l+1)^2(h^{2l+1}+(\Delta t)^2)+3\sum_{k=r+1}^{d_v}\lambda_k.\nonumber
\end{eqnarray}
\end{Theorem}
\begin{proof}
Since $\sum_{j=1}^n\Delta t \|\be_r^j\|_0^2\le T \max_{1\le j\le n}\|\be_r^j\|_0^2$ and
\begin{eqnarray}\label{eq:lauso}
\sum_{j=1}^n\Delta t \|\bu_r^j-\bu^j\|_0^2&\le&
3\left(\sum_{j=1}^n\Delta t \|\be_r^j\|_0^2+\sum_{j=1}^n\Delta t\|P_r^v \bu_h^j-\bu_h^j\|_0^2\right.\nonumber\\
&&\quad\left.+\sum_{j=1}^n\Delta t\| \bu_h^j-\bu^j\|_0^2\right),
\end{eqnarray}
applying \eqref{eq:error4}, \eqref{eq:cota_pod_0} and \eqref{eq:cota_est} we finally obtain \eqref{eq:cota_finalLPS}.
\end{proof}
\begin{remark}
We observe that the error bound \eqref{eq:cota_finalLPS} has the component $$(\Delta t)^{-1} (h^{2l+1}+(\Delta t)^2)$$ that comes from the treatment of
the divergence term \eqref{eq:cota_er_7} when using non inf-sup stable elements. However, apart from this term, the $L^2$ error behaves in terms of the mesh size $h$ as the $L^2$ error of the direct method, i.e. as $h^{2l+1}$. This is not the case when one compares with $P_r^v \bu^n$. In that case, the $L^2$ error of the method is bounded by the $H^1$ norm of the error $P_r^v\bu^n-\bu^n$ and, consequently, can be bounded in terms of  the $H^1$  (instead of $L^2$) norm
of the direct method.
\end{remark}
\begin{remark}\label{rm:PresErrEstLPS}
For the method proposed in this section, denoting by
$$
Z_r^n=\sum_{j=1}^n \Delta t z_r^j,
$$
one can argue as in \cite{samuele_pod} to bound the pressure error in the norm defined in \cite{samuele_pod}:
$$
|||Z_r^n|||:=\sup_{\bvar_r\in {\cal \bV}^r}\frac{(Z_r^n,\nabla \cdot \bvar_r)}{\|\nabla \bvar_r\|_0}+s_{\rm pres}(Z_r^n,Z_r^n)^{1/2}.
$$
For the mixed finite element direct method \eqref{eq:gal_est} one has the discrete inf-sup condition, see \cite[Lemma 4.2]{ahmed_etal}:
$$
\forall q_h \in Q_h^l,\quad \|q_h\|_0\le \beta_0\left(\sup_{\bv_h\in {\boldsymbol X}_h^l}\frac{(q_h,\nabla \cdot \bv_h)}{\|\nabla \bv_h\|_0}
+s_{\rm pres}(q_h,q_h)^{1/2}\right).
$$
However, the above inequality has not been proved if one changes in the supremum ${\boldsymbol X}_h^l$ by ${\cal \bV}^r$, so that
in principle the norm $|||\cdot|||$ could be weaker than the $L^2$ norm.
\end{remark}
\section{A grad-div stabilized POD method from an inf-sup stable FOM}\label{sec:grad-div-ROM}
In this section we consider the case in which we start with snapshots of a direct method based on inf-sup stable elements. Since the snapshots
satisfy a discrete divergence-free condition in this case one can formulate the POD method with only velocity approximations. Following \cite{schneier} we will prove that a pressure approximation can be computed also in this case and we will prove error bounds for both the velocity and the pressure with constants independent on inverse powers of the viscosity.

Let $l \geq 2$, we consider the  MFE pair known as Hood--Taylor elements \cite{BF,hood0} $({\boldsymbol X}_h^l, Q_{h}^{l-1})$.
We recall the definition of the divergence-free space ${\boldsymbol V}_{h,l}$ \eqref{eq:V}.

For these elements a uniform inf-sup condition is satisfied (see \cite{BF}), that is, there exists a constant $\beta_{\rm is}>0$ independent of the mesh size $h$ such that
\begin{equation}\label{lbbh}
 \inf_{q_{h}\in Q_{h}^{l-1}}\sup_{\bv_{h}\in{\boldsymbol X}_h^l}
\frac{(q_{h},\nabla \cdot \bv_{h})}{\|\bv_{h}\|_{1}
\|q_{h}\|_{L^2/{\mathbb R}}} \geq \beta_{\rm{is}}.
\end{equation}
As a direct method we consider a Galerkin method with grad-div stabilization and the implicit Euler method in time.
Given  {${\bu}_h^0$ an approximation to $\bu^0$ in ${\boldsymbol X}_h^l$}, find $(\bu_h^{n+1},p_h^{n+1})\in {\boldsymbol X}_h^l\times Q_h^{l-1}$ such that {for $n\ge 0$}
\begin{eqnarray}\label{eq:gal_grad_div}
&&\left(\frac{\bu_h^{n+1}-\bu_h^n}{\Delta t},\bv_h\right)+\nu(\nabla \bu_h^{n+1},\nabla \bv_h)+b(\bu_h^{n+1},\bu_h^{n+1},\bv_h)-(p_h^{n+1},\nabla \cdot \bv_h)
\nonumber\\
&&\quad+
\mu(\nabla \cdot\bu_h^{n+1},\nabla \cdot \bv_h)=({\boldsymbol f}^{n+1},\bv_h) \quad \forall \bv_h\in {\boldsymbol X}_h^l,\\
&&(\nabla \cdot \bu_h^{n+1},q_h)=0 \quad \forall q_h\in Q_h^{l-1},\nonumber
\end{eqnarray}
where $\mu$ is the positive grad-div stabilization parameter.

It is well-known that considering the discrete divergence-free space $\bV_{h,l}$ we can remove the pressure from \eqref{eq:gal_grad_div} since
$\bu_h^{n+1}\in \bV_{h,l}$ satisfies
\begin{eqnarray}\label{eq:gal_grad_div2}
&&\left(\frac{\bu_h^{n+1}-\bu_h^n}{\Delta t},\bv_h\right)+\nu(\nabla \bu_h^{n+1},\nabla \bv_h)+b(\bu_h^{n+1},\bu_h^{n+1},\bv_h)
\nonumber\\
&&\quad+
\mu(\nabla \cdot\bu_h^{n+1},\nabla \cdot \bv_h)=({\boldsymbol f}^{n+1},\bv_h), \quad \forall \bv_h\in {\bV}_{h,l}.
\end{eqnarray}
For this method the following bounds hold, see \cite{NS_grad_div}
\begin{eqnarray}\label{eq:cota_grad_div}
\|\bu^n-\bu_h^n\|_0+ h\|\bu^n-\bu_h^n\|_1\le C(\bu,p,l+1) (h^{l}+\Delta t), \quad 1\le n\le M,
\end{eqnarray}
and
\begin{eqnarray}\label{eq:cota_grad_div_pre}
\left(\sum_{j=1}^n\Delta t \|p^j-p_h^j\|_0^2\right)^{1/2}\le C(\bu,p,l+1) (h^{l}+\Delta t), \quad 1\le n\le M,
\end{eqnarray}
where the constant $C(\bu,p,l+1)$ does not depend on inverse powers of $\nu$.

Arguing as before, we can get a priori bounds for the $L^2$ orthogonal projection $P_r^v \bu_h^n$. However, since now the FOM has a rate
of convergence of order $l$ instead of $l+1/2$, as the method in the previous section, there are some differences. We will assume $d=2$.
In case $d=3$ one can prove a priori bounds
but using cubics (or higher) elements. In the error bound \eqref{eq:uh_infty} we have to replace $h^{3/2}$ by $h$.
 In the error bound \eqref{eq:grad_uh_infty} we have to replace $h^{5/2}$ by $h^2$. Finally, in the bound \eqref{eq:uh_2d_dmenosuno} we have to change $h^{3/2}$ by $h^2$ and
 $C(\bu,p,2)$ by $C(\bu,p,3)$.

We now consider the  grad-div POD model. For $n\ge 1$, find $\bu_r^n\in {\cal \bV}^r$ such that
\begin{eqnarray}\label{eq:pod_method2}
&&\left(\frac{\bu_r^{n}-\bu_r^{n-1}}{\Delta t},\bvar\right)+\nu(\nabla \bu_r^n,\nabla\bvar)+b_h(\bu_r^n,\bu_r^n,\bvar)
+\mu(\nabla \cdot\bu_r^n,\nabla \cdot\bvar)\nonumber\\
&&\quad=(\bff^{n},\bvar),\quad \forall \bvar\in {\cal \bV}^r.
\end{eqnarray}
We observe that $\bu_r^n$ belongs to the discrete divergence-free space $\bV_{h,l}$.

It is easy to get
\begin{eqnarray}\label{eq:prov_prop2}
&&\left(\frac{P_r^v \bu^{n}_h-P_r^v \bu^{n-1}_h}{\Delta t},\bvar\right)+\nu(\nabla  P_r^v \bu^n_h,\nabla\bvar)+b_h(P_r^v \bu^n_h,P_r^v \bu^n_h,\bvar)
\nonumber\\
&&\quad+\mu(\nabla \cdot P_r^v \bu^n_h,\nabla \cdot\bvar)\nonumber\\
&&=(\bff^{n},\bvar)
 +\nu(\nabla\bbeta_h^n,\nabla\bvar)+\mu(\nabla \cdot\bbeta_h^n,\nabla \cdot \bvar)\\
&&\quad+b_h(P_r^v\bu^n_h,P_r^v \bu^n_h,\bvar)-b_h(\bu^{n}_h,\bu^{n}_h,\bvar),\quad \forall \bvar\in {\cal \bV}^r.\nonumber
\end{eqnarray}
Subtracting \eqref{eq:prov_prop2} from \eqref{eq:pod_method2} and denoting as before by $\be_r^n=\bu_r^n-P_r^v\bu_h^n$
we get
\begin{eqnarray}\label{eq:error_need}
&&\left(\frac{\be_r^n -\be_r^{n-1}}{\Delta t},\bvar\right)+\nu(\nabla  \be_r^n ,\nabla\bvar)+b_h(\be_r^n,\be_r^n,\bvar)
+\mu(\nabla \cdot \be_r^n,\nabla \cdot\bvar)\nonumber\\
&&=
 -\nu(\nabla\bbeta_h^n,\nabla\bvar)-\mu(\nabla \cdot\bbeta_h^n,\nabla \cdot \bvar)-b_h(P_r^v\bu^n_h,P_r^v \bu^n_h,\bvar)\nonumber\\
 &&\quad+b_h(\bu^{n}_h,\bu^{n}_h,\bvar),\quad \forall \bvar\in {\cal \bV}^r.
\end{eqnarray}
Taking $\bvar=\be_r^n$ it is easy to obtain
\begin{eqnarray}\label{eq:error2}
&&\frac{1}{2\Delta t}\left(\|\be_r^n\|_0^2-\|\be_r^{n-1}\|_0^2\right)+\nu\|\nabla \be_r^n\|_0^2)
+\mu\|\nabla \cdot \be_r^n\|_0^2\nonumber\\
&&\quad\le -\left(b_h(P_r^v  \bu^n_h,P_r^v \bu^n_h,\be_r^n)-b_h(\bu_r^n,\bu_r^n,\be_r^n)\right)-\nu(\nabla\bbeta_h^n,\nabla\be_r^n)
\nonumber\\
&&\quad-\mu(\nabla \cdot\bbeta_h^n,\nabla \cdot \be_r^n)-\left(b_h(P_r^v\bu^n_h,P_r^v \bu^n_h,\be_r^n)-b_h(\bu^{n}_h,\bu^{n}_h,\be_r^n)\right).
\end{eqnarray}
Arguing exactly as before we get
\begin{eqnarray}\label{eq:error2}
&&\|\be_r^n\|_0^2+\nu\sum_{j=1}^n\Delta t \|\nabla \be_r^j\|_0^2
+\mu\sum_{j=1}^n\Delta t\|\nabla \cdot \be_r^j\|_0^2\\
&&\le \|\be_r^0\|_0^2+\sum_{j=1}^n\Delta t \left(2C_{1,\rm inf}+\frac{C_{\rm inf}^2}{2\mu}+1\right)\|\be_r^j\|_0^2
+C(\nu+\mu+1)\sum_{j=1}^n\Delta t\|\nabla \bbeta_h^j\|_0^2.\nonumber
\end{eqnarray}
Denoting by
$$
C_u=2C_{1,\rm inf}+\frac{C_{\rm inf}^2}{2\mu}+1,
$$
assuming $\Delta t C_u\le 1/2$ and applying Gronwall's Lemma \cite[Lemma 5.1]{Hey-RanIV} and \eqref{inv_1} we obtain
\begin{eqnarray}\label{eq:error3_b}
&&\|\be_r^n\|_0^2+\nu\sum_{j=1}^n\Delta t \|\nabla \be_r^j\|_0^2+\mu\sum_{j=1}^n\Delta t\|\nabla \cdot \be_r^j\|_0^2\nonumber\\
&&\quad\le e^{2T C_u}\left(\|\be_r^0\|_0^2+C(\nu+\mu+1)\|S^v\|_0\sum_{k=r+1}^{d_v}\lambda_k\right).
\end{eqnarray}
\begin{remark}
Let us observe that the constants in the error bound \eqref{eq:error3_b} are independent on inverse powers of $\nu$ and also that they only depend on the eigenvalues $\lambda_k$. This means that the error of the POD method, as expected, is essentially the error
of the best approximation to the Galerkin approximation in the POD space, i.e., the error in $P_r^v\bu_h^n$, plus the POD error coming from the tail of eigenvalues.
\end{remark}
\begin{Theorem}
Let $\bu$ be the velocity in the Navier-Stokes equations \eqref{NS}, let $\bu_r$ be the grad-div POD
stabilized approximation defined in \eqref{eq:pod_method2}, and assume that the solution $(\bu,p)$ of \eqref{NS} is regular enough. Then, the
following bound holds
\begin{eqnarray}\label{eq:cota_finalSUPv}
\sum_{j=1}^n\Delta t \|\bu_r^j-\bu^j\|_0^2&\le& 3e^{2T C_u}\left(\|\be_r^0\|_0^2+
C(\nu+\mu+1)\|S^v\|_2\sum_{k=r+1}^{d_v} \lambda_k\right.\nonumber\\
&&\quad \left.+3TC(\bu,p,l)^2(h^{2l}+(\Delta t)^2)+3\sum_{k=r+1}^{d_v}\lambda_k\right).
\end{eqnarray}
\end{Theorem}
\begin{proof}
Applying \eqref{eq:lauso}, from  \eqref{eq:error3_b}, \eqref{eq:cota_pod_0} and \eqref{eq:cota_grad_div}
we easily obtain \eqref{eq:cota_finalSUPv}.
\end{proof}
Following \cite{schneier} we propose a way to compute a POD pressure approximation in this framework. Given a function $p_r\in{\cal W}^r$ we consider the following problem: find $\bw_h\in {\boldsymbol X}_h^l$ such that
\begin{eqnarray}\label{eq:supremizer}
(\nabla \bw_h,\nabla\bv_h)=-(\nabla \cdot \bv_h,p_r),\quad \forall \bv_h \in {\boldsymbol X}_h^l.
\end{eqnarray}
The supremizer enrichment algorithm consists of solving \eqref{eq:supremizer} for each basis function $\psi_k$, $k=1,\cdots,r$ (we are considering to work with an equal number of basis functions for both velocity and pressure). Then,
applying a Gram-Schmidt orthonormalization  procedure to the set of solutions, a set of basis functions is obtained $\left\{\bzeta_k\right\}$, $k=1,\ldots,r$. Denoting by
$$
{\boldsymbol S}^r=<\bzeta_1,\bzeta_2,\ldots,\bzeta_{r}>\subset{\boldsymbol V}_{h,l}^\bot\subset {\boldsymbol X}_h^l,
$$
the following in-sup stability condition holds for the spaces ${\boldsymbol S}^{r}$ and ${\cal W}^r$, see \cite[Lemma 4.2]{schneier}:
\begin{eqnarray}\label{eq:inf_sup_supre}
\beta_{r}= \inf_{\psi\in {\cal W}^r}\sup_{\bzeta\in{{\boldsymbol S}^{r}}}
\frac{(\psi,\nabla \cdot \bzeta)}{\|\bzeta\|_{1}
\|\psi\|_{L^2/{\mathbb R}}} \geq \beta_{\rm{is}},
\end{eqnarray}
where $ \beta_{\rm{is}}$ is the constant in the inf-sup condition \eqref{lbbh}.

Using the space ${\boldsymbol S}^{r}$ a pressure $p_r^n\in {\cal W}^r$ can be computed satisfying for all $\bzeta\in {\boldsymbol S}^{r}$
\begin{eqnarray}\label{eq:pres}
(p_r^n,\nabla \cdot \bzeta)=\left(\frac{\bu_r^{n}-\bu_r^{n-1}}{\Delta t},\bzeta\right)+b_h(\bu_r^n,\bu_r^n,\bzeta)
+\mu(\nabla \cdot \bu_r^n,\nabla \cdot \bzeta)
-(\bff^{n},\bzeta).
\end{eqnarray}
Denoting by
$$
\|\bv\|_{{\boldsymbol S}^{r,*}}=\sup_{\bzeta\in {\boldsymbol S}^{r}}\frac{(\bv,\bzeta)}{\|\nabla \bzeta\|_0},
\quad
\|\bv\|_{{\cal \bV}^{r,*}}=\sup_{\bzeta\in {\cal \bV}^{r}}\frac{(\bv,\bzeta)}{\|\nabla \bzeta\|_0},
$$
by $0\leq\alpha<1$ the constant in the strengthened Cauchy-Schwarz inequality between the spaces ${\cal \bV}^r$ and ${\boldsymbol S}^{r}$,
and by
\begin{eqnarray}\label{cotaCr}
C_r^{H^1}=\left\|\sum_{k=1}^r \nabla \bvar_k\right\|_0,
\end{eqnarray}
and letting $\bvar\in {\cal \bV}^{r}$, it holds (see \cite[Lemma 5.13]{schneier})
\begin{eqnarray}\label{eq:cota_dual}
\|\bvar\|_{{\boldsymbol S}^{r,*}}\le \alpha C_P C_r^{H^1}\|\bvar\|_{{\cal \bV}^{r,*}},
\end{eqnarray}
where $C_P$ is the constant in the Poincar\'e inequality.

Now, we observe that adding and subtracting $P_r^p p_h^n$ from \eqref{eq:gal_grad_div} we get
\begin{eqnarray}\label{eq:p_hres}
(P_r^p p_h^n,\nabla \cdot \bzeta)&=&\left(\frac{\bu_h^{n}-\bu_h^{n-1}}{\Delta t},\bzeta\right)+b_h(\bu_h^n,\bu_h^n,\bzeta)
+\mu(\nabla \cdot \bu_h^n,\nabla \cdot \bzeta)
-(\bff^{n},\bzeta)\nonumber\\
&&\quad+(P_r^p p_h^n-p_h^n,\nabla \cdot \bzeta), \quad \forall\bzeta\in {\boldsymbol S}^{r}.
\end{eqnarray}
Subtracting \eqref{eq:p_hres} from \eqref{eq:pres} and applying \eqref{eq:inf_sup_supre}, \eqref{eq:cota_dual} and \eqref{diver_vol}  we get
\begin{eqnarray}\label{eq:erpre1}
&&\|z_r^n\|_0\le \frac{1}{\beta_{r}}\left(\alpha C_P C_r^{H^1} \left\|\frac{(\bu_r^n-\bu_h^n)-(\bu_r^{n-1}-\bu_h^{n-1})}{\Delta t }\right\|_{{\cal \bV}^{r,*}}\right.\\
&&\quad \left.\sup_{\bzeta\in {\boldsymbol S}^{r}}\frac{b_h(\bu_r^n,\bu_r^n,\bzeta)-b_h(\bu_h^n,\bu_h^n,\bzeta)}{\|\nabla \bzeta\|_0}
+\mu  \|\nabla \cdot(\bu_r^n-\bu_h^n)\|_0+\|\xi_h^n\|_0\right).\nonumber
\end{eqnarray}
We will bound now the terms on the right-hand side of \eqref{eq:erpre1}. For the first one we observe that
\begin{eqnarray*}
\left\|\frac{(\bu_r^n-\bu_h^n)-(\bu_r^{n-1}-\bu_h^{n-1})}{\Delta t }\right\|_{{\cal \bV}^{r,*}}
=\left\|\frac{\be_r^n-\be_r^{n-1}}{\Delta t }\right\|_{{\cal \bV}^{r,*}}.
\end{eqnarray*}
Applying now \eqref{eq:error_need} we obtain
\begin{eqnarray}\label{eq:ertem}
&&\left\|\frac{(\bu_r^n-\bu_h^n)-(\bu_r^{n-1}-\bu_h^{n-1})}{\Delta t }\right\|_{{\cal \bV}^{r,*}}
\le \nu\|\nabla \be_r^n\|_0+\mu\|\nabla\cdot\be_r^n\|_0\nonumber\\
&&\quad+(\nu+\mu) \|\nabla\bbeta_h^n\|_0+\sup_{\bzeta\in {\cal \bV}^{r}}\frac{b_h(\bu_h^n,\bu_h^n,\bzeta)-b_h(\bu_r^n,\bu_r^n,\bzeta)}{\|\nabla \bzeta\|_0}.
\end{eqnarray}
To bound the last term on the right-hand side of \eqref{eq:ertem} we write
\begin{eqnarray*}
&&b_h(\bu_r^n,\bu_r^n,\bzeta)-b_h(\bu_h^n,\bu_h^n,\bzeta)=b_h(\bu_r^n,\bu_r^n-\bu_h^n,\bzeta)+b_h(\bu_r^n-\bu_h^n,\bu_h,\bzeta)
\nonumber\\
&&\quad=b_h(\bu_r^n,\bzeta,\bu_r^n-\bu_h^n)+b_h(\bu_r^n-\bu_h^n,\bu_h,\bzeta)\nonumber\\
&&\quad\le\|\bu_r^n-\bu_h^n\|_0\left(\|\bu_r^n\|_{L^\infty}\|\bzeta\|_1+\frac{1}{2}\|\nabla \cdot\bu_r^n\|_{L^{2d/(d-1)}}\|\bzeta\|_{L^{2d}} \right)
\nonumber\\
&&\quad+\frac{1}{2}\left((\bu_r^n-\bu_h^n)\cdot \nabla \bu_h,\bzeta\right)-\frac{1}{2}\left((\bu_r^n-\bu_h^n)\cdot \nabla\bzeta,
\bu_h^n\right)\nonumber\\
&&\quad\le\|\bu_r^n-\bu_h^n\|_0\left(\|\bu_r^n\|_{L^\infty}\|\bzeta\|_1+\frac{1}{2}\|\nabla \cdot\bu_r^n\|_{L^{2d/(d-1)}}\|\bzeta\|_{L^{2d}} \right)
\nonumber\\
&&\quad+\frac{1}{2}\|\bu_r^n-\bu_h^n\|_0\left(\|\nabla \bu_h\|_{L^{2d/(d-1)}}\|\bzeta\|_{L^{2d}}+\|\bu_h^n\|_{L^\infty}\|\bzeta\|_1\right).
\end{eqnarray*}
Now, we observe that applying \eqref{inv}, \eqref{eq:cotaPlinf2}
and \eqref{eq:error3_b} we obtain
\begin{eqnarray*}
&&\|\bu_r^n\|_{L^\infty}\le \|\be_r^n \|_{L^\infty}+\|P_r^v \bu_h^n\|_{L^\infty}\le C h^{-d/2}\|\be_r^n\|_0+C_{\rm inf}\nonumber\\
&&\le C_{\bu_r,\rm inf}:=C h^{-d/2}e^{T C_u}\left(\|\be_r^0\|_0^2+C(\nu+\mu+1)\|S^v\|_0\sum_{k=r+1}^{d_v}\lambda_k\right)^{1/2}+C_{\rm inf}.
\end{eqnarray*}
Applying exactly the same argument we also obtain
\begin{eqnarray*}
&&\|\nabla \cdot\bu_r^n\|_{L^{2d/(d-1)}}\le  C h^{-1/2}\|\be_r^n\|_1+C_{\rm ld}\nonumber\\
&&\le C_{\bu_r,\rm ld}:=C h^{-d/2}e^{T C_u}\|S^v\|_2^{1/2}\left(\|\be_r^0\|_0^2+C(\nu+\mu+1)\|S^v\|_0\sum_{k=r+1}^{d_v}\lambda_k\right)^{1/2}
\nonumber\\
&&\quad+C_{\rm ld}.
\end{eqnarray*}
With the above error bounds, Sobolev embeddings \eqref{sob1}, \eqref{eq:uh_infty} and \eqref{eq:grad_uh_infty} we finally obtain
\begin{eqnarray}\label{eq:nonli}
&&b_h(\bu_r^n,\bu_r^n,\bzeta)-b_h(\bu_h^n,\bu_h^n,\bzeta)
\le C\|\bu_r^n-\bu_h^n\|_0\|\bzeta\|_1
\end{eqnarray}
Going back to \eqref{eq:ertem} we reach
\begin{eqnarray*}
\left\|\frac{(\bu_r^n-\bu_h^n)-(\bu_r^{n-1}-\bu_h^{n-1})}{\Delta t }\right\|_{{\cal \bV}^{r,*}}
&\le& \nu\|\nabla \be_r^n\|_0+\mu\|\nabla\cdot\be_r^n\|_0+(\nu+\mu) \|\nabla\bbeta_h^n\|_0\nonumber\\
&&\quad+C\|\bu_r^n-\bu_h^n\|_0
\end{eqnarray*}
Taking into account that
\begin{eqnarray}\label{eq:otra_norma0}
\|\bu_r^n-\bu_h^n\|_0\le \|\be_r^n\|_0+\|P_r^v\bu_h^n-\bu_h^n\|_0,
\end{eqnarray}
we finally obtain
\begin{eqnarray}\label{eq:ertem2}
\left\|\frac{(\bu_r^n-\bu_h^n)-(\bu_r^{n-1}-\bu_h^{n-1})}{\Delta t }\right\|_{{\cal \bV}^{r,*}}
&\le& \nu\|\nabla \be_r^n\|_0+\mu\|\nabla\cdot\be_r^n\|_0+(\nu+\mu) \|\nabla\bbeta_h^n\|_0
\nonumber\\
&&\quad+\|\be_r^n\|_0+\|P_r^v\bu_h^n-\bu_h^n\|_0.
\end{eqnarray}
For the second term on the right-hand side of \eqref{eq:erpre1} we apply \eqref{eq:nonli} and \eqref{eq:otra_norma0}. Inserting also \eqref{eq:ertem2}
into \eqref{eq:erpre1} we finally reach
\begin{eqnarray}\label{eq:erpre2}
\|z_r^n\|_0&\le& \beta_{r}^{-1}\alpha C_P C_r^{H^1}\left( \nu\|\nabla \be_r^n\|_0+\mu\|\nabla\cdot\be_r^n\|_0+(\nu+\mu) \|\nabla\bbeta_h^n\|_0
\right)\nonumber\\
&&\quad+\beta_{r}^{-1}\alpha C_P C_r^{H^1}\left(\|\be_r^n\|_0+\|P_r^v\bu_h^n-\bu_h^n\|_0 \right)\\
&&\quad C\beta_{r}^{-1}\left(\|\be_r^n\|_0+\|P_r^v\bu_h^n-\bu_h^n\|_0+\mu  \|\nabla \cdot(\bu_r^n-\bu_h^n)\|_0+\|\xi_h^n\|_0\right)
.\nonumber
\end{eqnarray}
We observe that
\begin{eqnarray*}
\|\nabla \cdot(\bu_r^n-\bu_h^n)\|_0\le \|\nabla \cdot\be_r^n\|_0+\|\nabla (P_r^v\bu_h^n-\bu_h^n)\|_0.
\end{eqnarray*}
From \eqref{eq:erpre2}
we get
\begin{eqnarray}\label{eq:erpre3}
\sum_{j=1}^n\Delta t \|z_r^j\|_0^2&\le& C \alpha C_r^{H^1}\sum_{j=1}^n\Delta t\left( \nu\|\nabla \be_r^j\|_0^2+\mu\|\nabla\cdot\be_r^j\|_0^2\right)
\nonumber\\
&&\quad +C \alpha C_r^{H^1}(\nu+\mu)\sum_{j=1}^n\|\nabla(P_r^v\bu_h^j-\bu_h^j)\|_0^2\nonumber\\
&&\quad +C\left(1+\alpha C_r^{H^1}\right)\sum_{j=1}^n\Delta t \left(\|\be_r^j\|_0^2+\|P_r^v\bu_h^j-\bu_h^j\|_0^2\right)\nonumber\\
&&\quad + C \mu\sum_{j=1}^n\Delta t \left(\|\nabla\cdot\be_r^j\|_0^2+\|\nabla(P_r^v\bu_h^j-\bu_h^j)\|_0^2\right)\nonumber\\
&&\quad +C \sum_{j=1}^n\Delta t \|P_r^p p_h^j-p_h^j\|_0^2.
\end{eqnarray}
To conclude we apply \eqref{eq:cota_pod_0}, \eqref{eq:cota_pod_0_pre}, \eqref{inv_1} and \eqref{eq:error3_b} to \eqref{eq:erpre3}
\begin{eqnarray}\label{eq:erpre4}
&&\sum_{j=1}^n\Delta t \|z_r^j\|_0^2\nonumber\\
&&\le C(1+T)\left(1+\alpha C_r^{H^1}\right)e^{2T C_u}\left(\|\be_r^0\|_0^2+C(\nu+\mu+1)\|S^v\|_0\sum_{k=r+1}^{d_v}\lambda_k\right)
\nonumber\\
&&\quad+\left(C \alpha C_r^{H^1}(\nu+\mu)+\mu\right)\|S^v\|_2\sum_{k=r+1}^{d_v} \lambda_k\nonumber\\
&&\quad+C\left(1+\alpha C_r^{H^1}\right)\sum_{k=r+1}^{d_v} \lambda_k+C\sum_{k=r+1}^{d_p} \gamma_k.
\end{eqnarray}
\begin{Theorem}
Let $p$ be the pressure in the Navier-Stokes equations \eqref{NS}, let $p_r$ be the grad-div POD
stabilized pressure defined in \eqref{eq:pres}, and assume that the solution $(\bu,p)$ of \eqref{NS} is regular enough. Then, the
following bound holds
\begin{eqnarray}\label{eq:erpre4_ul}
&&\sum_{j=1}^n\Delta t \|p_r^j-p^j\|_0^2\nonumber\\
&&\le C(1+T)\left(1+\alpha C_r^{H^1}\right)e^{2T C_u}\left(\|\be_r^0\|_0^2+C(\nu+\mu+1)\|S^v\|_0\sum_{k=r+1}^{d_v}\lambda_k\right)
\nonumber\\
&&\quad+3\left(C \alpha C_r^{H^1}(\nu+\mu)+\mu\right)\|S^v\|_2\sum_{k=r+1}^{d_v} \lambda_k\nonumber\\
&&\quad+C\left(1+\alpha C_r^{H^1}\right)\sum_{k=r+1}^{d_v} \lambda_k+C\sum_{k=r+1}^{d_p} \gamma_k+3C(\bu,p,l)^2 (h^{l}+\Delta t)^2.
\end{eqnarray}
\end{Theorem}
\begin{proof}
Since applying triangle inequality we have
\begin{equation}\label{eq:cota_finalSUPp}
\sum_{j=1}^n\Delta t \|p_r^j-p^j\|_0^2\le 3\sum_{j=1}^n\Delta t \|z_r^j\|_0^2+3
\sum_{j=1}^n\Delta t \|P_h p_h^j-p_h^j\|_0^2+3\sum_{j=1}^n\Delta t \|p_h^j-p^j\|_0^2,
\end{equation}
the bound \eqref{eq:erpre4_ul}
 for the pressure follows from \eqref{eq:erpre4}, \eqref{eq:cota_pod_0_pre} and \eqref{eq:cota_grad_div_pre}.
\end{proof}

\input{numerics}

\section{Conclusions}\label{sec:Concl}

In this paper, POD stabilized reduced order methods for the numerical simulation of incompressible flows are proposed, analyzed and tested. In particular, we consider two approaches. In the first approach, called LPS-ROM, the standard discrete inf-sup condition is circumvented and POD modes are computed by a non inf-sup stable finite element method, called LPS-FEM \cite{nos_lps}. In the second approach, called grad-div-ROM, the standard discrete inf-sup condition is recovered through supremizer enrichment and POD modes are computed by an inf-sup stable finite element method, called grad-div-FEM \cite{NS_grad_div}. In both approaches, we build reduced order approximations for both velocity and pressure. In the first case, we consider as full order method a LPS finite element scheme with equal order interpolations, which stabilizes the gradient of both velocity and pressure. As for the corresponding reduced order POD method, we add the same kind of LPS for the gradient of both velocity and pressure than the FOM, together with online grad-div stabilization. In the second case, we consider as full order method an inf-sup stable Galerkin method with grad-div stabilization and for the corresponding reduced order POD method we also apply grad-div stabilization. In the latter case, since the velocity snapshots (and of consequence the POD velocity modes) are discretely divergence-free, the pressure can be removed from the formulation of the reduced order POD velocity, so that we use a momentum equation recovery approach to recover the online pressure from a velocity-only ROM, based on supremizer enrichment of the reduced velocity.

The main contribution of the present paper is the numerical analysis of the fully discrete LPS-ROM and grad-div-ROM applied to the unsteady incompressible Navier-Stokes equations,
where rigorous error bounds with constants independent on inverse powers of the viscosity parameter are proved for both methods. To our knowledge, this is the first time this kind of sharp viscosity independent bounds are obtained for stabilized POD-ROM of incompressible flows. In this respect, the present study can be seen as an improvement of the numerical analysis performed in \cite{schneier} and \cite{samuele_pod}.

Numerical experiments have been conducted to illustrate the compared performances of the proposed schemes and assess their accuracy and efficiency. Within this framework, we have also proposed an adaptive in time algorithm for the online grad-div parameter used both in the LPS-ROM and the grad-div-ROM, by adjusting dissipation arising from the online grad-div stabilization term in order to better match the FOM energy. In both cases, using a small equal number of POD velocity-pressure modes already provides comparable reliable approximations, close to the FOM results, and theoretical considerations suggested by the numerical analysis are recovered in practice. Actually, the performed analysis helped to find good a priori error indicators that, at least for small $r$ (of interest in practice), almost match the computed errors over predictive time intervals. Also, the adaptive in time algorithm for the online grad-div parameter proved to significantly improve the long time accuracy of both ROM. Although they provide similar global results, an increased accuracy can be observed for the LPS-ROM that better matches the corresponding FOM local quantities of interest such as the drag coefficient, especially. In terms of efficiency, note that the LPS-ROM circumvents the standard discrete inf-sup condition for the POD velocity-pressure spaces, whose fulfillment can be rather expensive and inefficient in realistic CFD applications \cite{Rozza15}, since it could require the application of the suprimizer enrichment strategy, used here for the grad-div-ROM, in a multi-parameter dependent setting. A cheaper and efficient alternative could be to use an approximate supremizer enrichment procedure \cite{Rozza15}, for which however it is not possible to rigorously show that the standard discrete inf-sup condition is satisfied and it is only possible to rely on heuristic criteria to check it in a post-processing stage. Finally, the velocity modes for the LPS-ROM do not have to be necessarily weakly divergence-free, which allows to use snapshots generated for instance with penalty or projection-based stabilized methods, as the LPS-FEM used in this paper. This is not the case of the grad-div-ROM considered here and, for instance, of ROM based on a pressure Poisson equation approach \cite{IliescuJohn14,schneier,RozzaStabile18} for which the velocity snapshots, and hence the POD velocity modes must be at least weakly divergence-free.

\bibliographystyle{abbrv}
\bibliography{references_sta,Biblio_STAB-POD-ROM_NSE}
\end{document}

config: arxiv


TITLE: One assumption in the proof of one result of Hahn-Banach theorem.
QUESTION [2 upvotes]: Theorem: Let $X$ be a locally convex space, let $M$ be a linear subspace of $X$, and let $f\in M^*$. Then there exists $h\in X^*$ such that $h(x)=f(x)$ for all $x\in M$.
I saw one sentence in the proof that I don't understand. It is:
Replacing $M$ by $\overline{M}$ and replacing $f$ by its unique continuous linear extension to $\overline{M}$, we can assume that $M$ is closed in $X$.
My question is what is the reason to replace $M$ and $f$ and why we can assume that $M$ is closed in $X$?
Thank you in advance!

REPLY [1 votes]: First, let's handle the technical problem (why can we assume $M$ is closed and $f$ is continuous on $M$?)  Recall that the topology on any locally convex space is generated by some family of seminorms $p_\alpha$.  Thus, a function $f$ is continuous on $M$ if and only if
$$|f(x-y)| \leq \sum_{i=1}^n c_i p_{\alpha_i}(x-y)$$
for soem finite $n$ and $\alpha_1,\cdots,\alpha_n$.  Now, let $x \in \overline{M}$ and $(x_\kappa)_\kappa$ a Cauchy net in $M$ converging to $x$.  Then, $f(x_\kappa)$ is a Cauchy net in $\mathbb{R}$, so it converges to some limit, call it $f(x)$.
Now, to see that this extension of $f$ is continuous, let $(x_\mu)_\mu$ be another Cauchy net converging to $x$.  Then,
$$\begin{align*}|f(x-x_\mu)| &\leq |f(x) - f(x_\kappa)| + |f(x_\kappa - x_\mu)|\\
& \leq |f(x) - f(x_\kappa)| + \sum_{i=1}^n c_i p_{\alpha_i}(x_\kappa - x_\mu)\\
&\leq |f(x) - f(x_\kappa)| + \sum_{i=1}^n c_i p_{\alpha_i}(x - x_\kappa) + \sum_{i=1}^n c_i p_{\alpha_i}(x - x_\mu)\end{align*}$$
The first term on the right can be made arbitrarily small by choosing $\kappa$ appropriately by the definition of $f(x)$.  Likewise, the second and third terms can be made small in the limit (of nets) since $(x_\kappa)_\kappa$ and $(x_\mu)_\mu$ are both Cauchy nets converging to $x$.
As to why we want to do this, it helps to remember the proof strategy: we want to argue that if $M$ is a strict subspace of $X$ with a continuous linear functional $f$ defined on it, then there always exists a superspace $N \supsetneq M$ and a continuous functional $g$ with $g|_{M} = f$ (Zorn's lemma then tells us that $h$ exists).  We construct $N$ as a codimension $1$ extension of $M$: that is, we choose some $x \not\in M$, define $N = M + \operatorname{span}(x)$, and define $f(x)$ somehow and extend by linearity.
If $M$ is closed, then we are free to choose $f(x)$ however we like (and various refined versions of the theorem choose $f(x)$ so that the global extension $h$ has some nice geometric/analytic properties).  If $M$ is open, however, we are not be able to choose $f(x)$ freely if $x \in \partial M$, so we would have to define $f(x)$ by continuity anyways.


TITLE: Show that $λ(N)=\bigcap_{m=1}^{\infty}(U_m) = 0$
QUESTION [0 upvotes]: Let $A=\{a_1, a_2, . . .\}$ be a countable subset of R and let λ be the Lebesgue measure on
$(\mathbb R,\mathcal M)$. Define for each $k ∈ \mathbb N$
$$U_k =\bigcup_{n=1}^{\infty}(a_n−\frac{1}{2^{n+k}},a_n+\frac{1}{2^{n+k}})$$
Let $N =\bigcap_{m=1}^{\infty}(U_m)$
Show that $λ(N) = 0$.
I'm not sure how to start really:
I assume $$λ(\bigcup_{k=1}^{m}(U_k)) = λ(U_m) \leq \sum_{k=1}^{m}\sum_{n=1}^{\infty}(a_n−\frac{1}{2^{n+k}},a_n+\frac{1}{2^{n+k}})$$
by subadditivity. How do I show that $λ(N)=\bigcap_{m=1}^{\infty}(U_m) = 0$?
Please explain step by step as I am not experienced with this

REPLY [1 votes]: For every $k\in\{1,2,\dots\}$ we have:$$0\leq \lambda\left(N\right)\leq\lambda\left(U_{k}\right)\leq\sum_{n=1}^{\infty}\lambda\left(\left(a_{n}-\frac{1}{2^{n+k}},a_{n}+\frac{1}{2^{n+k}}\right)\right)=\sum_{n=1}^{\infty}\frac{2}{2^{n+k}}=2^{1-k}$$
This can only be true if: $$\lambda\left(N\right)=0$$