# Error estimates for finite element discretizations of the instationary
Navier-Stokes equations

Boris Vexler  Chair of Optimal Control, Technical University of Munich, School
of Computation Information and Technology, Department of Mathematics ,
Boltzmannstraße 3, 85748 Garching b. Munich, Germany (vexler@tum.de).  and
Jakob Wagner  Chair of Optimal Control, Technical University of Munich, School
of Computation Information and Technology, Department of Mathematics ,
Boltzmannstraße 3, 85748 Garching b. Munich, Germany<EMAIL_ADDRESS>ORCID: https://orcid.org/0000-0001-8510-9790.

###### Abstract.

In this work we consider the two dimensional instationary Navier-Stokes
equations with homogeneous Dirichlet/no-slip boundary conditions. We show
error estimates for the fully discrete problem, where a discontinuous Galerkin
method in time and inf-sup stable finite elements in space are used. Recently,
best approximation type error estimates for the Stokes problem in the
$L^{\infty}(I;L^{2}(\Omega))$, $L^{2}(I;H^{1}(\Omega))$ and
$L^{2}(I;L^{2}(\Omega))$ norms have been shown. The main result of the present
work extends the error estimate in the $L^{\infty}(I;L^{2}(\Omega))$ norm to
the Navier-Stokes equations, by pursuing an error splitting approach and an
appropriate duality argument. In order to discuss the stability of solutions
to the discrete primal and dual equations, a specially tailored discrete
Gronwall lemma is presented. The techniques developed towards showing the
$L^{\infty}(I;L^{2}(\Omega))$ error estimate, also allow us to show best
approximation type error estimates in the $L^{2}(I;H^{1}(\Omega))$ and
$L^{2}(I;L^{2}(\Omega))$ norms, which complement this work.

###### Key words and phrases:

Navier-Stokes, transient, instationary, finite elements, discontinuous
Galerkin, error estimates, best approximation, fully discrete

###### 1991 Mathematics Subject Classification:

35Q30, 65M60, 65M15, 65M22, 76D05, 76M10

##  1\. Introduction

In this paper, we consider the instationary Navier-Stokes equations in two
space dimensions with homogeneous boundary conditions, i.e.,

$\left\\{\begin{aligned} \partial_{t}u-\nu\Delta u+(u\cdot\nabla)u+\nabla
p&=f&&\text{in }I\times\Omega,\\\ \nabla\cdot u&=0&&\text{in
}I\times\Omega,\\\ u(0)&=u_{0}&&\text{in }\Omega,\\\ u&=0&&\text{on
}I\times\partial\Omega.\end{aligned}\right.$ (1)

Here $\nu>0$ denotes the viscosity, $I=(0,T]\subset\mathbb{R}$ a bounded,
half-open interval for some fixed finite endtime $T>0$, and
$\Omega\subset\mathbb{R}^{2}$ a bounded convex polygonal domain. The equations
are discretized in time by a discontinuous Galerkin (dG) method, i.e., the
solution is approximated by piecewise polynomials in time, defined on
subintervals of $I$, without any requirement of continuity at the time nodes,
see, e.g., [24, 49]. The parameter indicating the time discretization will be
denoted by $k$ and corresponds to the length of the largest subinterval in the
partition of $I$. To discretize in space, we use inf-sup stable pairs of
finite element spaces for the velocity and pressure components. The parameter
indicating the spatial discretization will be denoted by $h$ and corresponds
to the largest diameter of cells in the mesh.
Due to the variational formulation of the dG time discretization, this
discretization scheme is particularly suited for treating optimal control
problems. See, e.g., [13, 14] for optimal control problems governed by the
Navier-Stokes equations, also [42, 37] for optimal control of general
parabolic problems. While in [13, 14] the focus was put on low order schemes,
recently the authors of [2] analyzed dG schemes of arbitrary order for the
Navier-Stokes equations. Another advantage of the dG time discretization is
the fact, that the maximal parabolic regularity, exhibited by parabolic
problems, is preserved on the discrete level, and moreover can be extended to
the limiting cases $L^{1}$ and $L^{\infty}$ in time, at the expense of a
logarithmic factor, see [35, Theorems 11 & 12]. The natural energy norm for
the Navier-Stokes equations is the norm of the space
$L^{\infty}(I;L^{2}(\Omega)^{2})\,\cap\,L^{2}(I;H^{1}(\Omega)^{2})$. Indeed by
formally testing (1) with the solution $u$, one obtains

$\|u\|_{L^{\infty}(I;L^{2}(\Omega))}+\|u\|_{L^{2}(I;H^{1}(\Omega))}\leq
C\left(\|u_{0}\|_{L^{2}(\Omega)}+\|f\|_{L^{1}(I;L^{2}(\Omega))}\right).$

This bound is also preserved on the discrete level, i.e., holds for the fully
discrete solution $u_{kh}$, see Theorem 4.13. Our main goal in writing this
paper, was the investigation of the discretization error in terms of the
$L^{\infty}(I;L^{2}(\Omega))$-norm. In particular, such estimates are required
for the analysis of optimal control problems, subject to state constraints
pointwise in time, see [42], as the corresponding Lagrange multiplier in this
case is a measure in time. There have been numerous approaches in the
literature, deriving error estimates for the Navier-Stokes equations. In [13,
Theorem 4.7], the same combination of dG-cG discretization schemes was used,
and for $f\in L^{2}(I;L^{2}(\Omega)^{2})$ an estimate

$\|u-u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq C(\sqrt{k}+h)$ (2)

was shown. Additional terms arise, when changes in the spatial mesh on
different time intervals are permitted. Under the much stricter assumption
$f\in W^{1,\infty}(I;L^{2}(\Omega)^{2})$ and for the implicit Euler time
discretization, in [31] the estimate

$\|u-u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq C(k+h^{2})$ (3)

was shown. It was extended to the Crank-Nicholson scheme in [32] under the
assumption of $f\in W^{2,\infty}(I;L^{2}(\Omega)^{2})$, yielding an error
estimate at the time nodes of order $\mathcal{O}(k^{2}+h^{2})$. Beyond the
references mentioned above, there are many works analyzing the stationary
equations, e.g., [27, 45], semidiscrete equations in space, e.g., [9, 20], or
semidiscrete equations in time, e.g., [23, 46, 4]. Fully discrete error
estimates for stabilized discretization schemes can be found e.g., in [2, 34,
18, 5]. The error estimates are most often derived assuming the necessary
regularity of $u$, such that the discretization schemes can exhibit their full
approximative power. The main drawback of the results in the literature is the
fact, that usually the errors in $L^{\infty}(I;L^{2}(\Omega))$ and
$L^{2}(I;H^{1}(\Omega))$ are estimated in a combined fashion. Thus the
estimate always has to account for the spatial error in the $H^{1}$ norm,
yielding an order reduction for the estimate of the error in the spatial
$L^{2}$ norm. The goal of this paper is to prove an error estimate which can
be formulated as a best approximation type error estimate, which thus
estimates the error in the $L^{\infty}(I;L^{2}(\Omega))$ norm in an isolated
manner. Such an estimate for the instationary Stokes equations, has been
derived recently in [7]. More specifically, for velocity and pressure fields
$(w,r)$, solving the instationary Stokes equations on the continuous level,
and their fully discrete approximations $(w_{kh},r_{kh})$, it holds

$\|w-w_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\ln\left(\frac{T}{k}\right)\left(\inf_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\in
V_{kh}}\|w-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|w-R^{S}_{h}(w,r)\|_{L^{\infty}(I;L^{2}(\Omega))}\right),$

where $V_{kh}$ is the space of discretely divergence free space-time finite
element functions. The operator $R^{S}_{h}$ denotes a stationary Stokes
projection, see [7] or Section 5 for a formal definition. Using this Stokes
result, we will show that a best approximation type result also holds for the
nonlinear Navier-Stokes equations. This main result is stated in Theorem 5.6.
In Corollary 5.8, for $f\in L^{\infty}(I;L^{2}(\Omega)^{2})$, we then obtain
an estimate in terms of $\mathcal{O}(l_{k}(k+h^{2}))$, where $l_{k}$ denotes a
logarithmic term depending on $k$. This result provides a better order of
convergence compared to the estimate (2), shown in [13]. The order of
convergence in the estimate (3), that was presented in [31] is comparable, but
requires a much stronger regularity assumption. The main tools used in this
paper for proving the proposed $L^{\infty}(I;L^{2}(\Omega))$ error estimate
are an error splitting approach and a bootstrapping argument, to apply the
corresponding error estimate for the Stokes equations to the first part of the
error. In order to apply such an argument, understanding the precise
regularity of the occuring nonlinear term $(u\cdot\nabla)u$ is crucial. The
second part of the error will be estimated by a duality argument. This is
possible due to the variational nature of the dG time discretization. We will
derive a stability result for a discrete dual equation. For this result, a
specially adapted version of a discrete Gronwall lemma, Lemma 4.11, will be
presented. For the analysis of the discrete dual problem, we require an
estimate in the $L^{2}(I;H^{1}(\Omega))$ norm, which is why we formulate this
result first. We shall then show the error estimate in
$L^{\infty}(I;L^{2}(\Omega))$ with the tools presented above. The proof of the
error estimate in the $L^{2}(I;L^{2}(\Omega))$ norm is then straightforward
and concludes our work. Summarizing, our main results read

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\|u-u_{kh}\|_{L^{2}(I;H^{1}(\Omega))}}$
$\displaystyle\leq C\left(\sqrt{k}+h\right)$ $\displaystyle\text{if }f\in
L^{2}(I;L^{2}(\Omega)^{2})$ $\displaystyle\text{ and }u_{0}\in V,$
$\displaystyle\|u-u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}$ $\displaystyle\leq
C\ln(T/k)^{2}\left(k+h^{2}\right)$ $\displaystyle\text{if }f\in
L^{\infty}(I;L^{2}(\Omega)^{2})$ $\displaystyle\text{ and }u_{0}\in V\cap
H^{2}(\Omega)^{2},$ $\displaystyle\|u-u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}$
$\displaystyle\leq C\left(k+h^{2}\right)$ $\displaystyle\text{if }f\in
L^{2}(I;L^{2}(\Omega)^{2})$ $\displaystyle\text{ and }u_{0}\in V,$

and can be found in Theorems 4.16, 5.6 and 5.7 and Corollaries 5.8 and 5.9.
All three results are, up to logarithmic terms, optimal in terms of order of
convergence and in terms of required regularity. The structure of this paper
will be as follows. First, we fix some notation and function spaces in Section
2. We proceed in Section 3 by stating the appropriate weak formulations of (1)
with and without pressure and recall some known regularity results. We
conclude the section with an analysis of the regularity properties of the
nonlinear term $(u\cdot\nabla)u$. Section 4 will be devoted to the spatial and
temporal discretizations. We present a version of a discrete Gronwall lemma,
which allows us to treat problems with right hand sides that are only $L^{1}$
in time, and show stability results for fully discrete primal equations. With
this result, we derive an error estimate for the Navier-Stokes equations in
$L^{2}(I;H^{1}(\Omega))$. Lastly, in Section 5, we show stability results for
a discrete dual problem, which in the end allows us to show the error
estimates in the $L^{\infty}(I;L^{2}(\Omega))$ and $L^{2}(I;L^{2}(\Omega))$
norms.

##  2\. Preliminary

For a convex, polygonal domain $\Omega\subset\mathbb{R}^{2}$, $1\leq
p\leq\infty$ and $k\in\mathbb{N}$, we denote by $L^{p}(\Omega)$,
$W^{k,p}(\Omega)$, $H^{k}(\Omega)$ and $H^{1}_{0}(\Omega)$ the usual Lebesgue
and Sobolev spaces. The inner product on $L^{2}(\Omega)$ will be denoted by
$\left(\cdot,\cdot\right)_{\Omega}$. The space $L^{2}_{0}(\Omega)$ is the
subspace of $L^{2}(\Omega)$, consisting of all functions, that have zero mean.
For $s\in\mathbb{R}\backslash\mathbb{N}$, $s>0$ the fractional order
Sobolev(-Slobodeckij) space $W^{s,p}(\Omega)$ is defined, see, e.g., [21], as

$W^{s,p}(\Omega):=\left\\{\,v\in W^{\lfloor
s\rfloor,p}(\Omega):\sum_{|\alpha|=\lfloor
s\rfloor}\iint_{\Omega\times\Omega}\dfrac{|D^{\alpha}v(x)-D^{\alpha}v(y)|^{p}}{|x-y|^{(s-\lfloor
s\rfloor)p+2}}dxdy<+\infty\,\right\\}.$

In case $p=2$ we again use the notation $H^{s}(\Omega)$. Note that in this
case $H^{s}(\Omega)$ can equivalently be obtained via real or complex
interpolation of the integer degree spaces $H^{k}(\Omega)$. This is due to the
fact, that in the Hilbert space setting, all resulting Bessel potential spaces
$H^{s}_{2}(\Omega)$, Besov spaces $B^{s}_{2,2}(\Omega)$ and
Sobolev-(Slobodeckij) spaces $H^{s}(\Omega)$ coincide, see [50, pp. 12,39].
For $X$ being any function space over $\Omega$, we denote by $X^{*}$ its
topological dual space, and abbreviate the duality pairing by
$\left\langle\cdot,\cdot\right\rangle_{\Omega}$. We will also use the notation
$H^{1}_{0}(\Omega)^{*}=H^{-1}(\Omega)$. The structure of the Stokes and
Navier-Stokes equations requires also the definition of some vector valued
spaces, consisting of divergence free vector fields. We denote by
$\nabla\cdot$ the divergence operator and introduce the spaces

$V:=\overline{\left\\{\,v\in C^{\infty}_{0}(\Omega)^{2}:\ \nabla\cdot
v=0\,\right\\}}^{H^{1}(\Omega)}\quad\text{and}\quad
H:=\overline{\left\\{\,v\in C^{\infty}_{0}(\Omega)^{2}:\ \nabla\cdot
v=0\,\right\\}}^{L^{2}(\Omega)}.$

Note that instead of the definition via closures, in the case of $\Omega$
being bounded and Lipschitz, these spaces are alternatively characterized in
the following way, see [48, Chapter 1, Theorems 1.4 & 1.6]:

$V=\left\\{\,v\in H^{1}_{0}(\Omega)^{2}:\ \nabla\cdot
v=0\,\right\\}\quad\text{and}\quad H=\left\\{\,v\in L^{2}(\Omega)^{2}:\
\nabla\cdot v=0,\ u\cdot n=0\text{ on }\partial\Omega\,\right\\},$

where by $u\cdot n$ we denote the normal trace of the vector field $u$. To
improve readability, whenever vectorial spaces like $H^{1}(\Omega)^{2}$ would
arise in the subscript of some norm, we shall drop the outer superscript
$(\cdot)^{2}$. For a Banach space $X$ and $I=(0,T]$ we denote by $L^{p}(I;X)$
the Bochner space of $X$ valued functions, for which the following norm is
finite

$\|v\|_{L^{p}(I;X)}=\left(\int_{I}\|v(t)\|_{X}^{p}\ dt\right)^{1/p},$

with the usual convention when $p=+\infty$. It holds
$L^{p}(I;L^{p}(\Omega))\cong L^{p}(I\times\Omega)$, and for $p=2$, we denote
the inner product by $\left(\cdot,\cdot\right)_{I\times\Omega}$. Whenever $X$
is separable and $1\leq p<\infty$, it holds $(L^{p}(I;X))^{*}\cong
L^{p^{*}}(I;X^{*})$, where $1/p+1/p^{*}=1$. The duality pairing for such
spaces will be denoted by
$\left\langle\cdot,\cdot\right\rangle_{I\times\Omega}$. By $W^{k,p}(I;X)$ and
$H^{k}(I;X)$ for $k\in\mathbb{N}$ we denote the spaces of functions $v$
satisfying $\partial_{t}^{j}v\in L^{p}(I;X)$, $j=0,...,k$.

##  3\. Navier-Stokes equations

We start by recalling some regularity results for the Navier-Stokes equations,
and we are going to prove some additional results, especially adapted to the
situation considered in this paper. Throughout this paper, we shall always
assume the convexity of $\Omega$. We will state explicitly, whenever results
also hold in a more general setting. It is a well known result, that for $f\in
L^{2}(I;V^{*})+L^{1}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in H$, there exists a
unique weak solution, i.e., the following Proposition holds, see [48, Chapter
3, Theorem 3.1 & Remark 3.1]:

######  Proposition 3.1.

Let $f\in L^{2}(I;V^{*})+L^{1}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in H$. Then
there exists a unique weak solution $u\in L^{2}(I;V)\cap
C(\bar{I};L^{2}(\Omega)^{2})$ of (1), satisfying

$\left\langle\partial_{t}u,v\right\rangle_{\Omega}+\nu\left(\nabla u,\nabla
v\right)_{\Omega}+\left((u\cdot\nabla)u,v\right)_{\Omega}=\left(f,v\right)_{\Omega}\qquad\text{for
all }v\in V$ (4)

in the sense of distributions on $I$, and $u(0)=u_{0}$. Moreover, there holds
an estimate

$\|u\|_{L^{\infty}(I;L^{2}(\Omega))}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sqrt{\nu}}\|u\|_{L^{2}(I;V)}\leq
C\left(\|u_{0}\|_{H}+\|f\|_{L^{2}(I;V^{*})+L^{1}(I;L^{2}(\Omega))}\right).$

Note that the constant $C$ above only depends on $\nu$ and $\Omega$, but is
independent of $T$, see [47, Theorems V.1.4.2, V.1.5.3, V.3.1.1]. It is well
known, that under the assumptions of Proposition 3.1, the nonlinearity
satisfies for $1\leq s,q<2$:

$(u\cdot\nabla)u\in
L^{s}(I;L^{q}(\Omega)^{2})\quad\text{whenever}\quad\frac{1}{s}+\frac{1}{q}\geq\frac{3}{2},$
(5)

see [47, Lemma V.1.2.1]. Equation 4 is the weak formulation of (1) in
divergence free spaces. The proof of the above proposition relies heavily on
the fact, that the trilinear form $c(\cdot,\cdot,\cdot)$ defined by

$c:{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{1}_{0}(\Omega)^{2}\times
H^{1}_{0}(\Omega)^{2}\times H^{1}_{0}(\Omega)^{2}}\rightarrow\mathbb{R},\quad
c\left(u,v,w\right)=\left((u\cdot\nabla)v,w\right)_{\Omega},$

posesses the properties summarized in the following lemma.

######  Lemma 3.2.

Let $\Omega\subset\mathbb{R}^{2}$ be an open Lipschitz domain, then there
holds the estimate

$\displaystyle\|v\|_{L^{4}(\Omega)}$ $\displaystyle\leq
C\|v\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|v\|_{H^{1}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }v\in H^{1}(\Omega),$ (6)

due to which, the trilinear form $c\left(\cdot,\cdot,\cdot\right)$ satisfies
for all $u,v,w\in H^{1}_{0}(\Omega)^{2}$:

$\displaystyle c\left(u,v,w\right)$
$\displaystyle\leq\|u\|_{L^{4}(\Omega)}\|\nabla
v\|_{L^{2}(\Omega)}\|w\|_{L^{4}(\Omega)},$ $\displaystyle c\left(u,v,u\right)$
$\displaystyle\leq C\|u\|_{L^{2}(\Omega)}\|\nabla u\|_{L^{2}(\Omega)}\|\nabla
v\|_{L^{2}(\Omega)}.$

Let further $\nabla\cdot u=0$. Then it holds

$c\left(u,v,w\right)=-c\left(u,w,v\right)\quad\text{and}\quad
c\left(u,v,v\right)=0.$

###### Proof.

The estimate for the $L^{4}(\Omega)$ norm can be found in [26, Lemma II.3.2]
and [1, Theorem 3]. The properties of the trilinear forms are then
consequences of Hölder’s inequality and integration by parts, and are shown,
e.g., in [26, Lemma IX.2.1]. ∎

In what follows, we often consider the trilinear form $c$ integrated in time,
which we denote by

$c\left(\\!\left(u,v,w\right)\\!\right):=\int_{I}c\left(u,v,w\right)\ dt.$

In analyzing the Navier-Stokes equations, the instationary Stokes equations
frequently arise as an auxiliary problem. For initial data $u_{0}\in H$ and
right hand side $f\in
L^{1}(I;L^{2}(\Omega)^{2}){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+L^{2}(I;V^{*})}$,
there exists a unique solution $w\in L^{2}(I;V)\cap
L^{\infty}(I;L^{2}(\Omega)^{2})$ to the Stokes equations

$\left\\{\begin{aligned}
\left\langle\partial_{t}w,v\right\rangle_{\Omega}+\nu\left(\nabla w,\nabla
v\right)_{\Omega}&=\left(f,v\right)_{\Omega}\qquad\text{for all }v\in V,\\\
w(0)&=u_{0},\end{aligned}\right.$ (7)

where the first line of (7) is understood in the sense of distributions on
$I$. We introduce the Stokes operator $A\colon D(A)\to H$ defined by

$\left(Aw,v\right)_{\Omega}=\nu\left(\nabla w,\nabla
v\right)_{\Omega}\qquad\text{for all }v\in V,$

with domain $D(A):=\left\\{\,v\in V:\Delta v\in L^{2}(\Omega)^{2}\,\right\\}$
and the projection operator $\mathbb{P}:L^{2}(\Omega)^{2}\to H$, defined by

$\left(\mathbb{P}w,v\right)_{\Omega}=\left(w,v\right)_{\Omega}\qquad\text{for
all }v\in H,$

which is called the Helmholtz or Leray projection. Note that with the vector-
valued Laplacian

$-\Delta\colon D(\Delta)\to L^{2}(\Omega)^{2},$

with domain $D(\Delta):=\left\\{\,v\in H^{1}_{0}(\Omega)^{2}:\Delta v\in
L^{2}(\Omega)^{2}\,\right\\}$, the Stokes operator also satisfies the
representation

$A=-\mathbb{P}\Delta.$

For convex $\Omega$, the domains of the operators introduced above satisfy the
representations

$D(\Delta)=H^{1}_{0}(\Omega)^{2}\cap H^{2}(\Omega)^{2}\qquad\text{and}\qquad
D(A)=V\cap H^{2}(\Omega)^{2},$

see [17] for the $H^{2}$ regularity of the Stokes operator. The Stokes
operator $A$ generates an analytic semigroup in $H$, see [7, 40], also [6] for
a detailled general analysis. One important feature of the Stokes problem is
the maximal parabolic regularity, which indicates, that both the time
derivative $\partial_{t}w$ and the Stokes operator $Aw$ individually inherit
certain regularity properties of $f$, see Proposition 3.4 below. For
homogeneous initial data, this consequence of the analyticity of the semigroup
has been shown in [19], see also [47, Chapter IV, Theorem 1.6.3]. Since our
analysis should also treat inhomogeneous initial data $u_{0}$, we need to
define the proper spaces for the initial data:

$V_{1-1/s}:=\left\\{\,v\in H:\ \|v\|_{V_{1-1/s}}<\infty\,\right\\},$

see also [7] and [6, Chapter 1, Section 3.3], where

$\|v\|_{V_{1-1/s}}:=\left(\int_{I}\|A\exp(-tA)v\|_{H}^{s}\
dt\right)^{1/s}+\|v\|_{H}.$

######  Remark 3.3.

Instead of using the spaces $V_{1-1/s}$ explicitly as a requirement for the
initial data, we can make use of the following imbedding results: For $1<s\leq
2$ it holds $V\hookrightarrow V_{1-1/s}$ and for $1<s<\infty$, it holds $V\cap
D(\Delta)\hookrightarrow V_{1-1/s}$, see [7, Remarks 2.8, 2.9].

The Stokes problem exhibits the following regularity properties, see [7,
Proposition 2.6].

######  Proposition 3.4 (Maximal parabolic regularity).

Let $1<s<\infty$, $f\in L^{s}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V_{1-1/s}$.
Then the solution $w$ of the Stokes equations (7) satisfies

$\|\partial_{t}w\|_{L^{s}(I;L^{2}(\Omega))}+\|Aw\|_{L^{s}(I;L^{2}(\Omega))}\leq
C\left(\|f\|_{L^{s}(I;L^{2}(\Omega))}+\|u_{0}\|_{V_{1-1/s}}\right).$

In the setting of Proposition 3.4, the Stokes problem (7) formulated in
divergence free spaces is equivalent to the following velocity-pressure
formulation: Find $(w,r)\in\left[L^{2}(I;H^{1}_{0}(\Omega)^{2})\cap
C(\bar{I};L^{2}(\Omega)^{2})\right]\times L^{s}(I;L^{2}_{0}(\Omega))$
satisfying

$\left\langle\partial_{t}w,v\right\rangle_{I\times\Omega}+\nu\left(\nabla
w,\nabla v\right)_{I\times\Omega}-\left(r,\nabla\cdot
v\right)_{I\times\Omega}+\left(q,\nabla\cdot
w\right)_{I\times\Omega}=\left(f,v\right)_{I\times\Omega}$ (8)

for all $(v,q)\in\left[L^{2}(I;H^{1}_{0}(\Omega)^{2})\cap
L^{\infty}(I;L^{2}(\Omega)^{2})\right]\times L^{2}(I;L^{2}_{0}(\Omega))$ and
$w(0)=u_{0}$, see [7, Theorem 2.10]. Furthermore, the maximal parabolic
regularity results of Proposition 3.4 imply the following estimate for the
pressure:

$\|r\|_{L^{s}(I;L^{2}(\Omega))}\leq
C\left(\|f\|_{L^{s}(I;L^{2}(\Omega))}+\|u_{0}\|_{V_{1-1/s}}\right).$

Without additional smoothness assumptions, the maximal parabolic regularity
for the Stokes problem does not immediately extend to the Navier-Stokes
equations. A partial result can be obtained by considering that the maximal
parabolic regularity of the Stokes operator was recently extended to the
$L^{p}(\Omega)$ setting, where $p$ in general depends on the smoothness of
$\Omega$. For a general Lipschitz domain $\Omega$, [25, Theorem 1.6] shows
that for some $\varepsilon>0$ and any $p$, such that
$|1/p-1/2|<1/4+\varepsilon$, the maximal parabolic regularity holds. With
this, we obtain the following result for the Navier-Stokes equations.

######  Theorem 3.5.

Let $f\in L^{s}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V_{1-1/s}$ for some $s>1$
and $u\in L^{2}(I;V)\cap L^{\infty}(I;L^{2}(\Omega)^{2})$ the unique solution
of (4). Then for $\gamma:=\min\\{s,4/3\\}$ it holds

$\partial_{t}u\ ,\ Au\in L^{\gamma}(I;L^{4/3}(\Omega)^{2}).$

Further, there exists a unique $p\in L^{\gamma}(I;L^{2}_{0}(\Omega))$, such
that

$\left\langle\partial_{t}u,v\right\rangle_{\Omega}+\nu\left(\nabla u,\nabla
v\right)_{\Omega}+\left((u\cdot\nabla)u,v\right)_{\Omega}-\left(p,\nabla\cdot
v\right)_{\Omega}+\left(\nabla\cdot
u,q\right)_{\Omega}=\left(f,v\right)_{\Omega}$

for all $v\in H^{1}_{0}(\Omega)^{2},q\in L^{2}_{0}(\Omega)$ in the sense of
distributions on $I$, and $u(0)=u_{0}$.

###### Proof.

Due to (5) it holds $(u\cdot\nabla)u\in L^{4/3}(I;L^{4/3}(\Omega)^{2})$. Since
$f\in L^{s}(I;L^{2}(\Omega))$, this implies $\tilde{f}:=f-(u\cdot\nabla)u\in
L^{\gamma}(I;L^{4/3}(\Omega)^{2})$. Since $|3/4-1/2|=1/4$, [25, Theorem 1.6]
shows that the Stokes problem posesses maximal parabolic regularity in
$L^{4/3}(\Omega)$, which applied to $u_{0}$ and $\tilde{f}$ then yields
$\partial_{t}u,Au\in L^{\gamma}(I;L^{4/3}(\Omega)^{2}).$The existence of a
pressure can then be proven as in [7, Theorem 2.10]. ∎

The above result only recovers parts of the regularity available for the
Stokes problem. Using $H^{2}$ regularity, we will now show improved regularity
results for the Navier-Stokes equations. Note that from now on we explicitly
require the convexity of $\Omega$, whereas Propositions 3.1 and 3.4 and
Theorem 3.5 also hold for general Lipschitz domains.

######  Theorem 3.6 ($H^{2}$ regularity).

Let $u_{0}\in V$ and $f\in L^{2}(I;L^{2}(\Omega)^{2})$. Then the weak solution
$u$ to the Navier-Stokes equations (4) satisfies the improved regularity

$u\in L^{2}(I;H^{2}(\Omega)^{2})\cap H^{1}(I;L^{2}(\Omega)^{2})\hookrightarrow
C({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I}};H^{1}(\Omega)^{2}),$

and there exist constants $C_{1},C_{2}>0$, depending on $\nu,\Omega$ but
independent of $T$, such that there hold the bounds

$\displaystyle\|u\|_{L^{\infty}(I;H^{1}(\Omega))}$ $\displaystyle\leq
C_{1}\left(\|u_{0}\|_{V}+\|f\|_{L^{2}(I\times\Omega)}\right)\exp\left(C_{2}\left(\|u_{0}\|_{H}^{4}+\|f\|_{L^{1}(I;L^{2}(\Omega))}^{4}\right)\right),$
$\displaystyle\|u\|_{L^{2}(I;H^{2}(\Omega))}$ $\displaystyle\leq
C_{1}\left(\|u_{0}\|_{V}+\|f\|_{L^{2}(I\times\Omega)}\right)\left(1+\|u_{0}\|_{H}^{2}+\|f\|_{L^{1}(I;L^{2}(\Omega))}^{2}\right)\exp\left(C_{2}\left(\|u_{0}\|_{H}^{4}+\|f\|_{L^{1}(I;L^{2}(\Omega))}^{4}\right)\right).$

###### Proof.

The proof of this result for $C^{2}$ domains can be found in [48, Chapter 3,
Theorem 3.10]. Instead of a $C^{2}$ boundary, we can also use the $H^{2}$
regularity for the Stokes operator on convex, polygonal domains, see, e.g.,
[17, Theorem 5.5] or [33, Theorem 2], to obtain the claimed regularity. The
norm bounds are obtained by the Gronwall lemma. ∎

Note that contrary to the $H^{2}$ regularity for the instationary Stokes
problem, for the Navier-Stokes problem, even the proofs of the regularities
$\partial_{t}u,Au\in L^{2}(I;L^{2}(\Omega)^{2})$, that are contained in the
above result, require convexity of the domain. We shall state a corresponding
estimate for $\|\partial_{t}u\|_{L^{2}(I\times\Omega)}$ after discussing the
regularity of the nonlinearity.

######  Remark 3.7.

By Hölder’s inequality, with this $H^{2}$ regularity result, and the imbedding
$H^{2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$, we immediately obtain

$(u\cdot\nabla)u\in L^{2}(I;L^{2}(\Omega)^{2}).$

The regularity $u\in
C({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I}};H^{1}(\Omega)^{2})$
almost yields that $u$ is uniformly bounded over the whole space time
cylinder. However, since in two dimensions $H^{1}(\Omega)\not\hookrightarrow
L^{\infty}(\Omega)$, the boundedness in space has to be shown via an
additional argument.

We first show, that the nonlinearity actually posesses more regularity, than
what was claimed in Remark 3.7.

######  Theorem 3.8.

Let the assertions of Theorem 3.6 be satisfied, i.e., $u_{0}\in V$ and $f\in
L^{2}(I;L^{2}(\Omega)^{2})$, and let $u$ be the unique solution to the Navier-
Stokes equations (4). Then the nonlinearity satisfies

$(u\cdot\nabla)u\in L^{s}(I;L^{2}(\Omega)^{2})\qquad\text{for any }1\leq
s<\infty,$

and for any $0<\delta\leq\min\\{2,s\\}$ there holds the estimate

$\|(u\cdot\nabla)u\|_{L^{s}(I;L^{2}(\Omega))}\leq
C\|u\|_{L^{2}(I;H^{2}(\Omega))}^{\frac{\delta}{s}}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2-\frac{\delta}{s}},$

where the norms of $u$ on the right hand side can be estimated by Theorem 3.6.

###### Proof.

The proof follows the ideas of [47, Theorem V.1.8.2], however we will use
interpolation spaces instead of fractional powers of the Stokes operator. With
Hölder’s inequality and the Sobolev imbedding
$H^{1+\frac{\delta}{s}}(\Omega)\hookrightarrow C(\overline{\Omega})$, it holds
for any $s<\infty$ and $\delta>0$:

$\|(u\cdot\nabla)u\|_{L^{2}(\Omega)}^{s}\leq\|u\|_{L^{\infty}(\Omega)}^{s}\|\nabla
u\|_{L^{2}(\Omega)}^{s}\leq
C\|u\|_{H^{1+\frac{\delta}{s}}(\Omega)}^{s}\|\nabla u\|_{L^{2}(\Omega)}^{s}.$

Since $\delta\leq s$, we can express the space
$H^{1+\frac{\delta}{s}}(\Omega)$ as interpolation space
$[H^{1}(\Omega),H^{2}(\Omega)]_{\delta/s}$, and obtain from [11, Theorem 1]
the estimate

$\|u\|_{H^{1+\frac{\delta}{s}}(\Omega)}\leq
C\|u\|_{H^{1}(\Omega)}^{1-\frac{\delta}{s}}\|u\|_{H^{2}(\Omega)}^{\frac{\delta}{s}}.$

All in all, we see that

$\|(u\cdot\nabla)u\|_{L^{2}(\Omega)}^{s}\leq
C\|u\|_{H^{2}(\Omega)}^{\delta}\|u\|_{H^{1}(\Omega)}^{2s-\delta},$

which is integrable in time, since $u\in L^{\infty}(I;H^{1}(\Omega)^{2})\cap
L^{2}(I;H^{2}(\Omega)^{2})$, by Theorem 3.6 and $\delta\leq 2$. With Hölder’s
inequality, we obtain the proposed estimate, which concludes the proof. ∎

######  Remark 3.9.

The previous result $(u\cdot\nabla)u\in L^{s}(I;L^{2}(\Omega)^{2})$ shows,
that the Navier-Stokes equations inherit the maximal parabolic regularity of
the Stokes problem, in cases where $f\in L^{s}(I;L^{2}(\Omega)^{2})$ and
$u_{0}\in V_{1-1/s}$ for some $2\leq s<\infty$. Especially it also holds
$Au,\partial_{t}u\in L^{s}(I;L^{2}(\Omega)^{2})$ with

$\|\partial_{t}u\|_{L^{s}(I;L^{2}(\Omega))}+\|Au\|_{L^{s}(I;L^{2}(\Omega))}\leq
C\left(\|u_{0}\|_{V_{1-1/s}}+\|f\|_{L^{s}(I;L^{2}(\Omega))}+\|(u\cdot\nabla)u\|_{L^{s}(I;L^{2}(\Omega))}\right).$

With Theorem 3.8, and slightly higher regularity of the data, we obtain the
boundedness of $u$ in the space-time cylinder:

######  Theorem 3.10.

Let $\varepsilon>0$, $f\in L^{2+\varepsilon}(I;L^{2}(\Omega)^{2})$ and
$u_{0}\in V_{1-1/(2+\varepsilon)}$. Then the unique weak solution $u$ to (1)
satisfies additionally

$u\in C(\bar{I}\times\bar{\Omega})^{2},$

and for $\varepsilon$ sufficiently small, there holds the estimate

$\|u\|_{L^{\infty}(I\times\Omega)}\leq
C\left(\|u_{0}\|_{V_{1-1/(2+\varepsilon)}}+\|f\|_{L^{2+\varepsilon}(I;L^{2}(\Omega))}+\|u\|_{L^{2}(I;H^{2}(\Omega))}^{\varepsilon}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2-\varepsilon}\right),$

where the norms of $u$ on the right hand side can be estimated by Theorem 3.6.

###### Proof.

Applying Theorem 3.8, we observe, that it holds especially $(u\cdot\nabla)u\in
L^{2+\varepsilon}(I;L^{2}(\Omega)^{2})$. For $\varepsilon$ small enough, such
that $\delta:=\varepsilon(2+\varepsilon)\leq 2$, there moreover holds

$\|(u\cdot\nabla)u\|_{L^{2+\varepsilon}(I;L^{2}(\Omega))}\leq
C\|u\|_{L^{2}(I;H^{2}(\Omega))}^{\varepsilon}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2-\varepsilon}.$

By a bootstrapping argument, since $f\in
L^{2+\varepsilon}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{2}(\Omega)^{2}})$
and $u_{0}\in V_{1-1/\varepsilon}$, we can apply the maximal parabolic
regularity of Proposition 3.4, and obtain together with $H^{2}$ regularity

$\partial_{t}u\in
L^{2+\varepsilon}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{2}(\Omega)^{2}}),\qquad
u\in
L^{2+\varepsilon}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{2}(\Omega)^{2}}).$

Hence $u\in W^{1,2+\varepsilon}(I;L^{2}(\Omega)^{2})\cap
L^{2+\varepsilon}(I;H^{2}(\Omega)^{2})$. With the definitions
$p:=2+\varepsilon$, $s:=\frac{1}{2+\varepsilon/2}$,
$\theta:=1-\frac{1}{2+\varepsilon/4}=\frac{1}{2}+\frac{2\varepsilon}{8+\varepsilon}$
it holds $1/p<s<1/2$ and $1/2<\theta<1-s$. We can thus apply [3, Theorem 3],
see also [22, Lemma 2.11 b)], and obtain

$u\in
C^{s-1/p}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I}};(L^{2}(\Omega),H^{2}(\Omega))_{\theta,1}^{2}).$

Applying [8, Theorems 3.4.1 & 6.4.5], and [21, Theorem 4.57], we moreover have

$\displaystyle u\in
C^{s-1/p}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I};(L^{2}(\Omega),H^{2}(\Omega))_{\theta,2}^{2}})=C^{\frac{\varepsilon}{8+6\varepsilon+\varepsilon^{2}}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I}};H^{1+\frac{4\varepsilon}{8+\varepsilon}}(\Omega)^{2})\hookrightarrow
C(\bar{I}\times\bar{\Omega})^{2}.$

Note that here we have used, that for the $L^{2}(\Omega)$ case, all
corresponding Sobolev-Slobodeckij, Besov, Bessel-potential and interpolation
spaces coincide, see, e.g., [41]. For an overview over the topic of function
spaces, see also [50]. From the used embeddings, we moreover have the proposed
estimate. This concludes the proof. ∎

######  Corollary 3.11.

Let the assertions of Theorem 3.10 be satisfied, i.e., $f\in
L^{2+\varepsilon}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V_{1-1/(2+\varepsilon)}$
for some $\varepsilon>0$. Then the nonlinear term in the Navier-Stokes
equations satisfies

$(u\cdot\nabla)u\in L^{\infty}(I;L^{2}(\Omega)^{2}),$

and there holds the estimate

$\|(u\cdot\nabla)u\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\left(\|u_{0}\|_{V_{1-1/(2+\varepsilon)}}+\|f\|_{L^{2+\varepsilon}(I;L^{2}(\Omega))}+\|u\|_{L^{2}(I;H^{2}(\Omega))}^{\frac{2}{2+\varepsilon}}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2-\frac{2}{2+\varepsilon}}\right)\|u\|_{L^{\infty}(I;H^{1}(\Omega))}.$

where the norms of $u$ on the right hand side can be estimated by Theorem 3.6.

###### Proof.

This is a direct consequence of Theorems 3.6 and 3.10 and application of
Hölder’s inequality. ∎

In the formulation of equation 4, we have used divergence free test functions.
Whenever we want to test the equation with functions, that are not divergence
free, we have to consider an alternative, equivalent formulation, that
includes the pressure. The following theorem guarantees, that we can freely
switch between the two formulations, when $u_{0}\in V$ and $f\in
L^{2}(I;L^{2}(\Omega)^{2})$.

######  Theorem 3.12.

Let the assertions of Theorem 3.6 be satisfied, i.e., $u_{0}\in V$ and $f\in
L^{2}(I;L^{2}(\Omega)^{2})$. Then there exists a unique solution $(u,p)$ with

$u\in L^{2}(I;H^{2}(\Omega)^{2})\cap
C({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bar{I}};V){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},\
\partial_{t}u\in L^{2}(I;L^{2}(\Omega)^{2})}\quad\text{and}\quad p\in
L^{2}(I;H^{1}(\Omega)\cap L^{2}_{0}(\Omega)),$

such that $u(0)=u_{0}$ and

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\partial_{t}u,v\right)_{I\times\Omega}}+\nu\left(\nabla
u,\nabla
v\right)_{I\times\Omega}+\left((u\cdot\nabla)u,v\right)_{I\times\Omega}-\left(p,\nabla\cdot
v\right)_{I\times\Omega}+\left(q,\nabla\cdot
u\right)_{I\times\Omega}=\left(f,v\right)_{I\times\Omega},$ (9)

for all $(v,q)\in L^{2}(I;H^{1}_{0}(\Omega)^{2})\times
L^{2}(I;L^{2}_{0}(\Omega))$. Further it holds

$\|p\|_{L^{2}(I;H^{1}(\Omega))}\leq
C\left(\|u_{0}\|_{V}+\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u\|_{L^{2}(I;H^{2}(\Omega))}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}\right),$

where the norms of $u$ on the right hand side can be estimated by Theorem 3.6.

###### Proof.

This result can be shown using Theorem 3.8 with the choice $s=2$, together
with a bootstrapping argument. Moving $(u\cdot\nabla)u\in
L^{2}(I;L^{2}(\Omega)^{2})$ to the right hand side and applying Proposition
3.4 yields $\partial_{t}u,Au\in L^{2}(I;L^{2}(\Omega)^{2})$. The pressure is
then obtained by following the same steps as [7, Theorem 2.10, Corollary
2.11], where its uniqueness is given by its construction. ∎

##  4\. Discretization

After discussing the continuous formulation of the Navier-Stokes equations, we
now turn towards their discretization.

###  4.1. Spatial discretization

Let $\\{\mathcal{T}_{h}\\}$ denote a family of quasi-uniform triangulations of
$\overline{\Omega}$ consisting of closed simplices. The index $h$ denotes the
maximum meshsize. We discretize the velocity $u$ by a discrete function space
$U_{h}\subset H^{1}_{0}(\Omega)^{2}$ and the pressure $p$ by the discrete
space $M_{h}\subset L^{2}_{0}(\Omega)$, where $(U_{h},M_{h})$ satisfy the
discrete, uniform LBB-condition

$\sup_{v_{h}\in U_{h}}\dfrac{(\nabla\cdot v_{h},q_{h})_{\Omega}}{\|\nabla
v_{h}\|_{L^{2}(\Omega)}}\geq\beta\|q_{h}\|_{L^{2}(\Omega)}\qquad\text{for all
}q_{h}\in M_{h},$

with a constant $\beta>0$ independent of $h$. Throughout this work, we will
assume the following approximation properties of the spaces $U_{h}$ and
$M_{h}$. This assumption is valid, e.g., for Taylor-Hood and MINI finite
elements, even on shape regular meshes, see [7, Assumption 7.2].

######  Assumption 4.1.

There exist interpolation operators $i_{h}\colon H^{2}(\Omega)^{2}\cap
H^{1}_{0}(\Omega)^{2}\to U_{h}$ and $r_{h}\colon L^{2}(\Omega)\to M_{h}$, such
that

$\displaystyle\|\nabla(v-i_{h}v)\|_{L^{2}(\Omega)}$ $\displaystyle\leq
ch\|\nabla^{2}v\|_{L^{2}(\Omega)}$ $\displaystyle\text{for all }v\in
H^{2}(\Omega)^{2}\cap H^{1}_{0}(\Omega)^{2},$
$\displaystyle\|q-r_{h}q\|_{L^{2}(\Omega)}$ $\displaystyle\leq ch\|\nabla
q\|_{L^{2}(\Omega)}$ $\displaystyle\text{for all }q\in H^{1}(\Omega).$

The (vector valued) discrete Laplacian $\Delta_{h}:U_{h}\to U_{h}$ is defined
by

$\left(\nabla u_{h},\nabla
v_{h}\right)_{\Omega}=-\left(\Delta_{h}u_{h},v_{h}\right)_{\Omega}\quad\text{for
all }v_{h}\in U_{h}.$

We introduce the space $V_{h}$ of discretely divergence free functions as

$V_{h}:=\left\\{v_{h}\in U_{h}:\ \left(\nabla\cdot
v_{h},q_{h}\right)_{\Omega}=0\ \text{ for all }q_{h}\in M_{h}\right\\}.$

The $L^{2}$ projection onto this space will be denoted by
$\mathbb{P}_{h}\colon{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{2}(\Omega)^{2}}\to
V_{h}$, satisfying

$\left(\mathbb{P}_{h}v,\phi_{h}\right)_{\Omega}=\left(v,\phi_{h}\right)_{\Omega}\quad\text{for
all }\phi_{h}\in V_{h},$

and allows us to introduce the discrete Stokes operator $A_{h}\colon V_{h}\to
V_{h},A_{h}v_{h}=-\mathbb{P}_{h}\Delta_{h}v_{h}$. Having defined these
discrete spaces and operators, we can now consider the Ritz projection for the
stationary Stokes problem. For any $(w,r)\in H^{1}_{0}(\Omega)^{2}\times
L^{2}(\Omega)$, the projection
$\left(R^{S}_{h}(w,r),R^{S,p}_{h}(w,r)\right)\in U_{h}\times M_{h}$ is defined
by

$\displaystyle\left(\nabla\left(w-R^{S}_{h}(w,r)\right),\nabla\phi_{h}\right)_{\Omega}-\left(r-R^{S,p}_{h}(w,r),\nabla\cdot\phi_{h}\right)_{\Omega}$
$\displaystyle=0\quad\text{for all }\phi_{h}\in U_{h}$ (10)
$\displaystyle\left(\nabla\cdot\left(w-R^{S}_{h}(w,r)\right),\psi_{h}\right)_{\Omega}$
$\displaystyle=0\quad\text{for all }\psi_{h}\in M_{h}.$

In case that $w$ is discretely divergence free, i.e., $\left(\nabla\cdot
w,\psi_{h}\right)_{\Omega}=0$ for all $\psi_{h}\in M_{h}$, it holds
$R_{h}^{S}(w,r)\in V_{h}$. Note that the space $V_{h}$ is in general not a
subspace of the space $V$ of pointwise divergence free functions. This means,
that on the discrete space $V_{h}$, the form $c\left(\cdot,\cdot,\cdot\right)$
does not posess the anti symmetry properties shown in Lemma 3.2. Hence we
define, as in [31, 16], an anti symmetric variant:

$\hat{c}:{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{1}_{0}(\Omega)^{2}\times
H^{1}_{0}(\Omega)^{2}\times
H^{1}_{0}(\Omega)^{2}}\to\mathbb{R},\quad\hat{c}\left(u,v,w\right)=\frac{1}{2}c\left(u,v,w\right)-\frac{1}{2}c\left(u,w,v\right).$
(11)

Analogously to the trilinear form
$c\left(\\!\left(\cdot,\cdot,\cdot\right)\\!\right)$, we will use the notation

$\hat{c}\left(\\!\left(u,v,w\right)\\!\right):=\int_{I}\hat{c}\left(u,v,w\right)\
dt.$

######  Remark 4.2.

Note that, due to Lemma 3.2, we can equivalently replace
$c(\cdot,\cdot,\cdot)$ in the continuous formulation of the Navier-Stokes
equations (4) by $\hat{c}(\cdot,\cdot,\cdot)$, as the two forms coincide on
$V$.

By its definition, $\hat{c}\left(\cdot,\cdot,\cdot\right)$ now has the
following antisymmetric properties on the space $V_{h}$, which will later
allow us to show the stability of the fully discrete solutions.

######  Lemma 4.3.

The trilinear form $\hat{c}\left(\cdot,\cdot,\cdot\right)$ satisfies

$\displaystyle\hat{c}\left(u_{h},v_{h},w_{h}\right)$ $\displaystyle\leq
C\|\nabla u_{h}\|_{L^{2}(\Omega)}\|\nabla v_{h}\|_{L^{2}(\Omega)}\|\nabla
w_{h}\|_{L^{2}(\Omega)}$ $\displaystyle\text{for all }u_{h},v_{h},w_{h}\in
U_{h},$ $\displaystyle\hat{c}\left(u_{h},v_{h},w_{h}\right)$
$\displaystyle=-\hat{c}\left(u_{h},w_{h},v_{h}\right)$ $\displaystyle\text{for
all }u_{h},v_{h},w_{h}\in U_{h},$
$\displaystyle\hat{c}\left(u_{h},v_{h},v_{h}\right)$ $\displaystyle=0$
$\displaystyle\text{for all }u_{h},v_{h}\in U_{h}.$

###### Proof.

The last two identities are a direct consequence of the definition of
$\hat{c}\left(\cdot,\cdot,\cdot\right)$. The first estimate follows from Lemma
3.2 and the imbedding $H^{1}_{0}(\Omega)\hookrightarrow L^{4}(\Omega)$. ∎

Note that due to the above lemma, formally we are still allowed to switch the
second and third argument of $\hat{c}\left(\cdot,\cdot,\cdot\right)$. The
original form $c\left(\cdot,\cdot,\cdot\right)$ however had a strict
disctinction between the two arguments, as it contains the gradient of the
second argument, but only the function values of the third argument. This is
of importance when estimating the form in terms of its arguments. In
$\hat{c}\left(\cdot,\cdot,\cdot\right)$ gradients occur in the second and
third argument, thus switching the arguments does not allow us to obtain
improved estimates. For this reason, we state the following lemma, which
allows us to switch the second and third arguments of
$c\left(\cdot,\cdot,\cdot\right)$ by introducing an additional term, even if
the first argument is not (pointwise) divergence free.

######  Lemma 4.4.

Let
$u,v,w\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{1}_{0}(\Omega)^{2}}$.
Then it holds

$c\left(u,v,w\right)=-c\left(u,w,v\right)-\left(\nabla\cdot u,v\cdot
w\right)_{\Omega}.$

###### Proof.

The proof is simply an application of integration by parts, and can also be
seen, e.g., from [2, Equation 2.9]. ∎

We conclude this subsection on the space discretization by recalling some
important interpolation estimates. On the continuous level, applying (6) to
the first order derivatives, and using $H^{2}$ regularity, yields

$\|\nabla w\|_{L^{4}(\Omega)}\leq C\|\nabla
w\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|Aw\|_{L^{2}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }w\in H^{2}(\Omega)^{2}.$ (12)

Using the Stokes operator on the right hand side, instead of second order
derivatives, allows us to translate this result to the discrete setting. This
is facilitated by the following result, showing that for discretely divergence
free functions, the discrete Laplacian $\Delta_{h}$ can be bounded in terms of
the discrete Stokes operator $A_{h}$:

$\|\Delta_{h}w_{h}\|_{L^{2}(\Omega)}\leq
C\|A_{h}w_{h}\|_{L^{2}(\Omega)}\qquad\text{for all }w_{h}\in V_{h},$ (13)

see [30, Corollary 4.4] or [28, Lemma 4.1]. With this, we can translate (12)
to the discrete setting, by considering for some fixed $w_{h}\in V_{h}$ the
solution $w\in H^{1}_{0}(\Omega)^{2}$ to the continuous problem

$\left(\nabla w,\nabla
v\right)_{\Omega}=\left(-\Delta_{h}w_{h},v\right)_{\Omega}\qquad\text{for all
}v\in H^{1}_{0}(\Omega)^{2}.$

By the stability of the Poisson Ritz projection in $W^{1,4}(\Omega)$, (12) and
(13), we then obtain the discrete version of (12).

$\|\nabla w_{h}\|_{L^{4}(\Omega)}\leq C\|\nabla
w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }w_{h}\in V_{h}.$ (14)

Analogously, the Gagliardo-Nirenberg inequality,

$\|w\|_{L^{\infty}(\Omega)}\leq
C\|w\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|Aw\|_{L^{2}(\Omega)}^{\frac{1}{2}},$
(15)

which is a consequence of [1, Theorem 3] together with $H^{2}$ regularity, has
a discrete analogon. It can be shown using the standard discrete Gagliardo-
Nirenberg inequality

$\|w_{h}\|_{L^{\infty}(\Omega)}\leq
C\|w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|\Delta_{h}w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }w_{h}\in U_{h},$

which was proven in [29, Lemma 3.3]. The proof stated there for smooth domains
remains the same for convex domains. The discrete version of (15) is then
again obtained by applying (13) and reads

$\|w_{h}\|_{L^{\infty}(\Omega)}\leq
C\|w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }w_{h}\in V_{h}.$ (16)

Straightforward calculations, using the definition of $A_{h}$, also give

$\|\nabla w_{h}\|_{L^{2}(\Omega)}\leq
C\|w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}w_{h}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\qquad\text{for
all }w_{h}\in V_{h}.$ (17)

With these considerations regarding the spatial discretization, we can now
consider the fully discrete Navier-Stokes equations by also discretizing in
time.

###  4.2. Temporal discretization

For discretization in time, we employ the discontinuous Galerkin method of
order q (dG(q)), which is also used, e.g., in [7, 16, 24]. The time interval
$I=(0,T]$ is partitioned into $M$ half-open sub-intervals
$I_{m}=(t_{m-1},t_{m}]$ with $0=t_{0}<t_{1}<t_{2}<...<t_{M}=T$. We denote each
timestep by $k_{m}=t_{m}-t_{m-1}$ and for fixed $M$ the maximal timestep by
$k:=\max_{1\leq m\leq M}k_{m}$, as well as the minimal one by
$k_{\min}:=\min_{1\leq m\leq M}k_{m}$. If we want to emphasize that $I_{m}$
belongs to a discretization level $k$, we denote it by $I_{m,k}$. We make some
standard assumptions on the properties on the time discretization:

  1. (1)

There are constants $C,\beta>0$ independent of $k$, such that

$k_{\min}\geq Ck^{\beta}.$

  2. (2)

There is a constant $\kappa>0$ independent of $k$, such that for all
$m=1,2,...,M-1$

$\kappa^{-1}\leq\frac{k_{m}}{k_{m-1}}\leq\kappa.$

  3. (3)

It holds $k\leq\frac{T}{4}$.

A dG(q) function with values in a given Banach space $\mathcal{B}$ is then
given as a function in the space

$X^{q}_{k}(\mathcal{B}):=\left\\{\,v\in L^{2}(I;\mathcal{B}):\
v|_{I_{m}}\in\mathcal{P}_{q}(I_{m};\mathcal{B})\quad\text{for all }1\leq m\leq
M\,\right\\},$

where on each $I_{m}$ the space $\mathcal{P}_{q}(I_{m};\mathcal{B})$ is given
as the space of polynomials in time up to degree $q$ with values in
$\mathcal{B}$:

$\mathcal{P}_{q}(I_{m};\mathcal{B})=\left\\{\,v\in L^{2}(I_{m};\mathcal{B}):\
\exists\ v_{0},...,v_{q}\in\mathcal{B}\ \text{s.t.}\
v=\sum_{j=0}^{q}v_{j}t^{j}\,\right\\}.$

Note that no continuity is required at the time nodes $t_{m}$, which is why we
use the following standard notations for one sided limits and jump terms:

$v_{m}^{+}:=\lim_{\varepsilon\to 0^{+}}v(t_{m}+\varepsilon),\qquad
v_{m}^{-}:=\lim_{\varepsilon\to
0^{+}}v(t_{m}-\varepsilon),\qquad[v]_{m}:=v_{m}^{+}-v_{m}^{-}.$

We introduce the compact notations

$V_{kh}:=X^{q}_{k}(V_{h}),\qquad U_{kh}:=X^{q}_{k}(U_{h}),\qquad
M_{kh}:=X^{q}_{k}(M_{h}),\qquad Y_{kh}:=U_{kh}\times M_{kh}.$

Having defined these dG spaces, we introduce the following projection operator
in time: $\pi_{\tau}\colon C(I;L^{2}(\Omega))\to X^{q}_{k}(L^{2}(\Omega))$
defined by

$\left\\{\begin{aligned}
\left(\pi_{\tau}v-v,\varphi\right)_{I_{m}\times\Omega}&=0\qquad\quad\text{for
all }\varphi\in\mathcal{P}_{q-1}(I_{m};L^{2}(\Omega)),\text{ if }q>0,\\\
\pi_{\tau}v(t_{m}^{-})&=v(t_{m}^{-}),\end{aligned}\right.$ (18)

for all $m=1,2,...,M$. In case $q=0$, the projection operator is defined
solely by the second condition.

######  Remark 4.5.

In this paper we will restrict ourselves to the two lowest cases $q=0$ and
$q=1$. Since we work in a setting of low regularity of the right hand side
$f$, the error estimates would not benefit from higher order schemes. Also,
since the Navier-Stokes equations already pose a challenging large system to
solve, higher order schemes in many applications are not feasible from the
standpoint of computational cost.

We can now introduce the time-discretized formulation of the Navier-Stokes
equations. We define the time-discrete bilinear form for the transient Stokes
equations as in [7] by

$\mathfrak{B}(u,v):=\sum_{m=1}^{M}\left(\partial_{t}u,v\right)_{I_{m}\times\Omega}+\nu\left(\nabla
u,\nabla
v\right)_{I\times\Omega}+\sum_{m=2}^{M}\left([u]_{m-1},v^{+}_{m-1}\right)_{\Omega}+\left(u_{0}^{+},v_{0}^{+}\right)_{\Omega}.$

Since we frequently will test some discrete equations with their respective
solutions, let us recall the following lemma.

######  Lemma 4.6.

For any $v_{k}\in X_{k}^{q}(L^{2}(\Omega))$ it holds

$\displaystyle\left(\partial_{t}v_{k},v_{k}\right)_{I_{m}\times\Omega}+\left([v_{k}]_{m-1},v_{kh,m-1}^{+}\right)_{\Omega}$
$\displaystyle=\frac{1}{2}\left(\|v_{k,m}^{-}\|_{L^{2}(\Omega)}^{2}+\|[v_{k}]_{m-1}\|_{L^{2}(\Omega)}^{2}-\|v_{k,m-1}^{-}\|_{L^{2}(\Omega)}^{2}\right),$
$\displaystyle-\left(v_{k},\partial_{t}v_{k}\right)_{I_{m}\times\Omega}-\left(v_{k,m}^{-},[v_{k}]_{m}\right)_{\Omega}$
$\displaystyle=\frac{1}{2}\left(\|v_{k,m-1}^{+}\|_{L^{2}(\Omega)}^{2}+\|[v_{k}]_{m}\|_{L^{2}(\Omega)}^{2}-\|v_{k,m}^{+}\|_{L^{2}(\Omega)}^{2}\right).$

###### Proof.

For the first equality, we can express the integral over the time derivatives
via

$\left(\partial_{t}v_{k},v_{k}\right)_{I_{m}\times\Omega}=\frac{1}{2}\|v_{k,m}^{-}\|^{2}-\frac{1}{2}\|v_{k,m-1}^{+}\|^{2}.$

Writing $v_{k,m-1}^{+}=[v]_{k,m-1}+v_{k,m-1}^{-}$ and recombining terms gives
the first identity. The proof of the second equality works completely
anologous. ∎

The fully discrete formulation of the transient Navier-Stokes equations, using
the anti symmetrized trilinear form $\hat{c}\left(\cdot,\cdot,\cdot\right)$,
introduced in (11), is now given as: Find $u_{kh}\in V_{kh}$, such that

$\mathfrak{B}(u_{kh},v_{kh})+\hat{c}\left(\\!\left(u_{kh},u_{kh},v_{kh}\right)\\!\right)=\left(f,v_{kh}\right)_{I\times\Omega}+\left(u_{0},v_{kh,0}^{+}\right)_{\Omega}\quad\text{for
all }v_{kh}\in V_{kh}.$ (19)

Since in general $V_{h}\not\subset V$, the discrete solution $u_{kh}\in
V_{kh}$ is not divergence free, and thus, we are not allowed to use it as a
test function for the divergence-free continuous formulation (4). Because of
this, we introduce an equivalent formulation with pressure: Find
$(u_{kh},p_{kh})\in Y_{kh}$, such that for all $(v_{kh},q_{kh})\in Y_{kh}$ it
holds

$B((u_{kh},p_{kh}),(v_{kh},q_{kh}))+\hat{c}\left(\\!\left(u_{kh},u_{kh},v_{kh}\right)\\!\right)=\left(f,v_{kh}\right)_{I\times\Omega}+\left(u_{0},v_{kh,0}^{+}\right)_{\Omega},$
(20)

where the mixed bilinear form $B$ is defined by

$\displaystyle B((u,p),(v,q)):=$
$\displaystyle\sum_{m=1}^{M}\left(\partial_{t}u,v\right)_{I_{m}\times\Omega}+\nu\left(\nabla
u,\nabla
v\right)_{I\times\Omega}+\sum_{m=2}^{M}\left([u]_{m-1},v^{+}_{m-1}\right)_{\Omega}+\left(u_{0}^{+},v_{0}^{+}\right)_{\Omega}$
$\displaystyle-\left(\nabla\cdot v,p\right)_{I\times\Omega}+\left(\nabla\cdot
u,q\right)_{I\times\Omega}.$

######  Theorem 4.7.

The two formulations (19) and (20) are equivalent in the sense that, if
$u_{kh}\in V_{kh}$ satisfies (19), then there exists $p_{kh}\in M_{kh}$ such
that $(u_{kh},p_{kh})$ solves (20). Conversely, if $(u_{kh},p_{kh})$ satisfies
(20), then $u_{kh}$ is an element of $V_{kh}$ and satisfies (19).

###### Proof.

This can be shown by using the same arguments as [7, Proposition 4.3]. ∎

###  4.3. Stokes error estimates

Before further analyzing the fully discrete Navier-Stokes equations, let us
recall the discrete formulation of the Stokes equations (8), and the available
error estimates that we wish to extend to the Navier-Stokes equations in this
work. The fully discrete formulation of the instationary Stokes equation
reads: Find $(w_{kh},r_{kh})\in Y_{kh}$ satisfying

$B((w_{kh},r_{kh}),(\phi_{kh},\psi_{kh}))=\left(u_{0},\phi_{kh,0}^{+}\right)_{\Omega}+\left(f,\phi_{kh}\right)_{I\times\Omega}\qquad\text{for
all }(\phi_{kh},\psi_{kh})\in Y_{kh}.$ (21)

In the recent contributions [7, 36], best approximation type error estimates
for the Stokes problem were shown in the norms of
$L^{\infty}(I;L^{2}(\Omega))$, $L^{2}(I;L^{2}(\Omega))$ and
$L^{2}(I;H^{1}(\Omega))$. There hold the following results, formulated in
terms of best approximation error terms and the errors of the projection in
time $\pi_{\tau}$, defined in (18), and the Stokes Ritz projection
$R_{h}^{S}$, defined in (10).

######  Proposition 4.8 ([36, Theorems 6.1& 6.3]).

Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$, let $(w,r)$ and
$(w_{kh},r_{kh})$ be the continuous and fully discrete solutions to the Stokes
problems (8) and (21). Then for any $\chi_{kh}\in V_{kh}$, there holds

$\displaystyle\|w-w_{kh}\|_{L^{2}(I\times\Omega))}$ $\displaystyle\leq
C\left(\|w-\chi_{kh}\|_{L^{2}(I\times\Omega)}+\|w-\pi_{\tau}w\|_{L^{2}(I\times\Omega)}+\|w-R_{h}^{S}(w,r)\|_{L^{2}(I\times\Omega)}\right)$
$\displaystyle\|\nabla(w-w_{kh})\|_{L^{2}(I\times\Omega))}$ $\displaystyle\leq
C\left(\|\nabla(w-\chi_{kh})\|_{L^{2}(I\times\Omega)}+\|\nabla(w-\pi_{\tau}w)\|_{L^{2}(I\times\Omega)}+\|\nabla(w-R_{h}^{S}(w,r))\|_{L^{2}(I\times\Omega)}\right).$

######  Proposition 4.9 ([7, Corollary 6.4]).

Let $f\in L^{s}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V_{1-1/s}$ for some $s>1$,
and let $(w,r)$ and $(w_{kh},r_{kh})$ be the continuous and fully discrete
solutions to the Stokes equations (8) and (21). Then for any $\chi_{kh}\in
V_{kh}$, there holds

$\|w-w_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\left(\ln\frac{T}{k}\right)\left(\|w-\chi_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|w-R_{h}^{S}(w,r)\|_{L^{\infty}(I;L^{2}(\Omega))}\right).$

Having shown these error estimates for the Stokes equations, the natural
question arises, whether these results can be extended to the Navier-Stokes
equations. We first give a positive answer for the $L^{2}(I;H^{1}(\Omega))$
error in Theorem 4.16. The main result of this work is the error estimate in
the $L^{\infty}(I;L^{2}(\Omega))$ norm, presented in Theorem 5.6. With the
same techniques, the proof of the $L^{2}(I;L^{2}(\Omega))$ error estimate is
then straightforward, and we state the result in Theorem 5.7.

###  4.4. Existence and stability results

We begin our analysis of the fully discrete Navier-Stokes equations by
presenting some stability results, followed by an existence result of fully
discrete solutions. To show these, let us first recall the following version
of a discrete Gronwall lemma, which is stated in [32, Lemma 5.1] for a
constant timestep $k$, but its proof can easily be adapted to the setting of
variable timesteps $k_{m}$.

######  Lemma 4.10.

Let $\\{k_{n}\\},\\{a_{n}\\},\\{b_{n}\\},\\{c_{n}\\},\\{\gamma_{n}\\}$ be
sequences of nonnegative numbers and $B\geq 0$ a constant, such that for each
$n\in\mathbb{N}_{0}$ it holds

$a_{n}+\sum_{m=0}^{n}k_{m}b_{m}\leq\sum_{m=0}^{n}k_{m}\gamma_{m}a_{m}+\sum_{m=0}^{n}k_{m}c_{m}+B,$

and $k_{m}\gamma_{m}<1$ for all $m\in\\{0,...,n\\}$. Then with
$\sigma_{m}:=(1-k_{m}\gamma_{m})^{-1}$ it holds

$a_{n}+\sum_{m=0}^{n}k_{m}b_{m}\leq\exp\left(\sum_{m=0}^{n}k_{m}\sigma_{m}\gamma_{m}\right)\cdot\left(\sum_{m=0}^{n}k_{m}c_{m}+B\right).$

###### Proof.

By choosing $\widetilde{k}=1$, $\widetilde{\gamma}_{m}=k_{m}\gamma_{m}$ and
$\widetilde{c}_{m}=k_{m}c_{m}$, the quantities with $\widetilde{\cdot}$
correspond to the notation of [32, Lemma 5.1]. The result then directly
follows from this redefinition. ∎

Continuous and discrete Gronwall lemmas are stated in many different forms in
the literature, see, e.g., [38] and the references therein for an overview
over different generalizations of the original lemma. Note that, since the sum
on the right hand side of the assumed bound goes up to $n$, the additional
assumption $k_{m}\gamma_{m}<1$ is needed. This is not the case in explicit
forms of discrete Gronwall lemmas, where the sum on the right goes only up to
$n-1$, which is the form most often considered in the literature, e.g., [43].
In the context of dG timestepping schemes, the sequences $\\{a_{n}\\}$ and
$\\{b_{n}\\}$ often correspond to squared norms at time nodes or over
subintervals. It therefore seems natural to also state the following version
of a Gronwall lemma, where we also include a sum over non-squared
contributions. To the best of the authors knowledge, a result like this has
not been explicitly used in the literature. In later sections of this work,
this lemma will facilitate the analysis of discrete problems with right hand
sides that are $L^{1}$ in time. Here the norms of the solution occur in a non
squared contribution. The following lemma shows, that the weights of the
squared sum enter exponentially into the estimate, whereas the weights of the
linear sum enter linearly in the estimate for $x_{n}$. It can be understood as
an adaptation of Bihari’s inequality to the discrete setting, see [10, 39]. To
improve readability, we drop the explicit mentioning of the timesteps $k_{m}$,
as we can always apply a transformation as done in the proof of Lemma 4.10.

######  Lemma 4.11.

Let $\\{x_{n}\\},\\{b_{n}\\},\\{c_{n}\\},\\{d_{n}\\},\\{\gamma_{n}\\}$ be
sequences of nonnegative numbers and $B\geq 0$ a constant, such that for each
$n\in\mathbb{N}_{0}$ it holds

$x_{n}^{2}+\sum_{m=0}^{n}b_{m}\leq\sum_{m=0}^{n}\gamma_{m}x_{m}^{2}+\sum_{m=0}^{n}d_{m}x_{m}+\sum_{m=0}^{n}c_{m}+B,$

and $\gamma_{m}<1$ for all $m\in\\{0,...,n\\}$. Then with
$\sigma_{m}:=(1-\gamma_{m})^{-1}$ it holds

$x_{n}^{2}+\sum_{m=0}^{n}b_{m}\leq
2\exp\left(2\sum_{m=0}^{n}\sigma_{m}\gamma_{m}\right)\cdot\left[\left(\sum_{m=0}^{n}d_{m}\right)^{2}+\sum_{m=0}^{n}c_{m}+B\right].$

###### Proof.

For $n\in\mathbb{N}_{0}$ define $\delta_{n}\geq 0$ such that

$x_{n}^{2}+\sum_{m=0}^{n}b_{m}+\delta_{n}=\sum_{m=0}^{n}\gamma_{m}x_{m}^{2}+\sum_{m=0}^{n}d_{m}x_{m}+\sum_{m=0}^{n}c_{m}+B,$
(22)

and set $X_{n}:=\sqrt{x_{n}^{2}+\sum_{m=0}^{n}b_{m}+\delta_{n}}$. We will show

$X_{n}^{2}\leq
2\exp\left(2\sum_{m=0}^{n}\sigma_{m}\gamma_{m}\right)\cdot\left[\left(\sum_{m=0}^{n}d_{m}\right)^{2}+\sum_{m=0}^{n}c_{m}+B\right]\qquad\text{for
all }n\in\mathbb{N}_{0},$

from which the assertion follows by the definition of $X_{n}$. Note that by
assumption it holds

$X_{n}^{2}-X_{n-1}^{2}=\gamma_{n}x_{n}^{2}+d_{n}x_{n}+c_{n}\geq
0\qquad\text{for all }n\in\mathbb{N},$

and thus the sequences $\\{X_{n}^{2}\\}$ and $\\{X_{n}\\}$ are monotonically
increasing. By $X_{n}\geq x_{n}$ and the monotonicity of $X_{n}$, from (22) we
obtain

$\displaystyle X_{n}^{2}$
$\displaystyle\leq\sum_{m=0}^{n}\gamma_{m}X_{m}^{2}+\sum_{m=0}^{n}d_{m}X_{m}+\sum_{m=0}^{n}c_{m}+B$
$\displaystyle\leq\left(\sum_{m=0}^{n}\gamma_{m}X_{m}+\sum_{m=0}^{n}d_{m}\right)X_{n}+\sum_{m=0}^{n}c_{m}+B.$
(23)

Let us now recall, that for $a,b>0$, an estimate $x^{2}\leq ax+b$ implies
$x\leq a+\sqrt{b}$. To see this, we start by computing the roots of the
quadratic polynomial, yielding $x\leq\frac{a}{2}+\sqrt{\frac{a^{2}}{4}+b}$.
The root of the polynomial can then be estimated by
$\frac{a}{2}+\sqrt{\frac{a^{2}}{4}+b}\leq a+\sqrt{b},$ which can be shown by
subtracting $a/2$ and squaring both sides. Applied to equation 23, we have
thus for every $n\in\mathbb{N}_{0}$ the estimate

$X_{n}\leq\sum_{m=0}^{n}\gamma_{m}X_{m}+\sum_{m=0}^{n}d_{m}+\sqrt{\sum_{m=0}^{n}c_{m}+B}=\sum_{m=0}^{n}\gamma_{m}X_{m}+\sum_{m=0}^{n}\tilde{c}_{m}+\tilde{B},$

where we have defined $\tilde{B}:=0$ as well as

$\tilde{c}_{0}:=\left(d_{0}+\sqrt{c_{0}+B}\right){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\geq
0},\quad\text{and}\quad\tilde{c}_{m}:=\left(d_{m}+\sqrt{\sum_{j=0}^{m}c_{j}+B}-\sqrt{\sum_{j=0}^{m-1}c_{j}+B}\right){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\geq
0}\text{ for }m>0.$

We can thus apply Lemma 4.10 to $X_{n}$ and obtain after resubstituting the
tilded quantities:

$X_{n}\leq\exp\left(\sum_{m=0}^{n}\gamma_{m}\sigma_{m}\right)\cdot\left(\sum_{m=0}^{n}\tilde{c}_{m}+\tilde{B}\right)\leq\exp\left(\sum_{m=0}^{n}\gamma_{m}\sigma_{m}\right)\cdot\left(\sum_{m=0}^{n}d_{m}+\sqrt{\sum_{m=0}^{n}c_{m}+B}\right).$

Squaring and estimating the square of the sum yields the result. ∎

With these technical lemmas, we can show stability of the discrete solutions
in different norms under different assumptions on $f$. Solutions to the
discrete problem also satisfy the same energy bounds as the weak solutions,
i.e., there holds the following proposition, see [16, Lemma 5.1, Theorem A.1].

######  Proposition 4.12 (Stability of discrete Navier-Stokes).

Let $f\in
L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{-1}(\Omega)^{2}})$,
$u_{0}\in H$ and $u_{kh}\in V_{kh}$ satisfy (19). Then there hold the bounds

$\displaystyle\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|u_{kh}\|_{L^{2}(I;H^{1}(\Omega))}\leq
C\left(\|u_{0}\|_{H}+\|f\|_{L^{2}(I;H^{-1}(\Omega))}\right),\quad\text{ if
}q=0,$
$\displaystyle\left(\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\right)^{\frac{1}{2}}+\|u_{kh}\|_{L^{2}(I;H^{1}(\Omega))}\leq
C\left(\|u_{0}\|_{H}+\|f\|_{L^{2}(I;H^{-1}(\Omega))}\right),\quad\text{ if
}q\geq 1,$

with a constant $C$ depending on the domain $\Omega$ and the viscosity $\nu$.

Note that for the case $q\geq 1$, contrary to the continuous setting of
Proposition 3.1, due to the exponent $\frac{1}{2}$ on the left hand side, the
above proposition states a bound for
$\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}$, which depends on the squared norms
of the data $u_{0}$ and $f$. For the two low order cases $q=0$ and $q=1$, and
$f\in L^{2}(I;L^{2}(\Omega)^{2})$, [16, Lemma 5.1] also shows an estimate of
the form

$\max_{1\leq m\leq
M}\|u_{kh,m}^{-}\|_{L^{2}(\Omega)}+\|u_{kh}\|_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{2}(I;H^{1}(\Omega))}}\leq
C_{1}e^{C_{2}T}\left(\|u_{0}\|_{H}+\|f\|_{L^{2}(I;L^{2}(\Omega))}\right).$

With our discrete Gronwall estimate Lemma 4.11, we can generalize this result
to $f\in L^{1}(I;L^{2}(\Omega)^{2})$. Furthermore, by using the version of
Gronwall’s lemma presented here, contrary to [16], the bound does not grow
exponentially in $T$. This is in agreement with the result of Proposition 3.1
in the continuous setting. We obtain the following result, which now yields an
estimate for the norms of $u_{kh}$, that is linear in terms of the data.

######  Theorem 4.13.

Let $f\in
L^{1}(I;L^{2}(\Omega)^{2})+L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{-1}(\Omega)^{2}})$,
$u_{0}\in H$ and $u_{kh}\in V_{kh}$ satisfy equation 19 for either $q=0$ or
$q=1$. Then there holds the bound

$\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sqrt{\nu}}\|u_{kh}\|_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{2}(I;H^{1}(\Omega))}}\leq
C\left(\|u_{0}\|_{H}+\|f\|_{L^{1}(I;L^{2}(\Omega))+L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{-1}(\Omega)})}\right),$

with a constant $C$ only depending on $\Omega,\nu$. If $f\in
L^{1}(I;L^{2}(\Omega)^{2})$, then $C$ only depends on $\Omega$.

###### Proof.

We first prove the result for $f\in L^{1}(I;L^{2}(\Omega)^{2})$ and remark at
the end, which modifications are needed to also cover the case, where $f$ is
partly in $L^{2}(I;H^{-1}(\Omega)^{2})$. Note that in the special case of
$f\in L^{2}(I;H^{-1}(\Omega)^{2})$ and $q=0$, this theorem is precisely the
first statement of [16, Lemma 5.1]. For notational simplicity, we use the
convention $u_{kh,0}^{-}:=u_{0}$ and accordingly
$[u_{kh}]_{0}=u_{kh,0}^{+}-u_{0}$. By testing (19) with $u_{kh}|_{I_{m}}$ and
using Lemma 4.3 we arrive on each time interval at:

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\partial_{t}u_{kh},u_{kh}\right)_{I_{m}\times\Omega}}+\nu\left(\nabla
u_{kh},\nabla
u_{kh}\right)_{I_{m}\times\Omega}+\left([u_{kh}]_{m-1},u_{kh,m-1}^{+}\right)_{\Omega}=\left(f,u_{kh}\right)_{I_{m}\times\Omega}.$

Applying Lemma 4.6 gives

$\frac{1}{2}\|u_{kh,m}^{-}\|_{L^{2}(\Omega)}^{2}+\frac{1}{2}\|[u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
u_{kh}\|_{L^{2}(I_{m}\times\Omega)}^{2}=\frac{1}{2}\|u_{kh,m-1}^{-}\|_{L^{2}(\Omega)}^{2}+\left(f,u_{kh}\right)_{I_{m}\times\Omega}.$

Multiplying by two and summing up the identity over the intervals $1,...,n$,
we obtain

$\|u_{kh,n}^{-}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\|[u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}+2\nu\|\nabla
u_{kh}\|^{2}_{L^{2}(I_{m}\times\Omega)}=\|u_{kh,0}^{-}\|_{L^{2}(\Omega)}^{2}+2\sum_{m=1}^{n}\left(f,u_{kh}\right)_{I_{m}\times\Omega}.$
(24)

From this point on, we have to treat the two cases $q=0$ and $q=1$ separately.
Case 1: If $q=0$, then it holds
$\|u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}=\|u_{kh,m}^{-}\|_{L^{2}(\Omega)}$.
Thus with Hölder’s inequality, the terms in the last sum of (24) can be
estimated as

$\left(f,u_{kh}\right)_{I_{m}\times\Omega}\leq\|f\|_{L^{1}(I_{m};L^{2}(\Omega))}\|u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}\leq\|f\|_{L^{1}(I_{m};L^{2}(\Omega))}\|u_{kh,m}^{-}\|_{L^{2}(\Omega)}.$

Hence, from (24) we obtain the following

$\|u_{kh,n}^{-}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\|[u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}+2\nu\|\nabla
u_{kh}\|^{2}_{L^{2}(I_{m}\times\Omega)}\leq\|u_{0}\|_{L^{2}(\Omega)}^{2}+2\sum_{m=1}^{n}\|f\|_{L^{1}(I_{m};L^{2}(\Omega))}\|u_{kh,m}^{-}\|_{L^{2}(\Omega)}.$

An application of Lemma 4.11 proves the assertion.
Case 2: If $q=1$, then the $L^{\infty}(I_{m})$ norm can be estimated by the
evaluation at the two endpoints of the interval:

$\|u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}\leq\max\\{\|u_{kh,m}^{-}\|_{L^{2}(\Omega)},\|u_{kh,m-1}^{+}\|_{L^{2}(\Omega)}\\}.$

With triangle inequality we can estimate the right sided limit in terms of a
left sided limit and a jump:

$\|u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}\leq\max\\{\|u_{kh,m}^{-}\|_{L^{2}(\Omega)},\|u_{kh,m-1}^{-}\|_{L^{2}(\Omega)}+\|[u_{kh}]_{m-1}\|_{L^{2}(\Omega)}\\}.$

Defining
$x_{n}^{2}:=\|u_{kh,n}^{-}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\|[u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}$
for $n=1,...,M$ yields from (24) the estimate

$x_{n}^{2}+2\nu\sum_{m=1}^{n}\|\nabla
u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\leq\|u_{0}\|_{L^{2}(\Omega)}^{2}+2\sum_{m=1}^{n}\|f\|_{L^{1}(I_{m};L^{2}(\Omega))}(x_{m}+x_{m-1}),$

or after an index shift

$x_{n}^{2}+2\nu\sum_{m=1}^{n}\|\nabla
u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\leq\|u_{0}\|_{L^{2}(\Omega)}^{2}+2\|f\|_{L^{1}(I_{1};L^{2}(\Omega))}\|u_{0}\|+2\sum_{m=1}^{n}\|f\|_{L^{1}(I_{m}\cup
I_{m+1};L^{2}(\Omega))}\cdot x_{m}.$

In order to shorten the notation, we have added here a term
$\|f\|_{L^{1}(I_{n+1};L^{2}(\Omega))}x_{n}$, where in case $n=M$ we use the
convention $I_{M+1}=\emptyset$. Again we can apply Lemma 4.11, which yields
the estimate

$x_{n}^{2}+2\nu\sum_{m=1}^{n}\|\nabla
u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\leq
2\left[\left(2\sum_{m=1}^{n}\|f\|_{L^{1}(I_{m}\cup
I_{m+1};L^{2}(\Omega))}\right)^{2}+\|u_{0}\|_{L^{2}(\Omega)}^{2}+2\|f\|_{L^{1}(I_{1};L^{2}(\Omega))}\|u_{0}\|\right].$

Using Young’s inequality, to estimate the term
$2\|f\|_{L^{1}(I_{1};L^{2}(\Omega))}\|u_{0}\|$, concludes the proof. In case
$f=f_{1}+f_{2}$, with
$f_{1}\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}(I;L^{2}(\Omega)^{2})}$,
$f_{2}\in
L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{-1}(\Omega)^{2}})$,
the contribution of $f_{1}$ can be treated as above. For the $f_{2}$
contribution, in (24) one applies Hölder’s and Young’s inequalities, and
absorbs the $\|\nabla u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}$ term to the left.
Note that in both cases $q=0,1$, in the estimates before the application of
Lemma 4.11, no $x_{m}^{2}$ terms are summed on the right hand side, thus no
exponential dependency is introduced in the final estimate. ∎

The techniques presented above now allow us to show the existence of solutions
to the discrete equations (19).

######  Theorem 4.14.

Let $u_{0}\in H$ and $f\in
L^{1}(I;L^{2}(\Omega)^{2})+L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{-1}(\Omega)^{2}})$.
Let further either $q=0$ or $q=1$. Then there exists a solution $u_{kh}\in
V_{kh}$ of (19).

###### Proof.

The existence can be shown by applying a standard fixpoint argument, using the
stability result of Theorem 4.13. ∎

######  Remark 4.15.

Note that uniqueness of solutions $u_{kh}$ to (19) is not known at this point.
We shall show this later in Theorem 4.19, as we first need to obtain a first
convergence result, that allows us to apply Lemma 4.11.

###  4.5. $L^{2}(I;H^{1}(\Omega))$ error estimates

With the stability results presented thus far, we can show a first error
estimate in the norm of $L^{2}(I;H^{1}(\Omega))$. For this norm, [16, Theorem
5.2] shows a result, estimating the error for the Navier-Stokes equations in
terms of the error of a Stokes problem. The discrete Stokes problem there is
defined with right hand side $\partial_{t}u-\nu\Delta u$, which corresponds to
$f-\nabla p-(u\cdot\nabla)u$, i.e., the pressure is included on the right.
This means, that when applying the corresponding orthogonality relations, a
pressure term remains. In the following result, we use a different right hand
side for the discrete Stokes problem, yielding the following error estimate
for the Navier-Stokes equations in $L^{2}(I;H^{1}(\Omega))$.

######  Theorem 4.16.

Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$, $u_{0}\in V$ and let $(u,p)$,
$(u_{kh},p_{kh})$ solve the continuous and fully discrete Navier-Stokes
equations (9) and (20) respectively. Further let $k$ be small enough. Then for
any $\chi_{kh}\in V_{kh}$, there holds

$\displaystyle\|\nabla(u-u_{kh})\|_{L^{2}(I\times\Omega)}$ $\displaystyle\leq
C\left(\|\nabla(u-\chi_{kh})\|_{L^{2}(I\times\Omega)}+\|\nabla(u-\pi_{\tau}u)\|_{L^{2}(I\times\Omega)}+\|\nabla(u-R_{h}^{S}(u,p))\|_{L^{2}(I\times\Omega)}\right).$

###### Proof.

The proof uses the same arguments as the one of [16, Theorem 5.2]. We repeat
the main steps, in order to motivate, why due to the choice of our
instationary Stokes projection, no error term for the pressures arises on the
right hand side. We denote by $e:=u-u_{kh}$ the error, which we want to
estimate, and introduce $(\tilde{u}_{kh},\tilde{p}_{kh})\in Y_{kh}$ as the
instationary Stokes-projection of $(u,p)$, i.e., the solution to

$B((\tilde{u}_{kh},\tilde{p}_{kh}),(\phi_{kh},\psi_{kh}))=\left(f,v_{kh}\right)_{I\times\Omega}-c\left(\\!\left(u,u,v_{kh}\right)\\!\right)+\left(u_{0},v_{kh,0}^{+}\right)_{\Omega}\quad\text{for
all }(\phi_{kh},\psi_{kh})\in Y_{kh}.$ (25)

Note that for the definition of this equation, we can equivalently choose $c$
or $\hat{c}$, as we have $u$ with $\nabla\cdot u=0$ in its first argument.
There exists a solution $(\tilde{u}_{kh},\tilde{p}_{kh})$ to this discrete
problem, since the assumptions on $f$ and $u_{0}$ yield $(u\cdot\nabla)u\in
L^{2}(I;L^{2}(\Omega)^{2})$, see Theorem 3.8. Furthermore, due to [16, Theorem
4.10], the solution $\tilde{u}_{kh}$ to the above discrete Stokes problem
satisfies a bound

$\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}\leq C,$ (26)

with $C$ depending on $f,u_{0}$, but independent of $k,h$. We split the
velocity error into two components $e=\xi+\eta_{kh}$ where
$\xi:=u-\tilde{u}_{kh}$ and $\eta_{kh}:=\tilde{u}_{kh}-u_{kh}$. The error in
the pressure will be denoted by $r:=p-p_{kh}$, and we will split it into the
same contributions, $r=\omega+\kappa_{kh}$, with $\omega:=p-\tilde{p}_{kh}$
and $\kappa_{kh}:=\tilde{p}_{kh}-p_{kh}$. The error of the Stokes projection
$\xi$ is then bounded by Proposition 4.8 since by Theorem 3.8 it holds
$f-(u\cdot\nabla)u\in L^{2}(I;L^{2}(\Omega)^{2})$. It remains to estimate
$\eta_{kh}$.  By definition, we have the following Galerkin orthogonalities:
for all $(\phi_{kh},\psi_{kh})\in Y_{kh}$ it holds

$\displaystyle
B((u-u_{kh},p-p_{kh}),(\phi_{kh},\psi_{kh}))+\hat{c}\left(\\!\left(u,u,\phi_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(u_{kh},u_{kh},\phi_{kh}\right)\\!\right)$
$\displaystyle=0,$ $\displaystyle B((\xi,\omega),(\phi_{kh},\psi_{kh}))$
$\displaystyle=0.$

Subtracting the two and choosing
$(\phi_{kh},\psi_{kh})=(\eta_{kh},\kappa_{kh})$ yields

$B((\eta_{kh},\kappa_{kh}),(\eta_{kh},\kappa_{kh}))+\hat{c}\left(\\!\left(u,u,\eta_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(u_{kh},u_{kh},\eta_{kh}\right)\\!\right)=0.$
(27)

The two trilinear forms in (27) satisfy the relation

$\hat{c}\left(\\!\left(u,u,\eta_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(u_{kh},u_{kh},\eta_{kh}\right)\\!\right)=\hat{c}\left(\\!\left(\xi,u,\eta_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\tilde{u}_{kh},\xi,\eta_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\eta_{kh},\tilde{u}_{kh},\eta_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(u_{kh},\eta_{kh},\eta_{kh}\right)\\!\right),$

Due to Lemma 4.3, the last term on the right hand side vanishes, and for the
remaining ones, we obtain by Hölder’s and Young’s inequalities

$\displaystyle\hat{c}\left(\\!\left(\xi,u,\eta_{kh}\right)\\!\right)$
$\displaystyle\leq\sum_{m=1}^{M}\|\xi\|_{L^{2}(I_{m};H^{1}(\Omega))}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega))}$
$\displaystyle\leq\sum_{m=1}^{M}C\|\xi\|_{L^{2}(I_{m};H^{1}(\Omega))}^{2}\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2}+\frac{\nu}{4}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega))}^{2},$
$\displaystyle\hat{c}\left(\\!\left(\xi,\tilde{u}_{kh},\eta_{kh}\right)\\!\right)$
$\displaystyle\leq\sum_{m=1}^{M}\|\xi\|_{L^{2}(I_{m};H^{1}(\Omega))}\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega))}$
$\displaystyle\leq\sum_{m=1}^{M}C\|\xi\|_{L^{2}(I_{m};H^{1}(\Omega))}^{2}\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}^{2}+\frac{\nu}{4}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega))},$
$\displaystyle\hat{c}\left(\\!\left(\eta_{kh},\tilde{u}_{kh},\eta_{kh}\right)\\!\right)$
$\displaystyle\leq\sum_{m=1}^{M}\int_{I_{m}}\|\eta_{kh}\|_{H^{1}(\Omega)}^{\frac{3}{2}}\|\eta_{kh}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|\tilde{u}_{kh}\|_{H^{1}(\Omega)}\
dt$
$\displaystyle\leq\sum_{m=1}^{M}C\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}^{4}k_{m}\|\eta_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}+\frac{\nu}{4}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega)}^{2}.$

Due to $u_{kh}$ and $\tilde{u}_{kh}$ both being discretely divergence free,
all pressure contributions in (27) vanish, and with the above estimates, after
absorbing all $\frac{\nu}{4}\|\eta_{kh}\|_{L^{2}(I_{m};H^{1}(\Omega))}$ terms
to the left, we obtain

$\displaystyle\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\partial_{t}\eta_{kh},\eta_{kh}\right)_{I_{m}\times\Omega}}+\frac{\nu}{4}\left(\nabla\eta_{kh},\nabla\eta_{kh}\right)_{I_{m}\times\Omega}+\sum_{m=1}^{M}\left([\eta_{kh}]_{m-1},\eta_{kh,m-1}^{+}\right)_{\Omega}$
$\displaystyle\leq
C\left(\|u\|_{L^{\infty}(I;H^{1}(\Omega))}^{2}+\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}^{2}\right)\|\xi\|_{L^{2}(I;H^{1}(\Omega))}^{2}+C\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}^{4}\sum_{m=1}^{M}k_{m}\|\eta_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}.$

To abbreviate notation we have used here $[\eta_{kh}]_{0}:=\eta_{kh,0}^{+}$.
Choosing $k$ small enough, and using the boundedness of
$\|\tilde{u}_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}$, due to (26) as well as of
$\|u\|_{L^{\infty}(I;H^{1}(\Omega))}$, shown in Theorem 3.6, allows us to
apply the discrete Gronwall Lemma 4.11 as in Theorem 4.13, which concludes the
proof. ∎

###  4.6. Uniqueness and strong stability of discrete solutions

We next show the uniqueness and  stability of the discrete solution $u_{kh}$
in stronger norms, comparable to the continuous result of Theorem 3.6. These
results will be obtained, by applying the discrete Gronwall Lemma 4.11, to
which end we need the two following technical lemmas, guaranteeing, that the
coefficients in the Gronwall lemma become arbitrarily small, uniformly in $m$,
as $(k,h)\to 0$.

######  Lemma 4.17.

Let $I\subset\mathbb{R}$ a bounded interval, $\varepsilon>0$ and $\zeta\in
L^{1}(I)$. Then

$\sup_{x\in\bar{I}}\int_{x}^{x+\varepsilon}|\zeta(s)|\
ds\xrightarrow{\varepsilon\to 0}0.$

To keep notation simple, we formally extend $\zeta$ to $[0,T+\varepsilon]$ by
values $0$ such that the integration is well defined.

###### Proof.

The proof relies on dominated convergence and Dini’s theorem, see, e.g., [12,
p. 125]. We first define for fixed $\varepsilon$ the function
$\sigma_{\varepsilon}:\bar{I}\to\mathbb{R},x\mapsto\int_{x}^{x+\varepsilon}|\zeta(s)|\
ds$. Note that for each $\varepsilon>0$ this function $\sigma_{\varepsilon}$
is continuous. To see this, let $x\in I$ be fixed and let $y\to x$. W.l.o.g.
let $y>x$:

$|\sigma_{\varepsilon}(x)-\sigma_{\varepsilon}(y)|\leq\left|\int_{x}^{y}|\zeta(s)|\
ds+\int_{x+\varepsilon}^{y+\varepsilon}|\zeta(s)|\
ds\right|=\left|\int_{I}\chi_{x,y}(s)|\zeta(s)|\ ds\right|$

where $\chi_{x,y}(s)=1$ if $s\in(x,y)\cup(x+\varepsilon,y+\varepsilon)$, and
$0$ otherwise. For a fixed $s$, the integrand $\chi_{x,y}(s)|\zeta(s)|\to 0$
as $y\to x$, thus the integrand converges pointwise to $0$. By the dominated
convergence theorem, this means that the integral converges to $0$. For fixed
$\varepsilon>0$ this shows the continuity of $\sigma_{\varepsilon}$. Moreover,
by the same argument we can show, that for fixed $x$ and $\varepsilon\to 0$ it
holds $\sigma_{\varepsilon}(x)\to 0$. Thus the sequence $\sigma_{\varepsilon}$
converges pointwise to $0$. Note moreover, that for $\delta>\varepsilon$, and
fixed $x$, it holds $\sigma_{\delta}(x)\geq\sigma_{\varepsilon}(x)$, as the
integration covers a larger interval. We thus have shown, that when
$\varepsilon\to 0$ monotonically, also this pointwise convergence of
$\sigma_{\varepsilon}(x)$ is monotone. Hence we can apply Dini’s theorem, see
[12, p. 125], to obtain

$\|\sigma_{\varepsilon}\|_{L^{\infty}(I)}\rightarrow 0,\quad\text{ as
}\varepsilon\rightarrow 0.$

∎

######  Lemma 4.18.

Let $u\in L^{2}(I;H^{1}_{0}(\Omega)^{2})$, and $u_{kh}\in U_{kh}$ such that
$\|u-u_{kh}\|_{L^{2}(I;H^{1}(\Omega))}\to 0$ as $(k,h)\to 0$. Then it holds

$\displaystyle\sup_{1\leq m\leq M}\int_{I_{m,k}}\|\nabla
u\|_{L^{2}(\Omega)}^{2}\ dt\xrightarrow{(k,h)\to
0}0\quad\text{and}\quad\sup_{1\leq m\leq M}\int_{I_{m,k}}\|\nabla
u_{kh}\|_{L^{2}(\Omega)}^{2}\ dt\xrightarrow{(k,h)\to 0}0.$

###### Proof.

We first show the statement for $u$. To this end, note that for $k=\max_{1\leq
m\leq M}|I_{m,k}|$

$\sup_{1\leq m\leq M}\int_{I_{m,k}}\|\nabla u\|_{L^{2}(\Omega)}^{2}\
dt\leq\sup_{x\in\bar{I}}\int_{x}^{x+k}\|\nabla u\|_{L^{2}(\Omega)}^{2}\
dt\xrightarrow{k\to 0}0$

by Lemma 4.17. In order to show the result for the discrete solution $u_{kh}$,
we cannot directly apply the previous lemma, as $u_{kh}$ depends on $k$. We
thus insert insert $\pm u$ and apply the triangle inequality to obtain

$\displaystyle\sup_{1\leq m\leq M}\int_{I_{m,k}}\|\nabla
u_{kh}\|_{L^{2}(\Omega)}^{2}\ dt$ $\displaystyle\leq 2\sup_{1\leq m\leq
M}\int_{I_{m,k}}\|\nabla u\|_{L^{2}(\Omega)}^{2}\
dt+2\int_{I}\|\nabla(u-u_{kh})\|_{L^{2}(\Omega)}^{2}\ dt.$

With the claim for $u$ and $u_{kh}\to u$ in $L^{2}(I;H^{1}(\Omega))$, we have
shown the claim for $u_{kh}$. ∎

With these technical lemmas, we can show two results, the uniqueness of
solutions $u_{kh}$ and their boundedness in stronger norms. We first obtain
the following uniqueness result.

######  Theorem 4.19.

Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$. Then for $(k,h)$ small
enough, the solution $u_{kh}$ to (19) is unique.

###### Proof.

To show uniqueness, we assume two solutions $u_{kh}^{1},u_{kh}^{2}$ and define
$e_{kh}:=u_{kh}^{1}-u_{kh}^{2}$. Testing the equations for $m=1,...,M$ with
$e_{kh}\chi_{m}$, where $\chi_{m}$ denotes the characteristic function of the
interval $I_{m}$, and subtracting them leads to

$\mathfrak{B}(e_{kh},e_{kh}\chi_{m})+\hat{c}\left(\\!\left(u_{kh}^{1},u_{kh}^{1},e_{kh}\chi_{m}\right)\\!\right)-\hat{c}\left(\\!\left(u_{kh}^{2},u_{kh}^{2},e_{kh}\chi_{m}\right)\\!\right)=0$

Adding and subtracting
$\hat{c}\left(\\!\left(u_{kh}^{1},u_{kh}^{2},e_{kh}\chi_{m}\right)\\!\right)$
and applying Lemma 4.3 yields

$\mathfrak{B}(e_{kh},e_{kh}\chi_{m})+\hat{c}\left(\\!\left(e_{kh},u_{kh}^{2},e_{kh}\chi_{m}\right)\\!\right)=0.$

Applying Lemma 4.6, the bilinear form can be written as

$\mathfrak{B}(e_{kh},e_{kh}\chi_{m})=\frac{1}{2}\left(\|e_{kh,m}^{-}\|_{L^{2}(\Omega)}^{2}+\|[e_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}-\|e_{kh,m-1}^{-}\|_{L^{2}(\Omega)}^{2}\right)+\nu\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}.$

From the definition of $\hat{c}$, the estimates of Lemma 3.2, and Hölder’s
inequality in time, we obtain

$\displaystyle\hat{c}\left(\\!\left(e_{kh},u_{kh}^{2},e_{kh}\chi_{m}\right)\\!\right)$
$\displaystyle=\frac{1}{2}\left[\left((e_{kh}\cdot\nabla)u_{kh}^{2},e_{kh}\right)_{I_{m}\times\Omega}-\left((e_{kh}\cdot\nabla)e_{kh},u_{kh}^{2}\right)_{I_{m}\times\Omega}\right]$
$\displaystyle\leq C\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}$
$\displaystyle\quad+C\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{\frac{1}{2}}\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{\frac{3}{2}}\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{\frac{1}{2}}\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{\frac{1}{2}}.$

We can apply Young’s inequality and obtain

$\displaystyle\hat{c}\left(\\!\left(e_{kh},u_{kh}^{2},e_{kh}\chi_{m}\right)\\!\right)$
$\displaystyle\leq\frac{\nu}{2}\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}+C\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\left(1+\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right)\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}.$

where the first summand can be absorbed into
$\mathfrak{B}(e_{kh},e_{kh}\chi_{m})$. Thus on each time interval, there holds
the bound

$\displaystyle\frac{1}{2}\left(\|e_{kh,m}^{-}\|_{L^{2}(\Omega)}^{2}+\|[e_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}-\|e_{kh,m-1}^{-}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\right)$ $\displaystyle\leq
C\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\left(1+\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right)\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}.$

Multiplying the derived inequality by 2 and summing up the inequalities for
$m=1,...,n$ yields for any $n\in\\{1,...,M\\}$

$\displaystyle\|e_{kh,n}^{-}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\left(\|[e_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
e_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\right)$ $\displaystyle\leq
C\sum_{m=1}^{n}\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\left(1+\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right)\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}.$

Here we have used $e_{kh,0}^{-}=0$ as $u_{kh}^{1}$ and $u_{kh}^{2}$ satisfy
the same initial condition. As in the proof of Theorem 4.13,
$\|e_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}$ can be bounded by evaluations
of one-sided limits at the time nodes. Due to $f\in
L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$, Theorem 4.16 holds, and thus
Lemma 4.18 yields $\|\nabla u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\to
0$ uniformly in $m$ for $(k,h)\to 0$. Together with Theorem 4.13, bounding
$\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}$, we can choose the
discretization fine enough, such that
$C(1+\|u_{kh}^{2}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2})\|\nabla
u_{kh}^{2}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}<\frac{1}{2}$ for all $m$. All in
all, as in Theorem 4.13, an application of the Gronwall Lemma 4.11 shows that
$e_{kh}=0$, concluding the proof of uniqueness. ∎

We now turn toward proving the stability of $u_{kh}$ in stronger norms. The
proof follows the steps of the continuous result, shown in [48, Chapter 3,
Theorem 3.10], using the discrete analogons of the inequalities presented in
Section 4.1, and the discrete Gronwall Lemma 4.11. For the application of the
lemma, we need coefficients, that become small, as $k\to 0$. These
coefficients depend on $u_{kh}$, and thus on $k,h$, hence we apply Theorem
4.16 together with Lemma 4.18, in order to have coefficients, that converge to
0 uniformly in $m$. There holds the following.

######  Theorem 4.20.

Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$, then for $(k,h)$ small
enough, the unique solution $u_{kh}\in V_{kh}$ to (19) satisfies

$\displaystyle\|u_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}+\|A_{h}u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}\leq$
$\displaystyle
C_{1}\exp{\left(C_{2}\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{2}\|\nabla
u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}^{2}\right)}$
$\displaystyle\times\left(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{0}\|_{H^{1}(\Omega)}\right),$

with constants $C_{1},C_{2}$ independent of $k,h$. The
$L^{\infty}(I;L^{2}(\Omega))$ and $L^{2}(I;H^{1}(\Omega))$ norms of $u_{kh}$
can be estimated by the results of Theorem 4.13.

###### Proof.

For $m=1,...,M$, we test (19) with $A_{h}u_{kh}|_{I_{m}}$, which yields with
the convention $[u_{kh}]_{0}:=u_{kh,0}^{+}-u_{0}$

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\partial_{t}u_{kh},A_{h}u_{kh}\right)_{I_{m}\times\Omega}}+\nu\left(\nabla
u_{kh},\nabla A_{h}u_{kh}\right)_{I_{m}\times\Omega}$
$\displaystyle+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left([u_{kh}]_{m-1},A_{h}u_{kh,m-1}^{+}\right)_{\Omega}}+\hat{c}\left(\\!\left(u_{kh},u_{kh},A_{h}u_{kh}\chi_{I_{m}}\right)\\!\right)$
$\displaystyle=\left(f,A_{h}u_{kh}\right)_{I_{m}\times\Omega},$

From the above identity, for $m=2,...,M$, the definition of $A_{h}$ gives

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\partial_{t}\nabla
u_{kh},\nabla
u_{kh}\right)_{I_{m}\times\Omega}}+\nu\|A_{h}u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$
(28)
$\displaystyle+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left([\nabla
u_{kh}]_{m-1},\nabla
u_{kh,m-1}^{+}\right)_{\Omega}}+\hat{c}\left(\\!\left(u_{kh},u_{kh},A_{h}u_{kh}\chi_{I_{m}}\right)\\!\right)$
$\displaystyle=\left(f,A_{h}u_{kh}\right)_{I_{m}\times\Omega}.$

Here the terms containing time derivatives and jumps can be combined according
to Lemma 4.6. Some care has to be taken on the first time interval. It holds

$\displaystyle\left(u_{0},A_{h}u_{kh,0}^{+}\right)_{\Omega}$
$\displaystyle=-\left(u_{0},\mathbb{P}_{h}\Delta_{h}u_{kh,0}^{+}\right)_{\Omega}=-\left(\mathbb{P}_{h}u_{0},\Delta_{h}u_{kh,0}^{+}\right)_{\Omega}=\left(\nabla\mathbb{P}_{h}u_{0},\nabla
u_{kh,0}^{+}\right)_{\Omega}$

Thus on the first time interval, with the same arguments as in the proof of
Lemma 4.6, it holds

$\left(\partial_{t}u_{kh},A_{h}u_{kh}\right)_{I_{1}\times\Omega}+\left(u_{kh,0}^{+}-u_{0},A_{h}u_{kh,0}^{+}\right)_{\Omega}=\frac{1}{2}\left(\|\nabla
u_{kh,1}^{-}\|_{L^{2}(\Omega)}^{2}+\|\nabla(u_{kh,0}^{+}-\mathbb{P}_{h}u_{0})\|_{L^{2}(\Omega)}^{2}-\|\nabla\mathbb{P}_{h}u_{0}\|_{L^{2}(\Omega)}^{2}\right).$

For a more compact notation, we shall write $[\nabla
u_{kh}]_{0}:=\nabla(u_{kh,0}^{+}-\mathbb{P}_{h}u_{0})$. Note the need to
include the projection on the right hand side. With this slight abuse of
notation, (28) also holds on the first time interval. Since $u_{0}\in V$ we
can use the stability of $\mathbb{P}_{h}$ in $H^{1}$ for continuously
divergence free functions, see [36, Lemma 5.4], yielding an estimate

$\|\nabla\mathbb{P}_{h}u_{0}\|_{L^{2}(\Omega)}\leq C\|\nabla
u_{0}\|_{L^{2}(\Omega)}.$ (29)

For the terms of (28) involving the right hand side $f$, it holds

$\left(f,A_{h}u_{kh}\right)_{I_{m}\times\Omega}\leq\|f\|_{L^{2}(I_{m};L^{2}(\Omega))}\|A_{h}u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}\leq
C\|f\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}+\frac{\nu}{4}\|A_{h}u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2},$

where we can absorb the $A_{h}u_{kh}$ term. We now turn towards estimating the
trilinear form remaining in (28), which poses the main difficulty of this
proof. Due to Lemma 4.4, the trilinear term satisfies the expression

$\hat{c}\left(\\!\left(u_{kh},u_{kh},A_{h}u_{kh}\chi_{I_{m}}\right)\\!\right)=\left((u_{kh}\cdot\nabla)u_{kh},A_{h}u_{kh}\right)_{I_{m}\times\Omega}+\frac{1}{2}\left(\nabla\cdot
u_{kh},u_{kh}\cdot A_{h}u_{kh}\right)_{I_{m}\times\Omega},$

which we can estimate with Hölder’s inequality by

$\hat{c}\left(\\!\left(u_{kh},u_{kh},A_{h}u_{kh}\chi_{I_{m}}\right)\\!\right)\leq
C\int_{I_{m}}\|u_{kh}\|_{L^{\infty}(\Omega)}\|\nabla
u_{kh}\|_{L^{2}(\Omega)}\|A_{h}u_{kh}\|_{L^{2}(\Omega)}\ dt.$

Using (16) and Young’s inequality, we obtain the estimate

$\hat{c}\left(\\!\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}u_{kh},u_{kh},A_{h}u_{kh}\chi_{I_{m}}}\right)\\!\right)\leq\int_{I_{m}}C\|u_{kh}\|_{L^{2}(\Omega)}^{2}\|\nabla
u_{kh}\|_{L^{2}(\Omega)}^{4}+\frac{\nu}{4}\|A_{h}u_{kh}\|_{L^{2}(\Omega)}^{2}\
dt.$

The latter term can be absorbed, hence we proceed by discussing the first one.
It holds

$\int_{I_{m}}C\|u_{kh}\|_{L^{2}(\Omega)}^{2}\|\nabla
u_{kh}\|_{L^{2}(\Omega)}^{4}\ dt\leq
C\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{2}\|\nabla
u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\|\nabla
u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}.$

Let us introduce
$\gamma_{m}:=C\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{2}\|\nabla
u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$. All in all, we have derived from
(28) an estimate of the form

$\displaystyle\frac{1}{2}\left(\|\nabla
u_{kh,m}^{-}\|_{L^{2}(\Omega)}^{2}+\|[\nabla
u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}-\|\nabla
u_{kh,m-1}^{-}\|_{L^{2}(\Omega)}^{2}\right)+\frac{1}{2}\nu\|A_{h}u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$
$\displaystyle\leq\|f\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}+\gamma_{m}\|\nabla
u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2},$

with the usual modifications on the first time interval. Multiplying this by 2
and summing up from $m=1,...,n$ yields for any $n\in\\{1,...,M\\}$ an estimate
of the form

$\displaystyle\|\nabla
u_{kh,m}^{-}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\left(\|[\nabla
u_{kh}]_{m-1}\|_{L^{2}(\Omega)}^{2}+\nu\|A_{h}u_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}\right)$
$\displaystyle\leq C\|\nabla
u_{0}\|_{L^{2}(\Omega)}^{2}+\sum_{m=1}^{n}\left(\|f\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}+2\gamma_{m}\|\nabla
u_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right),$

where due to (29) we have dropped the projection $\mathbb{P}_{h}$ of the
initial data on the right hand side. Since $f\in L^{2}(I;L^{2}(\Omega)^{2})$
and $u_{0}\in V$, Theorem 4.16 shows that $u_{kh}\to u$ in
$L^{2}(I;H^{1}(\Omega))$ as $(k,h)\to 0$. Hence, due to Lemma 4.18, and
Theorem 4.13, it holds $\gamma_{m}\to 0$ uniformly, as $(k,h)\to 0$. This
implies that by following the same steps as the proof of Theorem 4.13, we
obtain the result as a consequence of the discrete Gronwall Lemma 4.11. ∎

##  5\. Error estimates

###  5.1. Duality based best approximation type estimates

In this concluding section of our work, we will show the main results, which
are the $L^{\infty}(I;L^{2}(\Omega))$ and $L^{2}(I;L^{2}(\Omega))$ error
estimates. Their proofs are based on duality arguments, and hence we begin
this section by showing two stability results for discrete dual equations. Let
us first motivate the specific dual problem, that we will consider. Due to the
nonlinear structure of the Navier-Stokes equations, there does not hold a
Galerkin orthogonality with respect to the bilinear form $B$, i.e., for
solutions $(u,p)$ of (9) and their discrete counterparts $(u_{kh},p_{kh})$ of
(20), it holds for test functions $(v_{kh},q_{kh})\in Y_{kh}$:

$B((u-u_{kh},p-p_{kh}),(v_{kh},q_{kh}))=-\hat{c}\left(\\!\left(u,u,v_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(u_{kh},u_{kh},v_{kh}\right)\\!\right),$

where the right hand side is nonzero in general. Since we want to use this
orthogonality relation after testing the dual equation with $u-u_{kh}$, we
need to reformulate the trilinear terms, such that $u-u_{kh}$ occurs linearly.
To this end, we use the identity

$\hat{c}\left(\\!\left(u,u,v_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(u_{kh},u_{kh},v_{kh}\right)\\!\right)=\hat{c}\left(\\!\left(\overline{uu_{kh}},u-u_{kh},v_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(u-u_{kh},\overline{uu_{kh}},v_{kh}\right)\\!\right),$
(30)

where we linearize around the average of continuous and discrete solutions to
the Navier-Stokes equations

$\overline{uu_{kh}}:=\frac{1}{2}(u+u_{kh}).$ (31)

With these considerations, we have the following lemma:

######  Lemma 5.1.

Let $(u,p)$ be a solution to the Navier-Stokes equations (9), and
$(u_{kh},p_{kh})$ their discrete approximation solving (20). Let further
$\overline{uu_{kh}}$ denote the average of $u$ and $u_{kh}$ as defined by
(31). Then for any $(v_{kh},q_{kh})\in Y_{kh}$, it holds

$B((u-u_{kh},p-p_{kh}),(v_{kh},q_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},u-u_{kh},v_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(u-u_{kh},\overline{uu_{kh}},v_{kh}\right)\\!\right)=0.$

###### Proof.

This result is an immediate consequence of the definitions of solutions to (9)
and (20), together with the identity (30) ∎

This motivates the choice of $\overline{uu_{kh}}$ as linearization point for
setting up a dual equation:
Find $(z_{kh},\varrho_{kh})\in Y_{kh}$ such that for all
$(\phi_{kh},\psi_{kh})\in Y_{kh}$ it holds

$B((\phi_{kh},\psi_{kh}),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},\phi_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\phi_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)=\left\langle
g,\phi_{kh}\right\rangle_{I\times\Omega},$ (32)

where the right hand side $g$ will be chosen appropriately, see the proofs of
Theorems 5.6 and 5.7. Note that the right hand side of (32) implicitly
prescribes the final data $z_{kh,M}^{+}=0$. This dual equation will help us in
deriving the sought error estimates, which we will do in the following
section.

######  Remark 5.2.

To analyze this dual problem, it will be convenient, to have dual
represenations of $\mathfrak{B}$ and $B$ at hand, which are obtained by
partial integration on each $I_{m}$ and rearranging the terms. Note that in
this representation, the time derivative is applied to the second argument. It
holds

$\hskip
39.83385pt\mathfrak{B}(u,v)=-\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(u,\partial_{t}v\right)_{I_{m}\times\Omega}}+\nu\left(\nabla
u,\nabla
v\right)_{I\times\Omega}-\sum_{m=1}^{M-1}\left(u_{m}^{-},[v]_{m}\right)_{\Omega}+\left(u_{M}^{-},v_{M}^{-}\right)_{\Omega},$
(33) $\displaystyle B((u,p),(v,q))=$
$\displaystyle-\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(u,\partial_{t}v\right)_{I_{m}\times\Omega}}+\nu\left(\nabla
u,\nabla
v\right)_{I\times\Omega}-\sum_{m=1}^{M-1}\left(u_{m}^{-},[v]_{m}\right)_{\Omega}+\left(u_{M}^{-},v_{M}^{-}\right)_{\Omega}$
(34) $\displaystyle+\left(\nabla\cdot
u,q\right)_{I\times\Omega}-\left(\nabla\cdot v,p\right)_{I\times\Omega},$

see also [35, 7].

######  Remark 5.3.

Similar to the discrete Navier-Stokes equations, we can consider an equivalent
formulation for the discrete dual equation in discretely divergence free
spaces: Find $z_{kh}\in V_{kh}$ satisfying

$\mathfrak{B}(\phi_{kh},z_{kh})+\hat{c}\left(\\!\left(\overline{uu_{kh}},\phi_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\phi_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)=\left\langle
g,\phi_{kh}\right\rangle_{I\times\Omega}\qquad\text{for all }\phi_{kh}\in
V_{kh}.$ (35)

With the same argument as [7, Proposition 4.3], there holds: If $z_{kh}\in
V_{kh}$ solves (35), then there exists $\varrho_{kh}\in M_{kh}$ such that
$(z_{kh},\varrho_{kh})\in Y_{kh}$ solves (32). If $(z_{kh},\varrho_{kh})\in
Y_{kh}$ solves (32), then $z_{kh}\in V_{kh}$ and it solves (35).

We first show unique solvability of the discrete dual problem, and the
stability in $L^{\infty}(I;L^{2}(\Omega))\cap L^{2}(I;H^{1}(\Omega))$. Since
both $u$ and $u_{kh}$ occur in the formulation of the discrete problem, we
need both results of Lemma 4.18 to hold true.

######  Theorem 5.4.

Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$, $u_{0}\in V$ and $u$,$u_{kh}\in
L^{\infty}(I;L^{2}(\Omega)^{2})\cap L^{2}(I;H^{1}(\Omega)^{2})$ be solutions
to the weak and fully discretized Navier-Stokes equations (4) and (19) for
either $q=0$ or $q=1$. Then for $(k,h)$ small enough, problem (35) has a
unique solution $z_{kh}\in V_{kh}$ for any $g\in L^{1}(I;L^{2}(\Omega)^{2})$,
and there holds the bound

$\|z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|z_{kh}\|_{L^{2}(I;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}H^{1}(\Omega)})}\leq
K\left(\|\overline{uu_{kh}}\|_{L^{2}(I;H^{1}(\Omega))\cap
L^{\infty}(I;L^{2}(\Omega))}\right)\|g\|_{L^{1}(I;L^{2}(\Omega))},$

where $K:[0,+\infty)\to[0,+\infty)$ is a strictly monotonically increasing,
continuous nonlinear function, independent of $k,h$.

###### Proof.

On the continuous level and for $g\in L^{2}(I;H^{-1}(\Omega)^{2})$, the proof
of a corresponding estimate can be found in [15, Proposition 2.7]. We adapt it
to the discrete setting and to $g\in L^{1}(I;L^{2}(\Omega)^{2})$, making use
of the previously derived discrete Gronwall Lemma 4.11. We only have to prove
the norm bound, since problem (32) is a quadratic system of linear equations,
thus is solvable, if it is injective. The norm bound yields, that for right
hand side $g=0$, $z_{kh}=0$ is the only solution, thus the norm bound implies
existence and uniqueness. For ease of notation, we use the convention
$[z_{kh}]_{M}=-z_{kh,M}^{-}$. Testing equation 35 with $z_{kh}\chi_{I_{m}}$,
$m=1,...,M$, where by $\chi_{I_{m}}$ we denote the indicator function of the
subinterval $I_{m}$, yields:

$\displaystyle-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(z_{kh}\chi_{I_{m}},\partial_{t}z_{kh}\right)_{I_{m}\times\Omega}}$
$\displaystyle+\nu\left(\nabla z_{kh}\chi_{I_{m}},\nabla
z_{kh}\right)_{I\times\Omega}-\left(z_{kh,m}^{-},[z_{kh}]_{m}\right)_{\Omega}$
$\displaystyle+\hat{c}\left(\\!\left(\overline{uu_{kh}},z_{kh}\chi_{I_{m}},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)\\!=\\!\left(g,z_{kh}\chi_{I_{m}}\right)_{I\times\Omega}.$

Applying Lemma 4.6 and writing the inner product as norm yields

$\displaystyle\frac{1}{2}\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)}^{2}+\frac{1}{2}\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
z_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$ (36)
$\displaystyle+\hat{c}\left(\\!\left(\overline{uu_{kh}},z_{kh}\chi_{I_{m}},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)$
$\displaystyle=\frac{1}{2}\|z_{kh,m}^{+}\|_{L^{2}(\Omega)}^{2}+\\!{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(g,z_{kh}\right)_{I_{m}\times\Omega}}.$

We proceed by estimating the trilinear forms. According to its definition
(11), the first term vanishes, and for the second one, it holds

$\displaystyle\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)$
$\displaystyle=\frac{1}{2}c\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)-\frac{1}{2}c\left(\\!\left(z_{kh}\chi_{I_{m}},z_{kh},\overline{uu_{kh}}\right)\\!\right).$

After applying Hölder’s inequality in space, we obtain

$\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)\leq\frac{1}{2}\int_{I_{m}}\left(\|z_{kh}\|_{L^{4}(\Omega)}^{2}\|\nabla\overline{uu_{kh}}\|_{L^{2}(\Omega)}+\|z_{kh}\|_{L^{4}(\Omega)}\|\overline{uu_{kh}}\|_{L^{4}(\Omega)}\|\nabla
z_{kh}\|_{L^{2}(\Omega)}\right)\ dt.$

After estimating the $L^{4}$ norms by Lemma 3.2 and applying Hölder’s
inequality in time, we arrive at

$\displaystyle\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)\leq$
$\displaystyle C\Big{(}\|\nabla
z_{kh}\|_{L^{2}(I_{m}\times\Omega)}\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}\|\nabla\overline{uu_{kh}}\|_{L^{2}(I_{m}\times\Omega)}$
$\displaystyle+\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{\frac{1}{2}}\|\nabla
z_{kh}\|_{L^{2}(I_{m}\times\Omega)}^{\frac{3}{2}}\|\overline{uu_{kh}}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{\frac{1}{2}}\|\nabla\overline{uu_{kh}}\|_{L^{2}(I_{m}\times\Omega)}^{\frac{1}{2}}\Big{)}.$

An application of Young’s inequality yields

$\hat{c}\left(\\!\left(z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right)\leq\frac{\nu}{2}\|\nabla
z_{kh}\|_{L^{2}(I_{m}\times\Omega)}^{2}+C\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\left(1+\|\overline{uu_{kh}}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right)\|\nabla\overline{uu_{kh}}\|_{L^{2}(I_{m}\times\Omega)}^{2}.$

In order to abbreviate the notation, we introduce
$\gamma_{m}:=C\|\nabla\overline{uu_{kh}}\|_{L^{2}(I_{m}\times\Omega)}^{2}\left(1+\|\overline{uu_{kh}}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}\right).$We
insert the above estimate into (36), absorb terms, and multiply by 2, which
yields

$\displaystyle\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)}^{2}+\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
z_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$
$\displaystyle\leq\gamma_{m}\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}+\|z_{kh,m}^{+}\|_{L^{2}(\Omega)}^{2}+2{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(g,z_{kh}\right)_{I_{m}\times\Omega}}.$

After an application of Hölder’s inequality, we obtain

$\displaystyle\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)}^{2}+\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
z_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$
$\displaystyle\leq\gamma_{m}\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}+\|z_{kh,m}^{+}\|_{L^{2}(\Omega)}^{2}$
(37)
$\displaystyle\quad+2\|g\|_{L^{1}(I_{m};L^{2}(\Omega))}\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}.$

Since we have assumed $q=0$ or $q=1$, we can estimate the
$\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}$ terms by evaluations at the
right and left endpoints:
$\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}=\max\\{\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)},\|z_{kh,m}^{-}\|_{L^{2}(\Omega}\\}$.
With triangle inequality there hold the following estimates:

$\displaystyle\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}$
$\displaystyle\leq\max\\{\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)},\|z_{kh,m}^{+}\|_{L^{2}(\Omega)}+\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}\\},$
$\displaystyle\|z_{kh}\|_{L^{\infty}(I_{m};L^{2}(\Omega))}^{2}$
$\displaystyle\leq\max\\{\|z_{kh,m-1}^{+}\|_{L^{2}(\Omega)}^{2},2\|z_{kh,m}^{+}\|_{L^{2}(\Omega)}^{2}+2\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}\\}.$

We introduce
$x_{n}^{2}:=\|z_{kh,n-1}^{+}\|_{L^{2}(\Omega)}^{2}+\sum_{m=n}^{M}\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
z_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$. Note that by the terminal
condition for $z_{kh}$, it holds $x_{M+1}=0$. Then after summing up (37) from
$m=n$ to $m=M$, we have:

$x_{n}^{2}\leq\sum_{m=n}^{M}2\gamma_{m}(x_{m}^{2}+x_{m+1}^{2})+2\|g\|_{L^{1}(I_{m};L^{2}(\Omega))}(x_{m}+x_{m+1}).$

Shifting indices, we arrive at

$x_{n}^{2}\leq\sum_{m=n}^{M}2(\gamma_{m}+\gamma_{m-1})x_{m}^{2}+2\|g\|_{L^{1}(I_{m}\cup
I_{m-1};L^{2}(\Omega))}x_{m},$

where for $n=1$ we use the convention $I_{0}=\emptyset$. Hence we are in the
setting of Lemma 4.11, where formally we have to introduce an index
transformation $\tilde{n}=M-n$. In order to apply the lemma, we need to verify
$2(\gamma_{m}+\gamma_{m-1})<1$ for all $m$. Due to $f\in
L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$, Theorem 4.16 shows
$\|u-u_{kh}\|_{L^{2}(I;H^{1}(\Omega))}\to 0$ as $(k,h)\to 0$ and we can apply
Lemma 4.18 and obtain with triangle inequality, that $\sup_{1\leq m\leq
M}\|\nabla\overline{uu_{kh}}\|^{2}_{L^{2}(I_{m};L^{2}(\Omega))}\to 0$ as
$(k,h)\to 0$. Together with the bounds from Proposition 3.1 and Theorem 4.13
for $\|u\|_{L^{\infty}(I;L^{2}(\Omega))}$ and
$\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}$, we obtain that $\gamma_{m}\to 0$
uniformly in $n$ for $(k,h)\to 0$. Thus we can choose the discretization fine
enough, such that $\gamma_{m}<1/8$, and thus we obtain from Lemma 4.11

$\displaystyle\|z_{kh,n-1}^{+}\|_{L^{2}(\Omega)}^{2}+\sum_{m=n}^{M}\|[z_{kh}]_{m}\|_{L^{2}(\Omega)}^{2}+\nu\|\nabla
z_{kh}\|_{L^{2}(I_{m};L^{2}(\Omega))}^{2}$ (38) $\displaystyle\leq
C_{1}\exp\left(C_{2}\|\nabla\overline{uu_{kh}}\|^{2}_{L^{2}(I\times\Omega)}\left(1+\|\overline{uu_{kh}}\|^{2}_{L^{\infty}(I,L^{2}(\Omega))}\right)\right)\|g\|_{L^{1}(I;L^{2}(\Omega))}^{2}.$

∎

The next theorem, similar to Theorem 4.20, states a stability result for the
discrete dual solution in stronger norms, whenever the right hand side
posesses more regularity.

######  Theorem 5.5.

Let the assumptions of Theorem 5.4 hold true, i.e., let $f\in
L^{2}(I;L^{2}(\Omega)^{2})$, $u_{0}\in V$ and $u$,$u_{kh}$ be solutions to the
weak and fully discretized Navier-Stokes equations (4) and (19) for either
$q=0$ or $q=1$. Then for $(k,h)$ small enough, for any $g\in
L^{2}(I;L^{2}(\Omega)^{2})$, the unique solution $z_{kh}\in V_{kh}$ to (35)
satisfies the bound

$\|z_{kh}\|_{L^{\infty}(I;H^{1}(\Omega))}+\|A_{h}z_{kh}\|_{L^{2}(I;L^{2}(\Omega))}\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C_{u,u_{kh}}}\|g\|_{L^{2}(I\times\Omega)},$

with a constant $C_{u,u_{kh}}$ depending on $u,u_{kh}$ in the form

$C_{u,u_{kh}}=C_{1}\exp\left(C_{2}\left(\|Au\|_{L^{2}(I;L^{2}(\Omega))}^{2}\|u\|_{L^{\infty}(I;L^{2}(\Omega))}^{2}+\|A_{h}u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}^{2}\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{2}\right)\right),$

where the norms of $u,u_{kh}$ are bounded by Proposition 3.1 and Theorems 3.6,
4.20 and 4.13, due to the assumptions on $f$ and $u_{0}$.

###### Proof.

The proof follows the same steps as the proof the stability of $u_{kh}$ in
stronger norms, presented in Theorem 4.20. We begin by testing the dual
equation with $A_{h}z_{kh}\chi_{I_{m}}$ for arbitrary $m=1,...,M$, where
$\chi_{I_{m}}$ denotes the characteristic function of the time interval
$I_{m}$. The nonlinear terms that occur for the discrete dual equation are of
the form

$\hat{c}\left(\\!\left(\overline{uu_{kh}},A_{h}z_{kh}\chi_{I_{m}},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(A_{h}z_{kh}\chi_{I_{m}},\overline{uu_{kh}},z_{kh}\right)\\!\right).$

By Hölder’s inequality in space, (17) and the discrete Gagliardo-Nirenberg
inequality (16), these terms can be estimated by

$\displaystyle
C\int_{I_{m}}\|z_{kh}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}z_{kh}\|_{L^{2}(\Omega)}^{\frac{3}{2}}\left(\|\nabla\overline{uu_{kh}}\|_{L^{2}(\Omega)}+\|\overline{uu_{kh}}\|_{L^{\infty}(\Omega)}\right)dt.$

Applying the continuous and discrete Gagliardo-Nirenberg inequalities (15) and
(16), and (17) to $\overline{uu_{kh}}$, it remains

$\displaystyle
C\int_{I_{m}}\|z_{kh}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}z_{kh}\|_{L^{2}(\Omega)}^{\frac{3}{2}}\left(\|u\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|Au\|_{L^{2}(\Omega)}^{\frac{1}{2}}+\|u_{kh}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|A_{h}u_{kh}\|_{L^{2}(\Omega)}^{\frac{1}{2}}\right)dt.$

An application of Young’s inequality, absorbing terms and summing over the
subintervals allows us to conclude the proof. By Theorem 5.4 it holds
$z_{kh}\in L^{\infty}(I;L^{2}(\Omega)^{2})$ with a bound independent on $k,h$
and linear in $\|g\|_{L^{1}(I;L^{2}(\Omega))}$. Further, the terms involving
$u,u_{kh}$ are summable, since the $L^{\infty}(I;L^{2}(\Omega))$ norms of
$u,u_{kh}$ remain bounded via Proposition 3.1 and Theorem 4.13, and the
$Au,A_{h}u_{kh}$ terms are summable by Theorem 3.6 and Theorem 4.20, where the
latter holds true for $(k,h)$ small enough, due to $f\in
L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$. ∎

We now turn towards showing the main result of our work, i.e., the error
estimate for the Navier-Stokes equations in the $L^{\infty}(I;L^{2}(\Omega))$
norm. As in the proof of Theorem 4.16, we will split the error $u-u_{kh}$ into
an error for a Stokes problem, and a remainder term, which we will estimate
using the discrete dual equation (35), to which we can apply the results of
Theorem 5.4. Similar to the result for the Stokes equations of Proposition
4.9, the error estimate will consist of two terms with the first one being a
best approximation error, and the second one being the error of the stationary
Stokes Ritz projection introduced in (10). This result estimates the
$L^{\infty}(I;L^{2}(\Omega))$ norm in an isolated fashion, and thus does not
suffer from an order reduction, which is observed in results that estimate the
error norm combined with the $L^{2}(I;H^{1}(\Omega))$ norm.

######  Theorem 5.6.

Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$. Let $(u,p)$ be the
unique solution to the Navier-Stokes equations (9), and $(u_{kh},p_{kh})$ the
corresponding solution to the discretized equations (20) for a discontinuous
Galerkin method in time with order $q=0$ or $q=1$, for sufficiently small
discretization parameters $(k,h)$. Then there holds

$\|u-u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\left(\ln\frac{T}{k}\right)\left(\inf_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\in
V_{kh}}\|u-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|u-R_{h}^{S}(u,p)\|_{L^{\infty}(I;L^{2}(\Omega))}\right).$

###### Proof.

We use the same notation as in the proof of Theorem 4.16 and denote the
velocity error by $e:=u-u_{kh}$ and the pressure error by $r:=p-p_{kh}$. We
consider the Stokes projection $(\tilde{u}_{kh},\tilde{p}_{kh})$ of $(u,p)$,
solving (25), and the induced splitting of the error into $e=\xi+\eta_{kh}$,
$r=\omega+\kappa_{kh}$, where $\xi=u-\tilde{u}_{kh}$,
$\eta_{kh}=\tilde{u}_{kh}-u_{kh}$, $\omega=p-\tilde{p}_{kh}$ and
$\kappa_{kh}=\tilde{p}_{kh}-p_{kh}$.  We immediately obtain the estimate

$\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\left(\ln\frac{T}{k}\right)\left(\inf_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\in
V_{kh}}\|u-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\chi_{kh}}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|u-R_{h}^{S}(u,p)\|_{L^{\infty}(I;L^{2}(\Omega))}\right),$

by Proposition 4.9, since by Theorem 3.8 it holds $f-(u\cdot\nabla)u\in
L^{2}(I;L^{2}(\Omega)^{2})$. Thus to finalize the proof, we have to estimate
the $L^{2}(\Omega)$ norm of $\eta_{kh}$ pointwise in time. To this end, as in
[7, Theorem 6.2], we fix $\tilde{t}\in I$ and construct $\theta\in
C^{\infty}_{0}(I)$ in such a way, that $\operatorname{supp}\theta\subset
I_{m}$ where $m$ is chosen such that $\tilde{t}\in I_{m}$, and

$\left(\eta_{kh}(\tilde{t})\theta,\phi_{kh}\right)_{I\times\Omega}=\left(\eta_{kh}(\tilde{t}),\phi_{kh}(\tilde{t})\right)_{\Omega}\
\text{for all }\phi_{kh}\in U_{kh},\quad\|\theta\|_{L^{1}(I)}\leq C\text{
independent of }\tilde{t}\text{ and }k.$

For the construction of such a function $\theta$ serving the purpose of a
regularized Dirac measure, we refer to [44, Appendix A.5]. We then define the
dual solution $z_{kh}\in V_{kh}$ such that for all $\phi_{kh}\in V_{kh}$, it
satisfies

$\mathfrak{B}(\phi_{kh},z_{kh})+\hat{c}\left(\\!\left(\overline{uu_{kh}},\phi_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\phi_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)=\left(\eta_{kh}(\tilde{t})\theta,\phi_{kh}\right)_{I\times\Omega}.$

By Theorem 5.4 we have the existence, uniqueness and regularity of $z_{kh}$,
satisfying the bound

$\|z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|\nabla
z_{kh}\|_{L^{2}(I\times\Omega)}\leq
K\left(\|\overline{uu_{kh}}\|_{L^{2}(I;H^{1}(\Omega))\cap
L^{\infty}(I;L^{2}(\Omega))}\right)\|\eta_{kh}(\tilde{t})\theta\|_{L^{1}(I;L^{2}(\Omega))},$
(39)

for $(k,h)$ small enough. From Remark 5.3, we obtain the existence of an
associated pressure $\varrho_{kh}\in M_{kh}$, such that for all
$(\phi_{kh},\psi_{kh})\in Y_{kh}$, it holds

$B((\phi_{kh},\psi_{kh}),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},\phi_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\phi_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)=\left(\eta_{kh}(\tilde{t})\theta,\phi_{kh}\right)_{I\times\Omega}.$

Choosing the specific test functions
$(\phi_{kh},\psi_{kh})=(\eta_{kh},\kappa_{kh})\in Y_{kh}$, we have

$\displaystyle\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=\left(\eta_{kh}(\tilde{t})\theta,\eta_{kh}\right)_{I\times\Omega}$
$\displaystyle=B((\eta_{kh},\kappa_{kh}),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},\eta_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\eta_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)$
$\displaystyle=B((e-\xi,r-\omega),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},e-\xi,z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(e-\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right)$
$\displaystyle=B((e,r),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},e,z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(e,\overline{uu_{kh}},z_{kh}\right)\\!\right)$
$\displaystyle\phantom{==}-B((\xi,\omega),(z_{kh},\varrho_{kh}))-\hat{c}\left(\\!\left(\overline{uu_{kh}},\xi,z_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right).$

For $\xi$ we can use the Galerkin orthogonality with respect to $B$, i.e.,
from (25), we obtain

$B((\xi,\omega),(\phi_{kh},\psi_{kh}))=0\qquad\text{for all
}(\phi_{kh},\psi_{kh})\in Y_{kh}.$

Furthermore, due to Lemma 5.1 it holds

$B((e,r),(z_{kh},\varrho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},e,z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(e,\overline{uu_{kh}},z_{kh}\right)\\!\right)=0.$

Thus we see directly

$\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}=-\hat{c}\left(\\!\left(\overline{uu_{kh}},\xi,z_{kh}\right)\\!\right)-\hat{c}\left(\\!\left(\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right).$

We want to make use of the $L^{\infty}(I;L^{2}(\Omega))$ estimate of $\xi$ and
thus have to move it to an argument of $\hat{c}$ that has no spatial gradient
applied to it. Since $\hat{c}$ was obtained by anti-symmetrizing $c$, there
are gradients in the second and third argument of $\hat{c}$, which is why we
revert to the original trilinear form $c$. The first argument has no gradient,
and thus

$\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}=-\frac{1}{2}c\left(\\!\left(\overline{uu_{kh}},\xi,z_{kh}\right)\\!\right)+\frac{1}{2}c\left(\\!\left(\overline{uu_{kh}},z_{kh},\xi\right)\\!\right)-\hat{c}\left(\\!\left(\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right).$
(40)

Lemma 4.4 allows us to switch $\xi$ to the third argument, and we thus have

$\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}=\frac{1}{2}\left(\nabla\cdot\overline{uu_{kh}},z_{kh}\cdot\xi\right)_{I\times\Omega}+c\left(\\!\left(\overline{uu_{kh}},z_{kh},\xi\right)\\!\right)-\hat{c}\left(\\!\left(\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right).$

After application of Hölder’s inequality, we obtain

$\displaystyle\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}\leq$ $\displaystyle
C\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}\|1\|_{L^{4}(I;L^{\infty}(\Omega))}$
(41)
$\displaystyle\times\left(\|z_{kh}\|_{L^{4}(I;L^{4}(\Omega))}\|\nabla\overline{uu_{kh}}\|_{L^{2}(I;L^{4}(\Omega))}+\|\nabla
z_{kh}\|_{L^{2}(I\times\Omega)}\|\overline{uu_{kh}}\|_{L^{4}(I;L^{\infty}(\Omega))}\right).$

By Theorem 3.6 and Theorem 4.20, the solutions $u$,$u_{kh}$ satisfy the bound

$\|\overline{uu_{kh}}\|_{L^{\infty}(I;H^{1}(\Omega))}+\|u\|_{L^{2}(I;H^{2}(\Omega))}+\|A_{h}u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}\leq
C,$

with a constant depending on the data. Hölder’s inequality and the continuous
and discrete Gagliardo-Nirenberg inequalities (15) and (16) yield moreover

$\|\overline{uu_{kh}}\|_{L^{4}(I;L^{\infty}(\Omega))}\leq
C\left(\|u\|_{L^{\infty}(I;L^{2}(\Omega))}^{\frac{1}{2}}\|Au\|_{L^{2}(I;L^{2}(\Omega))}^{\frac{1}{2}}+\|u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{\frac{1}{2}}\|A_{h}u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}^{\frac{1}{2}}\right).$

From (12) and (14), we further obtain

$\|\nabla\overline{uu_{kh}}\|_{L^{2}(I;L^{4}(\Omega))}\leq C\left(\|\nabla
u\|_{L^{\infty}(I;L^{2}(\Omega))}^{\frac{1}{2}}\|Au\|_{L^{2}(I;L^{2}(\Omega))}^{\frac{1}{2}}+\|\nabla
u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{\frac{1}{2}}\|A_{h}u_{kh}\|_{L^{2}(I;L^{2}(\Omega))}^{\frac{1}{2}}\right).$

With the above estimates, we thus obtain from (41)

$\|\eta_{kh}(\tilde{t})\|^{2}_{L^{2}(\Omega)}\leq
CT^{\frac{1}{4}}\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}\left(\|z_{kh}\|_{L^{4}(I;L^{4}(\Omega))}+\|\nabla
z_{kh}\|_{L^{2}(I\times\Omega)}\right).$

With Lemma 3.2, we further obtain the estimate

$\|z_{kh}\|_{L^{4}(I;L^{4}(\Omega))}\leq\|z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}^{\frac{1}{2}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\|\nabla
z_{kh}\|_{L^{2}(I\times\Omega)}^{\frac{1}{2}}}.$

Hence, applying (39) yields

$\|\eta_{kh}(\tilde{t})\|_{L^{2}(\Omega)}^{2}\leq
CT^{\frac{1}{4}}\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}\|\eta_{kh}(\tilde{t})\theta\|_{L^{1}(I;L^{2}(\Omega))}.$
(42)

By definition of $\theta$ we can estimate
$\|\eta_{kh}(\tilde{t})\theta\|_{L^{1}(I;L^{2}(\Omega))}$ by
$C\|\eta_{kh}(\tilde{t})\|_{L^{2}(\Omega)}$. This allows us to divide (42) by
$\|\eta_{kh}(\tilde{t})\|_{L^{2}(\Omega)}$ which shows the bound

$\displaystyle\|\eta_{kh}(\tilde{t})\|_{L^{2}(\Omega)}\leq
CT^{\frac{1}{4}}\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}.$

As $\tilde{t}\in I$ was arbitrary, this shows

$\displaystyle\|\eta_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}$ $\displaystyle\leq
CT^{\frac{1}{4}}\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}.$

Making use of the previously derived bound for
$\|\xi\|_{L^{\infty}(I;L^{2}(\Omega))}$ and triangle inequality for
$e=\xi+\eta_{kh}$ concludes the proof. ∎

The above theorem is the main result of this work, and the development of the
discrete Gronwall lemma and analysis of the dual problem were the key
ingredients, in order to prove it. With these techniques established, it is
now straightforward, to also prove an error estimate in the
$L^{2}(I;L^{2}(\Omega))$ norm. If we follow the steps of the proof of Theorem
5.6 up to (40), we then have to estimate the occuring trilinear form in terms
of $\|\xi\|_{L^{2}(I;L^{2}(\Omega))}$, i.e., the $L^{2}(I;L^{2}(\Omega))$ norm
of the Stokes error. This implies that we need to estimate the occuring
trilinear terms by stronger norms of the dual state $z_{kh}$, which are
bounded by the results presented in Theorem 5.5 With these considerations, we
can show the following theorem.

######  Theorem 5.7.

Let $f\in L^{2}(I;L^{2}(\Omega)^{2})$ and $u_{0}\in V$. Further let $(u,p)$
and $(u_{kh},p_{kh})$ be the continuous and fully discrete solutions to the
Navier-Stokes equations (9) and (20), for sufficiently small discretization
parameters $(k,h)$. Then for any $\chi_{kh}\in V_{kh}$, there holds

$\displaystyle\|u-u_{kh}\|_{L^{2}(I\times\Omega)}$ $\displaystyle\leq
C\left(\|u-\chi_{kh}\|_{L^{2}(I\times\Omega)}+\|u-\pi_{\tau}u\|_{L^{2}(I\times\Omega)}+\|u-R_{h}^{S}(u,p)\|_{L^{2}(I\times\Omega)}\right).$

###### Proof.

To deduce the error estimate for the Navier-Stokes equations from the
corresponding Stokes result of Proposition 4.8, we follow a duality argument
similar to the proof of Theorem 5.6. We split the error in the same fashion
$u-u_{kh}=u-\tilde{u}_{kh}+\tilde{u}_{kh}-u_{kh}$ where
$\xi:=u-\tilde{u}_{kh}$ is estimated by the Stokes result, and for
$\eta_{kh}=\tilde{u}_{kh}-u_{kh}$ we consider $z_{kh}$ a solution to the dual
problem

$B((\phi_{kh},\psi_{kh}),(z_{kh},\rho_{kh}))+\hat{c}\left(\\!\left(\overline{uu_{kh}},\phi_{kh},z_{kh}\right)\\!\right)+\hat{c}\left(\\!\left(\phi_{kh},\overline{uu_{kh}},z_{kh}\right)\\!\right)=\left(\eta_{kh},\phi_{kh}\right)_{I\times\Omega}.$

As before, we test the dual equation with $\eta_{kh}$ and obtain after
elimination of terms by applying Galerkin orthogonalities and Lemma 4.4:

$\|\eta_{kh}\|_{L^{2}(I\times\Omega)}^{2}=\frac{1}{2}\left(\nabla\cdot\overline{uu_{kh}},z_{kh}\cdot\xi\right)_{I\times\Omega}+c\left(\\!\left(\overline{uu_{kh}},z_{kh},\xi\right)\\!\right)-\hat{c}\left(\\!\left(\xi,\overline{uu_{kh}},z_{kh}\right)\\!\right).$

Applying Hölder’s inequality yields

$\displaystyle\|\eta_{kh}\|_{L^{2}(I\times\Omega)}^{2}\leq$ $\displaystyle
C\|\xi\|_{L^{2}(I\times\Omega)}\|1\|_{L^{4}(I;L^{\infty}(\Omega))}$
$\displaystyle\times\left(\|\overline{uu_{kh}}\|_{L^{4}(I;L^{\infty}(\Omega))}\|\nabla
z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|z_{kh}\|_{L^{4}(I;L^{\infty}(\Omega))}\|\nabla\overline{uu_{kh}}\|_{L^{\infty}(I;L^{2}(\Omega))}\right).$

By Proposition 3.1, Theorem 3.6, Theorem 4.13 and Theorem 4.20 all terms
containing $u$, $u_{kh}$ are bounded. Together with the discrete Gagliardo-
Nirenberg inequality (16) and Young’s inequality, we obtain

$\displaystyle\|\eta_{kh}\|_{L^{2}(I\times\Omega)}^{2}\leq
C(u,u_{kh})\|\xi\|_{L^{2}(I\times\Omega)}T^{\frac{1}{4}}\left(\|A_{h}z_{kh}\|_{L^{2}(I;L^{2}(\Omega))}+\|\nabla
z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}+\|z_{kh}\|_{L^{2}(I;L^{2}(\Omega))}\right).$

Using Theorem 5.5, we can bound the norms of $z_{kh}$ and obtain

$\displaystyle\|\eta_{kh}\|_{L^{2}(I\times\Omega)}^{2}\leq
C(u,u_{kh})T^{\frac{1}{4}}\|\xi\|_{L^{2}(I\times\Omega)}\|\eta_{kh}\|_{L^{2}(I\times\Omega)}.$

Canceling terms concludes the proof. ∎

###  5.2. Explicit orders of convergence

Using the same arguments from before, instead of the best approximation type
estimate, we can also directly use the error estimate for the Stokes
projection, shown in [7, Theorem 7.4] and [36, Corollaries 6.2& 6.4] to obtain
the following corollaries, yielding explicit orders of convergence.

######  Corollary 5.8.

Let 4.1 be fulfilled. Further let $f\in L^{\infty}(I;L^{2}(\Omega)^{2})$,
$u_{0}\in V\cap H^{2}(\Omega)^{2}$ and let $u$, $u_{kh}$ be the continuous and
fully discrete solutions to the Navier-Stokes equations (4) and (19)
respectively.  Then there holds

$\|u-u_{kh}\|_{L^{\infty}(I;L^{2}(\Omega)))}\leq
C\left(\ln\frac{T}{k}\right)^{2}(k+h^{2})\left(\|f\|_{L^{\infty}(I;L^{2}(\Omega))}+\|u_{0}\|_{V\cap
H^{2}(\Omega)^{2}}+\|(u\cdot\nabla)u\|_{L^{\infty}(I;L^{2}(\Omega))}\right),$

where the constants $C$ depend continuously on
$\|f\|_{L^{2}(I;L^{2}(\Omega))}$ and $\|u_{0}\|_{V}$. The last term
$\|(u\cdot\nabla)u\|_{L^{\infty}(I;L^{2}(\Omega))}$ can be bounded in terms of
$\|f\|_{L^{\infty}(I;L^{2}(\Omega))}$ and $\|u_{0}\|_{V\cap
H^{2}(\Omega)^{2}}$ by Corollary 3.11.

###### Proof.

From Corollary 3.11 and Remark 3.3, we obtain $(u\cdot\nabla)u\in
L^{\infty}(I;L^{2}(\Omega)^{2})$. Hence this result is a direct consequence of
Theorem 5.6 and [7, Theorem 7.4]. ∎

######  Corollary 5.9.

Let 4.1 hold true. Further let $f\in L^{2}(I;L^{2}(\Omega)^{2})$, $u_{0}\in V$
and let $u$, $u_{kh}$ be the continuous and fully discrete solutions to the
Navier-Stokes equations (4) and (19) respectively. Then there hold the
estimates

$\displaystyle\|\nabla(u-u_{kh})\|_{L^{2}(I;L^{2}(\Omega))}$
$\displaystyle\leq C\left(k^{\frac{1}{2}}+h\right),\text{ and}$
$\displaystyle\|(u-u_{kh})\|_{L^{2}(I;L^{2}(\Omega))}$ $\displaystyle\leq
C\left(k+h^{2}\right),$

where the constants $C$ depend continuously on
$\|f\|_{L^{2}(I;L^{2}(\Omega))}$ and $\|u_{0}\|_{V}$.

###### Proof.

This result is a direct consequence of Theorem 4.16, Theorem 5.7 and [36,
Corollaries 6.2 & 6.4]. ∎

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