Fully discrete best approximation type estimates in L^∞(I;L^2(Ω)^d) for finite element discretizations of the transient Stokes equations

1 Introduction

In this paper we consider the following transient Stokes problem with no-slip boundary conditions,


$\partial_{t} \to u - Δ \to u + \nabla p$	$= \to f$	$in I \times Ω,$	(1a)
$\nabla \cdot \to u$	$= 0$	$in I \times Ω,$	(1b)
$\to u$	$= \to 0$	$on I \times \partial Ω,$	(1c)
$\to u (0)$	$= {\to u}_{0}$	$in Ω .$	(1d)

Throughout this work, we assume that $Ω \subset R^{d}$ , $d \in {2, 3}$ , is a bounded polygonal/polyhedral Lipschitz domain, $T > 0$ and $I = (0, T]$ . We will require some (weak) assumptions on the data, which essentially allow for a weak formulation including both velocity and pressure and for $\to u \in C (¯ I; L^{2} (Ω)^{d})$ . We consider fully discrete approximations of problem creftype 1, where we use compatible finite elements (i.e. satisfying a uniform inf-sup condition) for the space discretization and the discontinuous Galerkin method for the temporal discretization. Our goal is to obtain best approximation type results, that do not involve any additional regularity assumptions on the solution beyond the regularity which follows directly from the assumed data above. Such results are important in the analysis of PDE constrained optimal control problems that we have in mind. We refer, e.g., to Meidner et al. (2011), where such estimates are required for numerical analysis of an optimal control problem constrained by the heat equation with state constraints pointwise in time.

Our main result is of the following form

(2)

where ${\to u}_{τ h}$ is the fully discrete finite element approximation of the velocity $\to u$ , ${\to v}_{τ h}$ is an arbitrary function from the finite element approximation of the velocity spaces $X_{τ}^{w} ({\to V}_{h})$ , $R_{h}^{S}$ is the Ritz projection for the stationary Stokes problem and $ℓ_{τ}$ is a logarithmic term, explicitly given in the statements of the results, see Section 6.

The result creftype 2 links the approximation error for the fully discrete transient Stokes problem to the best possible approximation of a continuous solution $\to u$ in the discrete space $X_{τ}^{w} ({\to V}_{h})$ as well as the approximation of the stationary Stokes problem in ${\to V}_{h}$ . Such results go in hand with only natural assumptions on the problem data and thus are desirable in applications. For this result we do not require additional regularity of the domain allowing, e.g., for reentrant corners and edges. Moreover, we do not require the mesh to be quasi-uniform nor shape regular. Therefore, the result is also true for graded and even anisotropic meshes (provided the discrete inf-sup condition holds uniformly on such meshes). The application of creftype 2 in such cases would require corresponding results for the stationary Stokes problem to estimate $\to u - R_{h}^{S} (\to u, p)$ , see Section 7.

Under the additional assumption of convexity of $Ω$ and some approximation properties of the discrete spaces we prove error estimates of the form

∥ \to u - {\to u}_{τ h} ∥_{L^{\infty} (I; L^{2} (Ω))} \leq C ℓ_{τ} (τ + h^{2}) (∥ \to f ∥_{L^{\infty} (I; L^{2} (Ω))} + ∥ {\to u}_{0} ∥_{{\to V}^{2}}),

where ${\to V}^{2}$

is an appropriate space introduced in the next section. This estimate seems to be optimal (probably up to logarithmic terms) with respect to both the assumed regularity of the data and the order of convergence.

In the case of the heat equation, a similar estimate with respect to $L^{\infty} (I; L^{2} (Ω))$ is derived in Meidner et al. (2011) and for a non-autonomous parabolic problem in (Leykekhman & Vexler, 2018, Theorem 4.5). For corresponding estimates in the maximum norm in the case of the heat equation we refer to Eriksson & Johnson (1995); Leykekhman & Vexler (2016); Schatz et al. (1980); Meidner et al. (2011) and for the maximum norm of the gradient to Leykekhman & Vexler (2017b); Leykekhman & Wahlbin (2008); Thomée et al. (1989). Further results are also available in case of discretization only in space. For an overview and respective references we refer to Leykekhman & Vexler (2016, 2017b).

We are not aware of any best approximation max-norm estimates in time and space for the instationary Stokes problem creftype 1 in the literature. A result for the fully discrete problem in form of $L^{\infty} (I; L^{2} (Ω)^{d})$ estimates based on discontinuous Galerkin methods is provided in Chrysafinos & Walkington (2010), including an overview over related results for (semi-)discrete problems based on other discretization approaches. Recently the numerical behavior of a stabilized discontinuous Galerkin scheme for the Stokes problem has been analyzed in Ahmed et al. (2017). Furthermore, there are results for the fully discrete Navier-Stokes problem under moderate regularity assumptions in Heywood & Rannacher (1990). Here, we focus on an approach via a discontinuous Galerkin time stepping scheme similar to the approach in Chrysafinos & Walkington (2010); Leykekhman & Vexler (2017a). However, all the results mentioned above differ from ours in an essential way. We give a more detailed comparison of our result and the existing results from the literature in Section 7.

Our main technical tools are continuous and discrete maximal parabolic regularity results. On the continuous level we use the estimate

∥ \partial_{t} \to u ∥_{L^{s} (I; L^{2} (Ω))} + ∥ A \to u ∥_{L^{s} (I; L^{2} (Ω))} + ∥ p ∥_{L^{s} (I; L^{2} (Ω))} \leq \frac{C s^{2}}{s - 1} ∥ \to f ∥_{L^{s} (I; L^{2} (Ω))}

for $\to f \in L^{s} (I; L^{2} (Ω)^{d})$ , ${\to u}_{0} = 0$ , $1 < s < \infty$ and $A$ being the Stokes operator creftype 5, see Section 2.3 and Section 2.3 for the details and also for the formulation in the case ${\to u}_{0} \neq 0$ . This estimate holds on a general Lipschitz domain $Ω$ . Assuming in addition the convexity of $Ω$ , we have

∥ \partial_{t} \to u ∥_{L^{s} (I; L^{2} (Ω))} + ∥ \to u ∥_{L^{s} (I; H^{2} (Ω))} + ∥ \nabla p ∥_{L^{s} (I; L^{2} (Ω))} \leq \frac{}{C s^{2}} s - 1 ∥ \to f ∥_{L^{s} (I; L^{2} (Ω))},

see Section 2.3 and Section 2.3. On the discrete level, we provide the corresponding estimates which hold even in the limit cases $s = 1$ and $s = \infty$ at the expense of an logarithmic term. In a way, we extend the discrete maximal parabolic regularity results from Leykekhman & Vexler (2017a) to the Stokes problem. The resulting estimate is

∥ \partial_{t} {\to u}_{τ h} ∥_{L^{s} (I; L^{2} (Ω))} + ∥ A_{h} {\to u}_{τ h} ∥_{L^{s} (I; L^{2} (Ω))} \leq C ln \frac{T}{τ} ∥ \to f ∥_{L^{s} (I; L^{2} (Ω))},

where $A_{h}$ is the discrete Stokes operator, see Section 5 for details and the precise formulation. Under the convexity assumption for the domain $Ω$ , similar to the continuous case, we also obtain

∥ Δ_{h} {\to u}_{τ h} ∥_{L^{s} (I; L^{2} (Ω))} + ∥ \nabla p_{τ h} ∥_{L^{s} (I; L^{2} (Ω))} \leq ln \frac{T}{τ} ∥ \to f ∥_{L^{s} (I; L^{2} (Ω))},

where $Δ_{h}$ is the discrete Laplace operator, see Section 5 and Section 8 for details.

In the next section we introduce a framework of function spaces for the treatment of the stationary and transient Stokes problem, the Stokes operator and the resolvent problem. Moreover, we discuss the weak formulation and the regularity issues for creftype 1. In Section 3 we discuss the spatial discretization, introduce respective discrete spaces, operators and prove a discrete resolvent estimate. In Section 4 we present a full discretization of creftype 1 and show discrete smoothing and discrete maximal regularity results for the velocity in Section 5 based on the operator calculus discussed for the heat equation in Leykekhman & Vexler (2017a). This allows us to prove best approximation results for the velocity in Section 6. In Section 7 we apply the best approximation type results to prove error estimates and compare these result to the existing results in the literature. Finally in Section 8 we explore an expansion of the discrete maximal parabolic estimates to the pressure.

2 Results on the continuous level

In this section, we introduce the function spaces we require for the analysis of creftype 1 and state some of the main properties of these spaces. In the later sections we adopt a technique based on discrete maximal parabolic regularity from Leykekhman & Vexler (2017a), where we used an operator calculus for $- Δ$ and its finite element analog $- Δ_{h}$ . In order to modify the corresponding results, we will introduce the continuous and the discrete Stokes operator. Furthermore, we will require analysis for the resolvent of these operators. In our presentation we follow the notation and presentation of (Guermond & Pasciak, 2008, Section 1 and Section 2).

2.1 Function spaces and Stokes operator

In the following, we will use the usual notation to denote the Lebesgue spaces $L^{p}$ and Sobolev spaces $H^{k}$ and $W^{k, p}$ . The space $L_{0}^{2} (Ω)$ will denote a subspace of $L^{2} (Ω)$ with mean-zero functions. The inner product on $L^{2} (Ω)$ as well as on $L^{2} (Ω)^{d}$ is denoted by $(\cdot, \cdot)$ . To improve readability, we omit the superscript $d$ when having for example $L^{2} (Ω)^{d}$ appear as subscript to norms. We also introduce the following function spaces

V=\Set→v∈C∞0(Ω)d∇⋅→v=0,→V0=¯¯¯¯VL2,→V1=¯¯¯¯VH1,

(3)

where the notation in the last line denotes the completion of the space $V$ with respect to the $L^{2} (Ω)^{d}$ and $H^{1} (Ω)^{d}$ topology, respectively. Notice that functions in ${\to V}^{1}$ have zero boundary conditions in the trace sense. Alternatively we have

→V1=\set→v∈H10(Ω)d∇⋅→v=0

by (Galdi, 2011, Theorem III.4.1).

We define the vector-valued Laplace operator

- Δ : D (Δ) \to L^{2} (Ω)^{d},

where the domain $D (Δ)$ is understood with respect to $L^{2} (Ω)^{d}$ and is given as

D(Δ)=\set→v∈H10(Ω)dΔ→v∈L2(Ω).

If the domain $Ω$ is convex, then the standard $H^{2} (Ω)$ regularity implies $D (Δ) = H_{0}^{1} (Ω)^{d} \cap H^{2} (Ω)^{d}$ . In addition, we introduce the space ${\to V}^{2}$ as

{\to V}^{2} = {\to V}^{1} \cap D (Δ) .

We will also use the following Helmholtz decomposition (cf. (Temam, 1977, Chapter I, Theorem 1.4) and (Galdi, 2011, Theorem III.1.1))

L^{2} (Ω)^{d} = {\to V}^{0} \oplus \nabla (H^{1} (Ω) \cap L_{0}^{2} (Ω)) .

(4)

As usual we define the Helmholtz projection $P : L^{2} (Ω)^{d} \to {\to V}^{0}$ (often called the Leray projection) as the $L^{2}$ -projection from $L^{2} (Ω)^{d}$ onto ${\to V}^{0}$ . Using $P$ and $- Δ$ , we define the Stokes operator $A : {\to V}^{2} \to {\to V}^{0}$ as

A = - P Δ |_{{\to V}^{2}} .

(5)

The operator $A$ is a self adjoint, densely defined and positive definite operator on ${\to V}^{0}$ . We note that $D (A) = {\to V}^{2}$ . There holds

(A \to v, \to v) = ∥ \nabla \to v ∥_{L^{2} (Ω)}^{2} \geq λ_{0} ∥ \to v ∥_{L^{2} (Ω)}^{2} \to v \in {\to V}^{2},

where $λ_{0} > 0$

is the smallest eigenvalue of the Laplace operator

- Δ

given by

λ_{0} = inf v \in H_{0}^{1} (Ω) \frac{∥ \nabla v ∥_{L^{2} (Ω)}^{2}}{∥ v ∥_{L^{2} (Ω)}^{2}} .

(6)

Similar to the Laplace operator, for convex polyhedral domains $Ω$ we have the following $H^{2}$ regularity bound due to Dauge (1989); Kellogg & Osborn (1976)

∥ \to v ∥_{H^{2} (Ω)} \leq C ∥ A \to v ∥_{L^{2} (Ω)}, \forall \to v \in {\to V}^{2} .

(7)

2.2 Stokes resolvent problem

The key to our analysis is the spectral representation of the semigroup generated by $A$ . For that we consider the Stokes resolvent problem for $\to f \in L^{2} (Ω)^{d}$


$z \to u - Δ \to u + \nabla p$	$= \to f$	$in Ω,$	(8a)
$\nabla \cdot \to u$	$= 0$	$in Ω,$	(8b)
$\to u$	$= \to 0$	$on \partial Ω .$	(8c)

Here $z \in Σ_{θ, ¯ ω}$ , which is defined as

Σθ,¯ω=\setc∈Cc≠¯ω and % |arg(c−¯ω)|<θ.

(9)

The solution $(\to u, p)$ to creftype 8 is a complex valued function in $H_{0}^{1} (Ω)^{d} \times L_{0}^{2} (Ω)$ as complex valued function spaces with a hermitian inner product. In our situation we are interested in the case of $θ \in (π / 2, π)$ and $¯ ω \in [- λ_{0}, 0]$ with $λ_{0} > 0$ from creftype 6. The operator $A$ is sectorial. In particular, for every $θ \in (π / 2, π)$ there exists a constant $C = C_{θ}$ such that for all $z \in Σ_{θ, ¯ ω}$ with $¯ ω \in [- λ_{0}, 0]$ and $(\to u, p)$ being the solution of creftype 8 with $\to f \in L^{2} (Ω)^{d}$ there holds the following resolvent estimate

∥ \to u ∥_{L^{2} (Ω)} \leq \frac{C}{| z - ¯ ω |} ∥ P \to f ∥_{L^{2} (Ω)} .

(10)

Proof.

It is straightforward to check, that every self-adjoint positive operator, which is densely defined on a Hilbert space is sectorial. This applies to the operator $A - ¯ ω Id$ for all $¯ ω \in [- λ_{0}, 0]$ , which results in the resolvent estimate creftype 10. Below we provide a direct proof for the discrete version of the operator $A$ , see Section 3.2, which is also applicable here. ∎

Using the Stokes operator $A$ creftype 5 one can rewrite the resolvent estimate creftype 10 as

∥ (z + A)^{- 1} P \to f ∥_{L^{2} (Ω)} \leq \frac{C}{| z - ¯ ω |} ∥ P \to f ∥_{L^{2} (Ω)} .

(11)

The resolvent estimates in $L^{p}$ norms for $p \neq 2$ are also known. For example, for $d = 3$ on Lipschitz domains, Shen (2012) has shown a resolvent estimate for some interval of $p$ satisfying $| 1 / p - 1 / 2 | < 1 / 6 + ε$ , for $ε > 0$ . On smooth $C^{3}$ domains it is known to hold even for $p = \infty$ (cf. Abe et al. (2015)). However, the extension to non-smooth convex domains is still an open problem and it even appears in a collection of open problems (cf. (Maz’ya, 2018, Problem 66)).

2.3 Weak formulation and regularity

In this section we discuss the weak formulation and the regularity of the transient Stokes problem (1). We will use the notation $L^{s} (I; X)$ for the corresponding Bochner space with a Banach space $X$ . Moreover, we will use also the standard notation $H^{1} (I; X)$ . The inner product in $L^{2} (I; L^{2} (Ω)^{d})$ and in $L^{2} (I; L^{2} (Ω)^{d})$ is denoted by $(\cdot, \cdot)_{I \times Ω}$ . We will also use the notation $(f, g)_{I \times Ω}$ for the corresponding integral for $f \in L^{s} (I; L^{2} (Ω)^{d})$ and $g \in L^{s^{'}} (I; L^{2} (Ω)^{d})$ with $1 \leq s \leq \infty$ and the dual exponent $s^{'}$ .

For the application of Galerkin finite element methods in space and time we will require a space-time weak formulation of the transient Stokes equations with respect to both variables, velocity $\to u$ and pressure $p$ . In a standard variational setting, e.g., with $f \in L^{2} (I; ({\to V}^{1})^{'})$ or $f \in L^{1} (I; L^{2} (Ω)^{d})$ , this is not possible, since only distributional pressure can be expected in general, see Section 2.3 below. Therefore, we will first introduce the (standard) weak formulation on the divergence free space, then we discuss regularity issues and introduce a velocity-pressure weak formulation based on a slightly stronger assumption on the data. Let $\to f \in L^{1} (I; L^{2} (Ω)^{d})$ and ${\to u}_{0} \in {\to V}^{0}$ . Then there exists a unique solution $\to u \in L^{2} (I; {\to V}^{1}) \cap C (¯ I, {\to V}^{0})$ with $\partial_{t} \to u \in L^{1} (I; {\to V}^{0}) + L^{2} (I; ({\to V}^{1})^{'})$ fulfilling $\to u (0) = {\to u}_{0}$ and

⟨ \partial_{t} \to u, \to v ⟩ + (\nabla \to u, \nabla \to v)_{I \times Ω} = (\to f, \to v)_{I \times Ω} for all \to v \in L^{2} (I; {\to V}^{1}) \cap L^{\infty} (I; {\to V}^{0}) .

(12)

Here, $⟨ \partial_{t} \to u, \to v ⟩$ for $\partial_{t} \to u = {\to w}_{1} + {\to w}_{2} \in L^{1} (I; {\to V}^{0}) + L^{2} (I; ({\to V}^{1})^{'})$ and $\to v \in L^{2} (I; {\to V}^{1}) \cap L^{\infty} (I; {\to V}^{0})$ is understood as

⟨ \partial_{t} \to u, \to v ⟩ = ({\to w}_{1}, \to v)_{I \times Ω} + ⟨ {\to w}_{2}, \to v ⟩_{L^{2} (I; ({\to V}^{1})^{'}) \times L^{2} (I; {\to V}^{1})} .

There holds the estimate

∥ \nabla \to u ∥_{L^{2} (I; L^{2} (Ω))} + ∥ \to u ∥_{C (¯ I; L^{2} (Ω))} \leq C (∥ \to f ∥_{L^{1} (I; L^{2} (Ω))} + ∥ {\to u}_{0} ∥_{{\to V}^{0}}) .

Proof.

For the existence of the solution under the stated assumptions and with the corresponding regularity we refer to (Temam, 1977, Chapter III, Theorem 1.1) and its extension to the case $\to f \in L^{1} (I; L^{2} (Ω)^{d})$ on page 264. The notion of the solution in Temam (1977) is formulated in the almost everywhere sense on $I$ , from which the formulation creftype 12 follows directly by integration in time. The uniqueness of $\to u$ solving creftype 12 is also obtained in the standard manner choosing $\to v = \to u$ for $\to f = 0$ and ${\to u}_{0} = 0$ . ∎

Another possibility to formulate the notion of the weak solution is to assume $\to f \in L^{2} (I; ({\to V}^{1})^{'})$ . Since we require in the sequel the additional assumption $\to f \in L^{s} (I; L^{2} (Ω)^{d})$ for some $s > 1$ we prefer to use the formulation from Section 2.3. Under the assumptions from Section 2.3 the existence of the corresponding pressure $p$ can be shown only in the following distributional sense. There exists $P \in C (¯ I, L^{2} (Ω))$ such that the Stokes system holds in the distributional sense for $\to u$ solving creftype 12 and $p = \partial_{t} P$ . Especially one can not expect in general $p \in L^{1} (I \times Ω)$ , cf. (Temam, 1977, Chapter III, p. 267).

In the following, we will discuss some additional regularity for the solution. On the one hand we need a slightly more regularity in order to be able to introduce the pressure $p$ as a function, cf. Section 2.3. Moreover, additional regularity beyond $\to u \in C (¯ I; L^{2} (Ω)^{d})$ is required if we use the best approximation result creftype 2 for providing (optimal) error estimates, see Section 7. It is well known, cf. again Temam (1977), that in the sense of Section 2.3 the equation creftype 12 can be understood as an abstract parabolic problem

	$\partial_{t} \to u + A \to u$	$= P \to f for a.a. t \in I,$		(13)
	$\to u (0)$	$= {\to u}_{0},$		(13)

with the Stokes operator $A$ defined in creftype 5.

Furthermore, note that by Section 2.2, the operator $A$ is sectorial and thus a generator of an analytic semigroup (Lunardi, 1995, Definition 2.0.1, 2.0.2). For the Hilbert space setting it has then been shown in de Simon (1964) that this is equivalent to having a maximal regularity estimate of the form

∥ \partial_{t} \to u ∥_{L^{s} (I; L^{2} (Ω))} + ∥ A \to u ∥_{L^{s} (I; L^{2} (Ω))} \leq C_{s} ∥ \to f ∥_{L^{s} (I; L^{2} (Ω))}

(14)

for problem creftype 1 with ${\to u}_{0} = 0$ , $1 < s < \infty$ and $\to f \in L^{s} (I; L^{2} (Ω)^{d})$ . For more details, we refer to (Sohr, 2014, Chapter IV, Theorem 1.6.3). In Section 5 we derive a respective estimate for a fully discrete version of creftype 14, a so called discrete maximal parabolic regularity result based on ideas from Leykekhman & Vexler (2017a).

For proving (optimal) error estimates in Section 7 we will use the maximal parabolic estimate creftype 14 for $s \to \infty$ . To this end we will need precise dependence of the constant $C_{s}$ on $s$ from (Ashyralyev & Sobolevskii, 1994, Chapter 1, eq. (3.9), Theorem 3.2). Moreover, we require this regularity result also for the case of non-homogeneous initial conditions.

To state this result, we consider the space of initial data ${\to V}_{1 - \frac{1}{s}}^{0}$ for $1 < s < \infty$ as in (Ashyralyev & Sobolevskii, 1994, Chapter 1, Section 3.3). The Banach space ${\to V}_{1 - \frac{1}{s}}^{0}$ with the norm

∥ {\to v}_{0} ∥_{{\to V}_{1 - \frac{1}{s}}^{0}} = {(\int_{0}^{1} ∥ A exp (- t A) {\to v}_{0} ∥_{{\to V}^{0}}^{s} d t)}_{0}^{1 / s} + ∥ {\to v}_{0} ∥_{{\to V}^{0}}

(15)

contains all functions ${\to v}_{0} \in {\to V}^{0}$ such that for a solution $\to u$ to the transient Stokes problem with right-hand side $\to f = 0$ and initial data ${\to u}_{0} = {\to v}_{0}$ it holds $A \to u \in L^{s} (I; {\to V}^{0})$ .

Let $1 < s < \infty$ , $\to f \in L^{s} (I; L^{2} (Ω)^{d})$ and ${\to u}_{0} \in {\to V}_{1 - \frac{1}{s}}^{0}$ .Then the solution $\to u$ to the problem

	$\partial_{t} \to u + A \to u$	$= P \to f for a.a. t \in I,$		(16)
	$\to u (\to x, 0)$	$= {\to u}_{0},$		(17)

fulfills $\partial_{t} \to u, A \to u \in L^{s} (I; L^{2} (Ω)^{d})$ . Moreover, there is a constant $C$ independent on $s$ , $f$ and $u_{0}$ such that

(18)

Proof.

This follows from (Ashyralyev & Sobolevskii, 1994, Chapter 1, Theorems 3.2, 3.7) since $A$ is the generator of an analytic semigroup. ∎

If the domain is polyhedral/polygonal and convex then Section 2.3 provides $\to u \in L^{s} (I; H^{2} (Ω)^{d})$ and the estimate

∥ \partial_{t} \to u ∥_{L^{s} (I; L^{2} (Ω))} + ∥ \to u ∥_{L^{s} (I; H^{2} (Ω))} \leq \frac{C s^{2}}{s - 1} (∥ \to f ∥_{L^{s} (I; L^{2} (Ω))} + ∥ {\to u}_{0} ∥_{{\to V}_{1 - \frac{1}{s}}^{0}}) .

There holds ${\to V}^{1} ↪ {\to V}_{\frac{1}{2}}^{0}$ . This follows from the fact that for the homogeneous problem ( $\to f = 0$ ) with ${\to u}_{0} \in {\to V}^{1}$ the following estimate holds

∥ A \to u ∥_{L^{2} (I; L^{2} (Ω))} \leq 2 ∥ A^{\frac{1}{2}} \to u_{0} ∥_{L^{2} (Ω)} = 2 ∥ {\to u}_{0} ∥_{{\to V}^{1}} .

The above inequality is stated, e.g., in (Sohr, 2014, Chapter IV, Theorem 1.5.2). For the representation of the norm on ${\to V}^{1}$ by $A^{\frac{1}{2}}$ see, e.g., (Sohr, 2014, Chapter III, Lemma 2.2.1). Therefore, for the range $1 < s \leq 2$ it is sufficient to assume ${\to u}_{0} \in {\to V}^{1}$ for the estimate in Section 2.3.

If $u_{0} \in {\to V}^{2}$ there holds for every $1 < s < \infty$

∥ {\to u}_{0} ∥_{{\to V}_{1 - \frac{1}{s}}^{0}} \leq ∥ A {\to u}_{0} ∥_{L^{2} (Ω)} .

We can argue as follows. Since ${\to u}_{0} \in {\to V}^{2}$ , we have that $A$ commutes with $exp (- t A)$ (cf. (Sohr, 2014, Chapter II, eq. (3.2.19))) and due to the boundedness of $exp (- t A)$ in the operator norm (cf. (Sohr, 2014, Chapter IV, eq. (1.5.8))) we can conclude using definition creftype 15:

	$∥ {\to u}_{0} ∥_{{\to V}_{1 - \frac{1}{s}}^{0}}$	$= {(\int_{0}^{1} ∥ A exp (- t A) {\to u}_{0} ∥_{{\to V}^{0}}^{s} d t)}_{0}^{1 / s} + ∥ {\to u}_{0} ∥_{{\to V}^{0}}$		(19)
		$\leq {(\int_{0}^{1} ∥ exp (- t A) ∥_{{\to V}^{0} \to {\to V}^{0}}^{s} ∥ A {\to u}_{0} ∥_{{\to V}^{0}}^{s} d t)}_{0}^{1 / s} + ∥ {\to u}_{0} ∥_{{\to V}^{0}} \leq C ∥ A {\to u}_{0} ∥_{L^{2} (Ω)} .$		(20)

Therefore we have the following version of the maximal parabolic regularity estimate

∥ \partial_{t} \to u ∥_{L^{s} (I; L^{2} (Ω))} + ∥ A \to u ∥_{L^{s} (I; L^{2} (Ω))} \leq \frac{C s^{2}}{s - 1} (∥ \to f ∥_{L^{s} (I; L^{2} (Ω))} + ∥ A {\to u}_{0} ∥_{L^{2} (Ω)}) .

which we will use in particular for $s \to \infty$ .

The next theorem provides the space-time weak formulation in both variables, velocity and pressure. Please note, that no additional regularity of the domain is required and the assumption $f \in L^{1} (I; L^{2} (Ω)^{d})$ from Section 2.3 is only slightly strengthened to $\to f \in L^{s} (I; L^{2} (Ω)^{d})$ for some $s > 1$ .

Let $\to f \in L^{s} (I; L^{2} (Ω)^{d})$ for some $1 < s < \infty$ and ${\to u}_{0} \in {\to V}_{1 - \frac{1}{s}}^{0}$ .Then there exists a unique solution $(\to u, p)$ with

\to u \in L^{2} (I; {\to V}^{1}) \cap C (¯ I, {\to V}^{0}), \partial_{t} \to u, A \to u \in L^{s} (I; L^{2} (Ω)^{d}) and p \in L^{s} (I; L_{0}^{2} (Ω))

fulfilling $\to u (0) = {\to u}_{0}$ and

(\partial_{t} \to u, \to v)_{I \times Ω} + (\nabla \to u, \nabla \to v)_{I \times Ω} - (p, \nabla \cdot \to v)_{I \times Ω} + (\nabla \cdot \to u, ξ)_{I \times Ω} = (\to f, \to v)_{I \times Ω}

(21)

for all

\to v \in L^{2} (I; H_{0}^{1} (Ω)^{d}) \cap L^{\infty} (I; L^{2} (Ω)^{d}) and ξ \in L^{2} (I; L_{0}^{2} (Ω)) .

There holds the estimate

∥ p ∥_{L^{s} (I; L^{2} (Ω))} \leq \frac{C s^{2}}{s - 1} (∥ \to f ∥_{L^{s} (I; L^{2} (Ω))} + ∥ {\to u}_{0} ∥_{{\to V}_{1 - \frac{1}{s}}^{0}}) .

Proof.

We take the unique solution $\to u$ with $\partial_{t} \to u, A \to u \in L^{s} (I; L^{2} (Ω)^{d})$ from Section 2.3 and $\to u \in L^{2} (I; {\to V}^{1}) \cap C (¯ I, {\to V}_{0})$ from Section 2.3. To prove the existence of the corresponding pressure $p$ we consider for almost every $t \in I$ the element $\to g (t) \in H^{- 1} (Ω)^{d}$ ,

⟨ \to g (t), w ⟩ = (\to f (t), w) - (\partial_{t} \to u (t), w) - (\nabla \to u (t), \nabla w), w \in H_{0}^{1} (Ω)^{d},

i.e.,

\to g (t) = \to f (t) - \partial_{t} \to u (t) + Δ \to u (t) \in H^{- 1} (Ω)^{d} .

This element is well defined due to $\to f (t), \partial_{t} \to u (t) \in L^{2} (Ω)^{d}$ and $\to u (t) \in {\to V}^{1} ↪ H_{0}^{1} (Ω)$ for almost every $t \in I$ . Moreover, there holds by creftype 12

⟨ \to g (t), w ⟩ = 0 \forall w \in {\to V}^{1}

for almost all $t \in I$ , since creftype 12 holds also pointwise almost everywhere. Therefore, we can apply (Temam, 1977, Chapter I, Proposition 1.1), which ensures the existence of a distribution $p (t)$ with

\nabla p (t) = \to g (t)

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in the distributional sense for almost every $t \in I$ . By (Temam, 1977, Chapter I, Proposition 1.2) we have $p (t) \in L_{0}^{2} (Ω)$ and

∥ p (t) ∥_{L^{2} (Ω)} \leq C ∥ \to g (t) ∥_{H^{- 1} (Ω)} .

Using the definition of $\to g$ we obtain $p \in L^{s} (I; L_{0}^{2} (Ω))$ and

	$∥ p ∥_{L^{s} (I; L^{2} (Ω))} \leq C ∥ \to g ∥_{L^{s} (I; H^{- 1} (Ω))}$	$\leq C (∥ \to f ∥_{L^{s} (I; L^{2} (Ω))} + ∥ \partial_{t} \to u ∥_{L^{s} (I; L^{2} (Ω))} + ∥ \nabla \to u ∥_{L^{s} (I; L^{2} (Ω))})$
		$\leq \frac{C s^{2}}{s - 1} (∥ \to f ∥_{L^{s} (I; L^{2} (Ω))} + ∥ {\to u}_{0} ∥_{{\to V}_{1 - \frac{1}{s}}^{0}}),$

where we have used $∥ \nabla \to v ∥_{L^{2} (Ω)} \leq C ∥ A \to v ∥_{L^{2} (Ω)}$ for every $\to v \in {\to V}^{1}$ and Section 2.3. With this regularity we obtain from creftype 22 and the definition of $\to g$

(- p, \nabla \cdot \to v)_{I \times Ω} = (\to f, \to v)_{I \times Ω} - (\partial_{t} \to u, \to v)_{I \times Ω} - (\nabla \to u, \nabla \to v)_{I \times Ω}

for all $\to v \in L^{2} (I; H_{0}^{1} (Ω)^{d}) \cap L^{\infty} (I; L^{2} (Ω)^{d})$ . Furthermore, it holds

(\nabla \cdot \to u, ξ)_{I \times Ω} = 0 \forall ξ \in L^{2} (I; L_{0}^{2} (Ω))

by $\to u \in L^{2} (I; {\to V}^{1})$ . This results in the stated weak formulation. ∎

Let the assumptions of Section 2.3 be fulfilled. Let in addition the domain $Ω$ be convex. Then we have $p \in L^{s} (I, H^{1} (Ω))$ and the corresponding estimate holds

∥ p ∥_{L^{s} (I; H^{1} (Ω))} \leq \frac{C s^{2}}{s - 1} (∥ \to f ∥_{L^{s} (I; L^{2} (Ω))} + ∥ {\to u}_{0} ∥_{{\to V}_{1 - \frac{1}{s}}^{0}}) .

Proof.

By convexity of $Ω$ we obtain $\to u \in L^{s} (I; H^{2} (Ω)^{d})$ and the corresponding estimate, see Section 2.3. Then, we have

\to g (t) = \to f (t) - \partial_{t} \to u (t) + Δ \to u (t) \in L^{2} (Ω)^{d} .

for almost all $t \in I$ in the notation of the proof of Section 2.3. This leads to the desired regularity and to the estimate. ∎

For regularity beyond these estimates, we want to highlight (Heywood & Rannacher, 1982, Corollary 2.1), where the authors show that bounds for, e.g., $\nabla^{3} \to u$ , $\partial_{t t} \to u$ , go hand in hand with the need of the data ${\to u}_{0}$ , $\to f$ and initial pressure $p_{0}$ (defined as ${lim}_{t \to 0} p (t)$ ) satisfying a nonlocal compatibility condition for $t \to 0$ at the boundary, which is potentially difficult to verify.

3 Spatial discretization and discrete resolvent estimates

In this section we consider the discrete version of the operators presented in the previous section.

3.1 Spatial discretization

Let ${T_{h}}$ be a family of triangulations of $¯ Ω$ , consisting of closed simplices, where we denote by $h$ the maximum mesh-size. Let ${\to X}_{h} \subset H_{0}^{1} (Ω)^{d}$ and $M_{h} \subset L_{0}^{2} (Ω)$ be a pair of compatible finite element spaces, i.e., them satisfying a uniform discrete inf-sup condition,

sup {\to v}_{h} \in {\to X}_{h} \frac{(q_{h}, \nabla \cdot {\to v}_{h})}{∥ \nabla {\to v}_{h} ∥_{L^{2} (Ω)}} \geq β ∥ q_{h} ∥_{L^{2} (Ω)} \forall q_{h} \in M_{h},

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with a constant $β > 0$ independent of $h$ . We introduce the usual discrete Laplace operator $- Δ_{h} : {\to X}_{h} \to {\to X}_{h}$ by

(- Δ_{h} {\to z}_{h}, {\to v}_{h}) = (\nabla {\to z}_{h}, \nabla {\to v}_{h}), \forall {\to z}_{h}, {\to v}_{h} \in {\to X}_{h} .

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To define a discrete version of the Stokes operator $A$ , we first define the space of discretely divergence-free vectors ${\to V}_{h}$ as

→Vh=\Set→vh∈→Xh(∇⋅→vh,qh)=0∀qh∈Mh.

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Using this space we can define the discrete Leray projection $P_{h} : L^{1} (Ω)^{d} \to {\to V}_{h}$ to be the $L^{2}$ -projection onto ${\to V}_{h}$ , i.e.,

(P_{h} \to u, {\to v}_{h}) = (\to u, {\to v}_{h}) \forall {\to v}_{h} \in {\to V}_{h} .

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Using $P_{h}$ , we define the discrete Stokes operator $A_{h} : {\to V}_{h} \to {\to V}_{h}$ as $A_{h} = - P_{h} Δ_{h} |_{{\to V}_{h}}$ . By this definition we have that for ${\to u}_{h} \in {\to V}_{h}$ , $A_{h} {\to u}_{h} \in {\to V}_{h}$ fulfills

(A_{h} {\to u}_{h}, {\to v}_{h}) = (\nabla {\to u}_{h}, \nabla {\to v}_{h}), \forall {\to v}_{h} \in

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Proof.

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3 Spatial discretization and discrete resolvent estimates

3.1 Spatial discretization