In this paper we consider the following transient Stokes problem with no-slip boundary conditions,
Throughout this work, we assume that , , is a bounded polygonal/polyhedral Lipschitz domain, and . We will require some (weak) assumptions on the data, which essentially allow for a weak formulation including both velocity and pressure and for . 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
where is the fully discrete finite element approximation of the velocity , is an arbitrary function from the finite element approximation of the velocity spaces , is the Ritz projection for the stationary Stokes problem and 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 in the discrete space as well as the approximation of the stationary Stokes problem in . 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 , see Section 7.
Under the additional assumption of convexity of and some approximation properties of the discrete spaces we prove error estimates of the form
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 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 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
for , , and 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 . This estimate holds on a general Lipschitz domain . Assuming in addition the convexity of , we have
see Section 2.3 and Section 2.3. On the discrete level, we provide the corresponding estimates which hold even in the limit cases and 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
where is the discrete Stokes operator, see Section 5 for details and the precise formulation. Under the convexity assumption for the domain , similar to the continuous case, we also obtain
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 and its finite element analog . 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 and Sobolev spaces and . The space will denote a subspace of with mean-zero functions. The inner product on as well as on is denoted by . To improve readability, we omit the superscript when having for example appear as subscript to norms. We also introduce the following function spaces
where the notation in the last line denotes the completion of the space with respect to the and topology, respectively. Notice that functions in have zero boundary conditions in the trace sense. Alternatively we have
by (Galdi, 2011, Theorem III.4.1).
We define the vector-valued Laplace operator
where the domain is understood with respect to and is given as
If the domain is convex, then the standard regularity implies . In addition, we introduce the space as
As usual we define the Helmholtz projection (often called the Leray projection) as the -projection from onto . Using and , we define the Stokes operator as
The operator is a self adjoint, densely defined and positive definite operator on . We note that . There holds
is the smallest eigenvalue of the Laplace operatorgiven by
2.2 Stokes resolvent problem
The key to our analysis is the spectral representation of the semigroup generated by . For that we consider the Stokes resolvent problem for
Here , which is defined as
The solution to creftype 8 is a complex valued function in as complex valued function spaces with a hermitian inner product. In our situation we are interested in the case of and with from creftype 6. The operator is sectorial. In particular, for every there exists a constant such that for all with and being the solution of creftype 8 with there holds the following resolvent estimate
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 for all , which results in the resolvent estimate creftype 10. Below we provide a direct proof for the discrete version of the operator , see Section 3.2, which is also applicable here. ∎
The resolvent estimates in norms for are also known. For example, for on Lipschitz domains, Shen (2012) has shown a resolvent estimate for some interval of satisfying , for . On smooth domains it is known to hold even for (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 for the corresponding Bochner space with a Banach space . Moreover, we will use also the standard notation . The inner product in and in is denoted by . We will also use the notation for the corresponding integral for and with and the dual exponent .
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 and pressure . In a standard variational setting, e.g., with or , 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 and . Then there exists a unique solution with fulfilling and
Here, for and is understood as
There holds the estimate
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 on page 264. The notion of the solution in Temam (1977) is formulated in the almost everywhere sense on , from which the formulation creftype 12 follows directly by integration in time. The uniqueness of solving creftype 12 is also obtained in the standard manner choosing for and . ∎
Another possibility to formulate the notion of the weak solution is to assume . Since we require in the sequel the additional assumption for some we prefer to use the formulation from Section 2.3. Under the assumptions from Section 2.3 the existence of the corresponding pressure can be shown only in the following distributional sense. There exists such that the Stokes system holds in the distributional sense for solving creftype 12 and . Especially one can not expect in general , 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 as a function, cf. Section 2.3. Moreover, additional regularity beyond 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
with the Stokes operator defined in creftype 5.
Furthermore, note that by Section 2.2, the operator 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
for problem creftype 1 with , and . 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 . To this end we will need precise dependence of the constant on 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 for as in (Ashyralyev & Sobolevskii, 1994, Chapter 1, Section 3.3). The Banach space with the norm
contains all functions such that for a solution to the transient Stokes problem with right-hand side and initial data it holds .
Let , and .Then the solution to the problem
fulfills . Moreover, there is a constant independent on , and such that
This follows from (Ashyralyev & Sobolevskii, 1994, Chapter 1, Theorems 3.2, 3.7) since is the generator of an analytic semigroup. ∎
If the domain is polyhedral/polygonal and convex then Section 2.3 provides and the estimate
There holds . This follows from the fact that for the homogeneous problem () with the following estimate holds
The above inequality is stated, e.g., in (Sohr, 2014, Chapter IV, Theorem 1.5.2). For the representation of the norm on by see, e.g., (Sohr, 2014, Chapter III, Lemma 2.2.1). Therefore, for the range it is sufficient to assume for the estimate in Section 2.3.
If there holds for every
We can argue as follows. Since , we have that commutes with (cf. (Sohr, 2014, Chapter II, eq. (3.2.19))) and due to the boundedness of in the operator norm (cf. (Sohr, 2014, Chapter IV, eq. (1.5.8))) we can conclude using definition creftype 15:
Therefore we have the following version of the maximal parabolic regularity estimate
which we will use in particular for .
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 from Section 2.3 is only slightly strengthened to for some .
Let for some and .Then there exists a unique solution with
There holds the estimate
This element is well defined due to and for almost every . Moreover, there holds by creftype 12
in the distributional sense for almost every . By (Temam, 1977, Chapter I, Proposition 1.2) we have and
Using the definition of we obtain and
for all . Furthermore, it holds
by . This results in the stated weak formulation. ∎
Let the assumptions of Section 2.3 be fulfilled. Let in addition the domain be convex. Then we have and the corresponding estimate holds
For regularity beyond these estimates, we want to highlight (Heywood & Rannacher, 1982, Corollary 2.1), where the authors show that bounds for, e.g., , , go hand in hand with the need of the data , and initial pressure (defined as ) satisfying a nonlocal compatibility condition for 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 be a family of triangulations of , consisting of closed simplices, where we denote by the maximum mesh-size. Let and be a pair of compatible finite element spaces, i.e., them satisfying a uniform discrete inf-sup condition,
with a constant independent of . We introduce the usual discrete Laplace operator by
To define a discrete version of the Stokes operator , we first define the space of discretely divergence-free vectors as
Using this space we can define the discrete Leray projection to be the -projection onto , i.e.,
Using , we define the discrete Stokes operator as . By this definition we have that for , fulfills