A posteriori error estimations for mixed finite-element approximations to the Navier-Stokes equations
A posteriori estimates for mixed finite element discretizations of the Navier-Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier-Stokes equations can be reduced to the estimation of the error in a steady Stokes problem. As a consequence, any available procedure to estimate the error in a Stokes problem can be used to estimate the error in the nonlinear evolutionary problem. A practical procedure to estimate the error based on the so-called postprocessed approximation is also considered. Both the semidiscrete (in space) and the fully discrete cases are analyzed. Some numerical experiments are provided.
We consider the incompressible Navier–Stokes equations
in a bounded domain () with a smooth boundary subject to homogeneous Dirichlet boundary conditions on . In (1), is the velocity field, the pressure, and a given force field. For simplicity in the exposition we assume, as in , , , , , that the fluid density and viscosity have been normalized by an adequate change of scale in space and time.
Let and be the semi-discrete (in space) mixed finite element (MFE) approximations to the velocity and pressure , respectively, solution of (1) corresponding to a given initial condition
We study the a posteriori error estimation of these approximations in the and norm for the velocity and in the norm for the pressure. To do this for a given time , we consider the solution (, ) of the Stokes problem
We prove that and are approximations to and whose errors decay by a factor of faster than those of and ( being the mesh size). As a consequence, the quantities and , are asymptotically exact indicators of the errors and in the Navier-Stokes problem (1)–(2).
Furthermore, the key observation in the present paper is that () is also the MFE approximation to the solution of the Stokes problem (1). Consequently, any available procedure to a posteriori estimate the errors in a Stokes problem can be used to estimate the errors and which, as mentioned above, coincide asymptotically with the errors and in the evolutionary NS equations. Many references address the question of estimating the error in a Stokes problem, see for example , , , , , ,  and the references therein. In this paper we prove that any efficient or asymptotically exact estimator of the error in the MFE approximation to the solution of the steady Stokes problem (1) is also an efficient or asymptotically exact estimator, respectively, of the error in the MFE approximation to the solution of the evolutionary Navier-Stokes equations (1)–(2).
For the analysis in the present paper we do not assume to have more than second-order spatial derivatives bounded in up to initial time , since demanding further regularity requires the data to satisfy nonlocal compatibility conditions unlikely to be fulfilled in practical situations , . The analysis of the errors and follows closely  where MFE approximations to the Stokes problem (1) (the so-called postprocessed approximations) are considered with the aim of getting improved approximations to the solution of (1)–(2) at any fixed time . In this paper we will also refer to (, ) as postprocessed approximations although they are of course not computable in practice and they are only considered for the analysis of a posteriori error estimators. The postprocessed approximations to the Navier-Stokes equations were first developed for spectral methods in , , ,  and also developed for MFE methods for the Navier-Stokes equations in , , .
For the sake of completeness, in the present paper we also analyze the use of the computable postprocessed approximations of  for a posteriori error estimation. The use of this kind of postprocessing technique to get a posteriori error estimations has been studied in ,  and  for nonlinear parabolic equations excluding the Navier-Stokes equations. We refer also to  where the so-called Stokes reconstruction is used to a posteriori estimate the errors of the semi-discrete in space approximations to a linear time-dependent Stokes problem. We remark that the Stokes reconstruction of  is exactly the postprocessing approximation () in the particular case of a linear model.
In the second part of the paper we consider a posteriori error estimations for the fully discrete MFE approximations and , ( for ) obtained by integrating in time with either the backward Euler method or the two-step backward differentiation formula (BDF). For this purpose, we define a Stokes problem similar to (1) but with the right-hand-side depending now on the fully discrete MFE approximation (problem (69)–(70) in Section 4 below). We will call time-discrete postprocessed approximation to the solution of this new Stokes problem. As before, is not computable in practice and it is only considered for the analysis of a posteriori error estimation.
Observe that in the fully discrete case (which is the case in actual computations) the task of estimating the the error of the MFE approximation becomes more difficult due to the presence of time discretization errors , which are added to the spatial discretization errors . However we show in Section 4 that if temporal and spatial errors are not very different in size, the quantity correctly esimates the spatial error because the leading terms of the temporal errors in and get canceled out when subtracting , leaving only the spatial component of the error. This is a very convenient property that allows to use independent procedures for the tasks of estimating the errors of the spatial and temporal discretizations. We remark that the temporal error can be routinely controlled by resorting to well-known ordinary differential equations techniques. Analogous results were obtained in  for fully discrete finite element approximations to evolutionary convection-reaction-diffusion equations using the backward Euler method.
As in the semidiscrete case, a key point in our results is again the fact that the fully discrete MFE approximation to the Navier-Stokes problem (1)–(2) is also the MFE approximation to the solution of the Stokes problem (69)–(70). As a consequence, we can use again any available error estimator for the Stokes problem to estimate the spatial error of the fully discrete MFE approximations to the Navier-Stokes problem (1)–(2).
Computable mixed finite element approximations to , the so-called fully discrete postprocessed approximations, were studied and analyzed in  where we proved that the fully discrete postprocessed approximations maintain the increased spatial accuracy of the semi-discrete approximations. The analysis in the second part of the present paper borrows in part from . Also, we propose a computable error estimator based on the fully discrete postprocessed approximation of  and show that it also has the excellent property of separating spatial and temporal errors.
The rest of the paper is as follows. In Section 2 we introduce some preliminaries and notation. In Section 3 we study the a posteriori error estimation of semi-discrete in space MFE approximations. In Section 4 we study a posteriori error estimates for fully discrete approximations. Finally, some numerical experiments are shown in Section 5.
2 Preliminaries and notations
We will assume that is a bounded domain in , of class , for . When dealing with linear elements ( below) may also be a convex polygonal or polyhedral domain. We consider the Hilbert spaces
endowed with the inner product of and , respectively. For integer and , we consider the standard spaces, , of functions with derivatives up to order in , and . We will denote by the norm in , and will represent the norm of its dual space. We consider also the quotient spaces with norm .
We recall the following Sobolev’s imbeddings : For , there exists a constant such that
For , (3) holds with .
The following inf-sup condition is satisfied (see ), there exists a constant such that
where, here and in the sequel, denotes the standard inner product in or in .
Let be the projector onto . We denote by the Stokes operator on :
Applying Leray’s projector to (1), the equations can be written in the form
where for , in .
We shall use the trilinear form defined by
It is straightforward to verify that enjoys skew-symmetry:
Let us observe that for .
Let us consider for and the operators and , which are defined by means of the spectral properties of (see, e.g., [13, p. 33], ). Notice that is a positive self-adjoint operator with compact resolvent in . An easy calculation shows that
where, here and in what follows, when applied to an operator denotes the associated operator norm.
for some constants and . We shall also assume that there exists a constant such that
Finally, we shall assume that for some
so that, according to Theorems 2.4 and 2.5 in , there exist positive constants and such that the following bounds hold:
where and for some . Observe that for , we can take and . For simplicity, we will take these values of and .
Let , be a family of partitions of suitable domains , where is the maximum diameter of the elements , and are the mappings of the reference simplex onto .
Let , we consider the finite-element spaces
where denotes the space of polynomials of degree at most on . As it is customary in the analysis of finite-element methods for the Navier-Stokes equations (see e. g., , , , , ) we restrict ourselves to quasiuniform and regular meshes , so that as a consequence of [12, Theorem 3.2.6], the following inverse inequality holds for each
where , .
and the so-called mini-element  when , where , and . Here, is spanned by the bubble functions , , defined by , if and 0 elsewhere, where denote the barycentric coordinates of . For these elements a uniform inf-sup condition is satisfied (see ), that is, there exists a constant independent of the mesh grid size such that
We remark that our analysis can also be applied to other pairs of LBB-stable mixed finite elements (see [16, Remark 2.1]).
The approximate velocity belongs to the discrete divergence-free space
which is not a subspace of . We shall frequently write instead of whenever the value of plays no particular role.
Let be the discrete Leray’s projection defined by
We will use the following well-known bounds
We will denote by the discrete Stokes operator defined by
Let be the solution of a Stokes problem with right-hand side , we will denote by the so-called Stokes projection (see ) defined as the velocity component of solution of the following Stokes problem: find such that
The following bound holds for :
The proof of (16) for can be found in . For the general case, must be such that the value of satisfies . This can be achieved if, for example, is piecewise of class , and superparametric approximation at the boundary is used . Under the same conditions, the bound for the pressure is 
For a domain of class we also have the bound (see )
In what follows we will apply the above estimates to the particular case in which is the solution of the Navier–Stokes problem (1)–(2). In that case is the discrete velocity in problem (14)–(15) with . Note that the temporal variable appears here merely as a parameter, and then, taking the time derivative, the error bound (16) can also be applied to the time derivative of changing , by , .
where if and if .
see [29, Proposition 3.2].
3 A posteriori error estimations. Semidiscrete case
Let us consider the MFE approximation at any time to obtained by solving (21)–(22). We consider the postprocessed approximation in which is the solution of the following Stokes problem written in weak form
for all and . We remark that the MFE approximation to is also the MFE approximation to the solution of the Stokes problem (25)–(26). In Theorems 1 and 2 below we prove that the postprocessed approximation is an improved approximation to the solution of the evolutionary Navier-Stokes equations (1)–(2) at time . Although, as it is obvious, is not computable in practice, it is however a useful tool to provide a posteriori error estimates for the MFE approximation at any desired time . In Theorem 1 we obtain the error bounds for the velocity and in Theorem 2 the bounds for the pressure. The improvement is achieved in the norm when using the mini-element () and in both the and norms in the cases .
In the sequel we will use that for a forcing term satisfying (8) there exists a constant , depending only on , and , such that the following bound hold for :
The following inequalities hold for all and , see [29, (3.7)]:
The proof of Theorem 1 requires some previous results which we now state and prove.
We will use the fact that for , from where it follows that
where the constant is independent of .
Due to the equivalence (30) and and since for it is sufficient to prove
for or . We follow the proof [5, Lemma 3.1] where a different threshold assumption is assumed. We do this for , since the case is similar but yet simpler. We write
where . We first observe that
where, in the last inequality, we have used that thanks to Sobolev’s inequality (3) we have . Similarly,
The proof of the case in (34) is finished if we show that for , both and are bounded in terms of and the value in the threshold assumption (31). To do this, we will use the inverse inequality (11) and the fact that the Stokes projection satisfies that
for some constant (see for example the proof of Lemma 3.1 in ). We have
where in the last inequality we have applied (11), and, similarly,
In the sequel we consider the auxiliary function solution of
According to [16, Remark 4.2] we have
for some constant . The following lemma provides a superconvergence result.
Since for we have
where , it follows that
Applying (32) we have , so that taking into account that
it follows that
Since applying [16, Lemma 4.2] we obtain
Then by writing and applying (13) and (37), the proof is finished if we check that the threshold condition (31) holds for and . In view of (37), (13) and the inverse inequality (11) we have indeed that , for . In the case of the threshold condition holds due to (