Abstract
This paper is devoted to the study of numerical approximation for a class of two-dimensional Navier-Stokes equations with slip boundary conditions of friction type. The objective is to establish the well-posedness and stability of the numerical scheme in -norm and -norm for all positive time using the Crank-Nicholson scheme in time and the finite element approximation in space. The resulting variational structure dealing with is in the form of inequality, and obtaining -estimate is more involved because of the presence of the nondifferentiable term appearing at the boundary where slip occurs. We prove that the numerical scheme is stable in and -norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition. Finally, some numerical simulations are presented to illustrate our theoretical analysis.
1. Introduction
We consider the Navier-Stokes equations of viscous incompressible fluids: with the impermeability boundary condition and the slip boundary condition
On the remaining part of the boundary, , we assume Dirichlet boundary condition, i.e.,
Finally, the initial condition is given by
Here, is a bounded domain, with boundary . It is assumed that is made of two components , and with , and . is a positive quantity representing the viscosity coefficient, is the initial velocity, and is the barrier or threshold function. The velocity of the fluid is and stands for the pressure, while is the external force. Furthermore, is the outward unit normal to the boundary of , is the tangential component of the velocity , and is the tangential traction. Of course, is the Cauchy stress tensor, where is the identity matrix, and .
It can easily be shown that (4) is equivalent to where the symbol is the subdifferential of the real value function , with . We recall that if is the Hilbert space with , then,
It should be mentioned that different boundary conditions describe different physical phenomena. The slip boundary condition of friction type can be justified by the fact that frictional effects of the fluid at the pores of the solid can be very important. Many studies have focused on the properties of the solution of the resulting boundary value problem, for example, existence, uniqueness, regularity, and continuous dependence on data, for Stokes, Navier-Stokes, and Brinkman–Forchheimer equations under such boundary condition. Details can be found in [1–4] among others. In [5], a generalization of the boundary condition (4) is formulated and analyzed for the steady Stokes flow, while the case of Navier-Stokes equations has been examined in [6]. There are numerous works devoted to the development of efficient schemes for the nonstationary Navier-Stokes problem dealing with Dirichlet or periodic boundary conditions; some works can be found in [7–11]. It should also be mentioned that there are other works dealing with Navier-Stokes equations with time fractional derivatives (see for instance [12] and references therein). For the time fractional operators, details can be found in [13, 14].
The subject of the present work is to establish the well-posedness and stability of the numerical scheme on -norm and -norm for all positive times of the two-dimensional problems (1)-(4) using the Crank-Nicholson scheme in time and the finite element approximation in space. The resulting variational structure dealing with is in the form of inequality, and obtaining -estimate is more involved because of the presence of the nondifferentiable term appearing at the boundary where slip occurs.
2. Preliminaries and Variational Formulation
In this section, we introduce notations and some results that will be used throughout our work. We also formulate various weak formulations and discuss (recall) some existence results. For the mathematical setting of the problem, we need to introduce the following spaces:
is the dual space of , and the duality pairing between and is denoted by . Throughout the paper, we assume that is bounded, convex planar domain with polygonal boundary. As usual, stands for the function . Next, we introduce the Stokes operator by following the approach adopted in [15, 16]. We denote by the Helmholtz projection operator, which is bounded projection associated to the Helmholtz decomposition of . We define the Stokes operator as follows such that , with domain given as follows, . Now, assuming that is and is , then since , and one has where is the first eigenvalue of the Stokes operator . It should be noticed that thanks to (10), is a norm on equivalent to the usual -norm.
The Stokes operator is self-adjoint, positive with a compact inverse which is self-adjoint as a mapping from to .
We recall some classical bilinear and trilinear forms (see [17, 18])
We denote by a bilinear operator from into such that
The bilinear form satisfies the inf-sup conditions; i.e., there exists a positive constant such that
As a readily obtainable consequence of Korn’s inequality (11), there exists a positive constant such that
The trilinear form is continuous on and enjoys the following properties:
We will make reference to the following inequalities:
We assume that , and we set . We also assume that . With the above notations, we introduce the following variational formulation for (1)-(6): Find such that and for a.e. , with where .
Note that since the bilinear form satisfies the inf-sup condition (14), the variational inequality problem (23) is equivalent to the following:
Find such that
and for a.e. , with
The problem of existence and uniqueness of (24)-(25) can be stated as follows and has been proved in Kashiwabara [3].
Theorem 1. Assume Then, there exists a unique solution of problem (24)-(25) such that Let be a uniform partition of with a given time step . We consider a time discretization of (24)-(25) using the Crank-Nicolson scheme. Find such that and for all where . We want to show that the solution of (28)-(29) is uniformly bounded for all , in both the - and -norms. In what follows, we discretize in space and derived such a result assuming some kind of stability condition.
3. Numerical Scheme
For the spatial discretization, we introduce the general framework as in, e.g., [18, 19]. We consider a family of finite element spaces , each of which is endowed with two scalar products, and , with the corresponding norms, and which mimic the - and -norms. These norms are related as follows: where is independent of and is such that
We assume that the operator satisfies the same properties on as on . We also assume that a trilinear continuous form enjoys the same properties on with the constant independent of .
We introduce the so-called restriction operators and assume that, if , then, with the constant being independent of (see, e.g., [18]).
As for the temporal discretization, we consider the following scheme, a discrete version of (24)-(25): Find such that and for all
Remark 2. For existence and uniqueness of the solution of (34)-(35), one observes that the variational inequality (34)-(35) can be seen as a case of following modified variational formulation associated to the stationary Navier-Stokes equations with slip boundary condition type. where , , and . Following [6, 20, 21], (36) admits a unique solution .
4. The -Stability
We start this section by performing the stability analysis of the scheme (34)-(35) in and show that the solution is uniformly bounded, provided that a stability CFL-type condition is satisfied.
Lemma 3. Let be arbitrarily fixed and assume that and Then, for any integer , we have
Proof. We first establish the relation (52) below and next use it to handle the proof by induction. First, let and in (35); one has
that is
Using relation (20), we have
Using Cauchy-Schwarz inequality, (31) and (21), we write
and the right hand side of (44) is bounded as follows:
To bound the nonlinear term in (44), we write it as
Using (18) and (19), we have
Hence, using (16),(17), and (21) and recalling (31), we obtain the following bounds:
Gathering (44)-(51), we obtain
Note that according to CFL-condition (37), if
then
We now use the induction. It is clear that (38) and (39) hold for . Then assuming that (38) holds for , for , we see under the assumption of Lemma 3 that (39) holds for . Then (52), together with (39) and (37), implies
If we drop the last term on the left hand side and rewrite the remaining equation with replaced by and take the sum with , for some , we obtain
and hence, (41) and (42) hold for all .
Now using (30), relation (55) implies
Using recursively (57), we obtain
Thus, (38) holds for .
5. The -Stability
For proving the uniform bound of in for all , we first show that it is bounded on any finite interval of time. Then we extend the result to the infinite time using the discrete uniform Gronwall lemma.
Lemma 4. Let be arbitrarily fixed and assume that , and assume also that the CFL-condition (37) is satisfied. Assume that also satisfies Assume also that for some the following is true: where , is given by (65) and is given by (64). Then, where and are positive constants independent of and .
Proof. Let in (35); we obtain
that is
Using relations (16) and (21) and the uniform bound (39), we majorize the trilinear form as
where .
where .
Using Cauchy-Schwarz inequality and relation (21), we have that
Gathering relations (63)-(66), we find
from which we obtain
From (68), we have either
or
where
Let us show that with our assumption, (70) is impossible. Taking in (35), we find
Using (30) and (21), we bound the right hand side of (72) by
Since is a trilinear form, we can rewrite the nonlinear term in (72) as
and using property (17), we obtain the following bounds:
Employing (40), we bound the last term of the left hand side of (72) by
Gathering (72)-(76) and recalling (39), we obtain
and hence,
from which we find, using (60),
(79) contradicts (70), and therefore, we obtain
where .
Since (by (60)) and
we obtain, using (59) and (60) and the fact that ,
with appropriate choice of constants and .
To prove that scheme (35) is conditionally stable on a finite interval of time, we need the following discrete Gronwall lemma [22].
Lemma 5. Discrete Gronwall Lemma.
Given , an integer , and positive sequences , , and such that
we have
Proof. Using recursively (83), we derive and since for all , the conclusion of the lemma follows.
Proposition 6. Estimates on a finite interval of time.
Let and be fixed, and let . Assume that, besides the CFL-condition (37), also satisfies
where
is a monotonically increasing function in all its arguments and is given in (95) below and .
Then,
(a)Relation (58) holds for all (integer part of )(b)
Proof. Let and let be such that (37) and (86) are satisfied.
We will use induction on . If , assumption (86) implies
Thus, the conclusion (61) of Lemma 4 holds for . Now assume that (60) holds for , for some . Hence, (61) holds for . If we rewrite (61) as (83) with
and noting that, using (41), we have
and therefore,
Similarly, for , we have
Using (86) and recalling that , the last term of (83) can be bounded as
Then, Lemma 5 and relations (90)-(93) imply
where
Using (94) and recalling assumption (86), it is easily checked that condition (60) holds for , and by the same Lemma 5, we have (61) that holds for .
To prove the uniform bound of for all , we will repeatedly apply Proposition 6 on different intervals of time, considering different initial values, and we will need the following discrete uniform Gronwall lemma, a generalized version of the discrete uniform Gronwall lemma of Shen [22], whose proof can be found in [7].
Lemma 7. Discrete uniform Gronwall lemma.
Given , positive integers such that , , and positive sequences , and such that
Assume also that for any satisfying then we have
Theorem 8. Uniform bound of for all . Let , and assume that , where . Also let be arbitrarily fixed and assume that, besides the CFL-condition (37), also satisfies
where are defined above and is given in (107) below.
Then, we have
where is a continuous function defined on , increasing.
Moreover, there exists an such that
Proof. In order to derive uniform bounds for all , we apply Proposition 6 on successive intervals of time, with different initial values. On each interval considered, we obtain a bound which depends on the norm and on the length of the interval. Using the discrete uniform Gronwall lemma, we majorize the norm of the initial values by a constant , and recalling the fact that is an increasing function of its arguments, we obtain a bound independent on the initial value considered.
First using (33) and (99) and since is a decreasing function of its arguments, we can apply Proposition 6 with to obtain
To extend the bound (103) to , we apply again Proposition 6, namely, depends on the discrete initial value; we want to bound independent of and .
Rewrite (102) in the form of (96) with , and . Then, we apply Lemma 7 with , , to obtain the bound of . For , using (41), we have
Then, Lemma 7, together with the assumption , yields
Taking into account assumption (99) on the time step , relation (107), and the fact that is an increasing function of its arguments, we apply Proposition 6 with and initial data . We obtain that the relation (61) holds for all , and
Applying again Lemma 7 with , and , we obtain
Iterating the above procedure, we find
and recalling (103), we conclude
As for the beyond which is bounded independent of , we can evidently take (see (110)). This completes the proof of the theorem.
6. Numerical Experiments
Let us explain our numerical experiments. We assume , the boundary of which consists of two portions and given by
The time interval is given by with . For the triangulation of , we employ a uniform mesh, where denotes the division number of each side of the domain. The implementation is done by extending the Matlab code developed in [23, 24]. In all the examples presented, the velocity and pressure will be approximated by element. Let us consider
The initial condition is given by .
The functions and are chosen such that defined above is the solution of (1)-(5).
It is easy to verify that the solution satisfies on , on . By direct computations, we have
On the other hand, from the slip boundary conditions (5), we have then we find from (104) that for the given function :
Indeed, it is observable in Figure 1, slip and non-slip condition on the boundary. In fact in Figure 1(a), and we see the manifestation of the slip due to the adherence of the flow at the boundary, whereas in Figure 1(b), and no slip occurs.
(a)
(b)
To analyze the convergence rate, we simulated the same problem. Since we do not know the explicit exact solution when , we employ the approximate solutions with as the reference solutions , and we compute the -norm for velocity of the difference of the reference solution and the approximate solution .
For the convergence with respect to the mesh size , we choose and we solve problem (35) with different values of (). In Figure 2(a), we plot the of -errors against .
(a)
(b)
For the convergence with respect to the time step , is fixed () and we solve problem (35) with different time steps . Figure 2(b) shows the plots of -error norm against .
7. Conclusions
In this paper, we have proposed and analyzed the Crank-Nicolson scheme for the two-dimensional Navier-Stokes equations driven by slip boundary conditions of friction type. We established the well-posedness and stability of the numerical scheme in -norm and -norm for all positive time using the Crank-Nicholson scheme in time and the finite element method in space. We have proven that the numerical scheme is stable in and -norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the referees for their constructive remarks and for helpful comments that improved the quality of this paper. The part of this work is based on the first author’s thesis and has been submitted as repository at the University of Pretoria, South Africa.