Spectral discretization of the time - dependent Navier - Stokes problem with mixed boundary conditions

: In this work, we handle a time - dependent Navier - Stokes problem in dimension three with a mixed boundary conditions. The variational formulation is written considering three independent unknowns: vorticity, velocity, and pressure. We use the backward Euler scheme for time discretization and the spectral method for space discretization. We present a complete numerical analysis linked to this variational for - mulation, which leads us to a priori error estimate


Introduction
Many physical cases can be modeled by Navier-Stokes equations with mixed boundary conditions, for instance, a reservoir of water covered by a membrane or a flow in a piping network.Different types of nonstandard boundary conditions are suggested in the pioneering papers [12,13,24].In this paper, we consider non-stationary Navier-Stokes equations provided with a mixed boundary condition.On some parts of the boundary, we provide an homogeneous Dirichlet condition of the velocity.On the other part, the normal component of the velocity and the tangential components of the vorticity are given.
Let Ω be a bounded simply-connected open domain in 3 , with a Lipschitz continuous connected boundary Ω ∂ and T 0; [ ] an interval in , where T is a positive constant.We consider a partition without overlap of Ω ∂ into two connected parts Γ m and Γ, Ω Γ Γ and Γ Γ , where the index m " " in Γ m stands for membrane.We denote by x y z x , , ( ) and n the unit outward normal vector to Ω on its boundary Ω ∂ .We intend to work with the following time-dependent Navier where f represents a density of body forces and the viscosity ν is a positive constant.The unknowns are the velocity u and the pressure P of the fluid.
Using the basic idea in [22,31], by introducing the vorticity ω curl u = as a new unknown (see also [4,6,23]), the convection term can be written as follows: | | ∇ = × + and system (1) is fully equivalent as follows: where the dynamical pressure p is defined by: p P u 1 2 .

| | = +
We assume that Γ Γ m ∂ = ∂ a Lipschitz-continuous sub-manifold of Ω ∂ .A priori error analysis of the finite element discretization of the stationary Stokes and Navier-Stokes problem with mixed boundary conditions have first been performed in [5,14,31] and extended to a posteriori error analysis for the time-dependent Stokes and Navier-Stokes problem with mixed boundary conditions in [17,18].The case of the discretization by spectral, and spectral element methods, relying on this type of boundary conditions are studied in several papers (see [1,2,7,8,10,11,19,21]).
We propose in this work the discretization of the considered problem using Euler's implicit scheme with respect to the time variable combined with the spectral method with respect to the space variables.Unlike boundary conditions on normal component of velocity and tangential components of vorticity, which are defined on the whole boundary [4,6,30], mixed boundary conditions produce a lack of regularity of the solution especially for the vorticity.Consequently, we propose a new formulation inspired from [4,30] for the two-dimensional domain and [6] for the three-dimensional domain.We prove that the time semidiscrete problem has a solution under the conditions, where Γ m is of class 1,1 C or convex with sufficiently large viscosity.
For the spectral discretization, we assume that Ω is a cube, where Γ m is one of its faces.Since the vorticity is a potential-vector in three dimension, it is required to choose appropriate polynomial spaces, which are the spectral analogs of Nédélec's finite elements spaces on cubic three-dimensional meshes (see [29]).The full spectral discrete problem is constructed using the Galerkin method with numerical integration [15,16].We then prove its well-posedness.To deal with the nonlinear aspects of the problem, the theorem of Brezzi et al. [20] is used.We thus prove optimal error estimates for the vorticity and the velocity.The choice of the discrete space of the pressure gives us a non-optimal inf-sup condition.Hence, we have the luck of optimality on the approximation of the pressure.This article is organized as follows: Section 2 presents the variational formulation of the continuous problem.Section 3 is devoted to the well-posedness of time semi-discrete problem by using Euler's implicit scheme in time.The well-posedness of the full spectral discrete problem is detailed in Section 4. Finally, the analysis of the a priori error estimate for velocity, vorticity, and pressure is presented in Section 5.

The continuous variational formulation
Before writing the variational formulation of the problem (2), we begin by defining the following Sobolev spaces: which is a Banach space equipped with the following norm and semi-norm: is an Hilbert space equipped with the scalar product, on Ω, and Ω D( ) the space of indefinitely differentiable functions with a compact support in Ω.We consider H div, Ω ( ) the domain of the div operator as follows: provided with the norm: We remind (see [25,Chap I,Sec 2]) that the normal trace operator is defined from , such that for any scalar function w smooth enough, we have: Then, we define the kernel of the normal trace operator in H div, Ω ( ) as follows: We also consider H curl, Ω ( ) the domain of the curl operator: provided with the norm: The tangential trace operator is defined from H curl, Ω ( ) into H Ω 3 Then, the trivial definition of the velocity's space is This space is provided with the semi-norm We also consider the following spaces, which depend on time.Let S a separable Banach space.We consider T S 0, ; ) is a Banach space related to the norm: ∂ is the time partial derivative up the order j of the function v. Let also the spaces and H T S 0, ; s ( )is an Hilbert space when it is equipped with the following scalar product: Finally, we define also S L( ) the Banach space of the linear and continuous functions from S into provided with the norm Now, we suppose that the data f belong to the space L T 0, ; Ω 2 ( ( )) ′ , where Ω ( )′ is the dual space of Ω ( ).We denote by t t ωt ω t pt p t u u , , , , , ( ) ( ) ( ) ( ) ( ) ( ) = ⋅ = ⋅ = ⋅ and we consider the following variational formula- tion on the interval T where .,.
( ) is a solution of the problem (2) in the sense of distributions.
Proof 1.According to the definition of the space Ω ( ), each solution ω t t p t u , , ( ( ) ( ) ( )) of problem (7) satisfies equations (4) and (5) of problem (2).Equations (2) and (3) in problem (7) give us equations ( 2) in the first equation of problem (7).Then, we conclude the first equation of problem (2).Finally, we consider a regular function We note that a , ; . Then, its kernel is a closed subspace of Ω ( ), which coincides with the space of divergence free functions in Ω ( ).We also consider the kernel of the form c .,.; .
The existence of a solution for the problem (9) requires the continuity of the nonlinear term K , ; ( ) ⋅ ⋅ ⋅ , which requires the following assumption.
Assumption 1.We assume that the domain Ω has a 1,1 C boundary or is a polyhedron with no re-entrant corners inside Γ m .
( ) (see [9], Thm 2.17 for the proof).When Ω is a polyhedron, we recall that the space of restrictions of functions of Ω ( ) to Ω V ⧹ , where V is a neighborhood of the re-entrant corners of (see ( [14], Lem.2.5) for the proof and more details).So, to give a sense to the nonlinear term K , ; ( ) ⋅ ⋅ ⋅ , we suppose Assumption 1 satisfied.

The time semi-discrete problem
The aim of this section is the time discretization of problem (2) by the implicit Euler scheme.We consider a partition of the interval . The regularity parameter σ τ is defined as follows: , the time semi-discrete problem issued from Euler's implicit method is written as follows: Problem (11) is equivalent to the following variational formulation: where t f f ., , , )is solution of problems ( 12) and (13), we conclude that ω u , k k ( ) belongs to W and is solution of the following reduced problem: ˘, ;  , ; .
The main difficulty now is to prove that problem (12)-( 14) admits a solution.We observe that the form a ˘, ; and the functional F is linear and continuous on V. Now we show the two following properties of the form a ˘.,.; .( ).
and there exists α Proof 2. We prove the positivity of the form a ˘.,.; .( ).Assume that in W, we obtain Then v 0 = , which is in contradiction with the fact that v 0 V\{ } ∈ .Now we prove the inf-sup condition (16).
We integrate by part and since ω curl u u , div 0 .
Using (17) combined with the following inequality To go further in the proof that problems (12)-( 14) admit a solution, we investigate the properties of the trilinear form K .,.; .( ).
Lemma 2. If Assumption 1 is satisfied, the trilinear form K , ; Proof 3. From Hölder inequality and the fact that the space where C * only depends on the domain Ω. □ By using the following equality proved by Green formula we conclude for u V ∈ , the antisymmetry property of the trilinear form K , ; The existence of solution for problems (12)-( 14) is proved for a large enough viscosity ν with respect to the norm of the functional F.
Proposition 2. For any data f u , 0 ( ) belongs to . Knowing u k 1 − , if Assumption 1 holds, and there exists a constant C ◇ such that then problems (12)-( 14) Proof 4. Let the iterative sequence ω u , and ω u , Thanks to properties (15) and (16) of the form a ˘, ; ( ) ⋅ ⋅ ⋅ and the continuity of the trilinear form K .,.; .( ), problem (22) has a unique solution (see [11] for more details about the proof).We consider where C * is the continuity constant defined in (18).Let us now prove by induction on j 0 ≥ that the sequence ω u , )) is bounded by r k in the norm of the space W, j ω r u 0, .
The estimation (24) is clearly true for j 0 = , and we assume that it holds for the iteration j 1 − .If v u j k = in the problem (22), we obtain using properties (16) and (18) Cauchy-Schwarz inequality and the following Young inequality: Then, from (6) and the fact that the estimation (24) holds for the iteration j 1 − , we have , and combining (23) and (20), with C C 4 2 = ◇ * , we obtain (24).Furthermore, we show for any j 2 ≥ that .
= − − and using the continuity of the trilinear form K , ; ( ) ⋅ ⋅ ⋅ combined with 25, we obtain and by (24), we obtain Now thanks to (23), we conclude that ≥ is a Cauchy sequence in the closet space W, so that it converges to ω u , k k ( ) in W. By passing to the limit in (22), it is readily checked that ω u , k k ( ) is solution of problems (12)-( 14).In order to prove estimation (21), we take v u k = in the equality (14), since the trilinear form K , ; ( ) ⋅ ⋅ ⋅ verifies the antisymmetry property (19), we obtain By using Cauchy-Schwarz inequality combined with Young inequality (25) and the fact that u V k ∈ , we obtain Summing the inequality (26) over k, we conclude the estimate (21).
, such that (20) is satisfied.At each time step, knowing u k 1 − problems (12) and (13) has at most a solution ω p u , , To prove the existence of p k , we define for any v Ω ( ) ∈ the following functional: The functional Σ k is linear, continuous on Ω ( ) and vanishes on the space V.Then, using the inf-sup condition (10), there exists p k in L Ω 0 2 ( ) such that: ( )be two solutions of problems ( 12) and (13) such that ω ω , ; , ;  , ;  , ; .
The antisymmetry property of the trilinear form K .,.; .( ) gives that K ω u u , ; 0 Then by applying the continuity of K .,.; .

The full spectral discrete problem
In this section, we are interested to the spectral discretization of problems (12) and (13).Henceforth, we assume that The spectral discretization is done by the same way of the Nédélec's finite element method (see [29,Sec 2]).We introduce Ω n m s , , ( ) the space of the restriction on Ω of the polynomials of degree n in the x direction, m in the y direction, and s in the z direction.
Let N 2 ≥ be an integer, we define the space which approximates the space of the velocity Ω ( ) and the space of the discrete vorticity.For the approximation of pressure in the space L Ω 0 2 ( ), we consider the , which we will define explicitly later.For a better accuracy of the approximation of the trilinear form K .,.; .( ), we will do a over numeric integration (see [28] and [8]).We choose the integer M N ≥ equal to the integer part of , where α is a fixed real number in the interval 0, 1 of polynomials with degree n ≤ , then we have: We also recall the following important inequality (see [15]): Relying on this formula, we introduce the discrete scalar product on Ω M ( ) defined by: for continuous functions φ and ψ on Ω ¯φ ψ and we consider I N the Lagrange interpolating operator at the nodes ξ ξ ξ , , Assume that the data f and u 0 are, respectively, continuous on and Ω, we construct the discrete problem from (12) and (13) using the Galerkin method combined with numerical integration.
where the bilinear forms a ˘, ; By using Cauchy-Schwarz inequality and (31), it follows that the bilinear forms a ˘, ; − is linear and continuous on N .Moreover, as a consequence of the exactness of property (30), the bilinear forms b , ( ) ⋅ ⋅ and b , N ( ) . While the trilinear form K ., .;N ( ) is defined as follows: , .
Now in order to define the discrete space N of the pressure, we start by introducing the set of the spurious modes SP N of the discrete bilinear form b , N ( ) where λN [ ] stands for the integer part of λN .We set where the polynomial χ N is defined by: and L N is the Lagrange polynomial defined on the interval 1, 1 ] [ − .The proof of the next lemma is given in ( [7], Lem.a.1).
Lemma 3. The space SP N is spanned by the polynomials where ψ N runs through Now since SP N it has been characterized, the space N is chosen as the subspace of L Ω Ω N 0 2 1 ( ) ( ) ∩ − equal to the orthogonal of the space SP N SP .
We intend now to show the well-posedness of problems (33) and (34).We begin by introducing the discrete kernel V N of the bilinear form b , N ( ) This kernel coincides with V N ∩ , i.e., the space of divergence-free polynomials in N (see [7], Lem.3.2., for the proof).
Similarly, we introduce the discrete kernel W N of the form c , ; N ( ) ⋅ ⋅ ⋅ as follows: We remark that for any solution ω p u , ; Knowing ˘, ;  , ; .
The well-posedness of the discrete reduced problems (33)-( 40) is proved by the same arguments as for the continuous reduced problems (12)-( 14).We show the following two properties of the form a ˘, , N ( ) ⋅ ⋅ ⋅ .Lemma 4. For k K 1 ≤ ≤ , the positivity property of the form a ˘, ; N ( ) ⋅ ⋅ ⋅ holds: From the property (31), we obtain ω τ ν a u v curlv ˘, ; .
, we conclude that v 0 N = , using property (6) and the fact that v N is divergence free.□ The form a ˘, ; N ( ) ⋅ ⋅ ⋅ also verifies the following inf-sup condition.Lemma 5.For any On the other hand, setting ω curl u By Cauchy-Schwarz inequality combined with (31), .
We therefore prove the desired inf-sup condition (43).□ Proposition 3.For any data f u , 0 Proof 8. We define the mapping ϕ N from W N into its dual space by: .
We are equipped W N with the following norm: We check that ϕ N is continuous since the space W N is finite dimensional, and from the antisymmetry property (19) of the trilinear form K , ; N ( ) ⋅ ⋅ ⋅ , we obtain . N Then, by Young's inequality (25), we have   = and by summing over k, we obtain the desired result (45).□ Due to the definition of the space N in (37), we consider the following inf-sup condition proved in ([7], lem.a.6): There exists a positive constant γ independent of N such that q q γN q b v v , sup , .
Thanks to this inf-sup condition (47), the existence of the pressure follows the same proof as given in Theorem 1, then from Proposition 3, and we have the next full result.
For any data L T f 0, ; Ω 2 ( ( )) ∈ ′, we define the operator S such that SF k is the solution ω u , .

k k 1 =
where K is a positive integer.Let τ t t k − − , the time step, and the k-tuple τ τ τ τ , , , k 1 2 . It follows from assumption (1) and condition (20) that the pair ω (33)  and (34), the pair ω u , of the following reduced discrete problem:

=−
Spectral discretization of the time-dependent Navier-Stokes problem  1459 we check that ϕ on the sphere of W N with radius ϖ N .Then, by using Brouwer's fixed point theorem (see ([25], Chap.IV, Cor.1.1)), we conclude that problems (33)-(40) have a solution ω u , in the first equality of (34), and considering the antisymmetry property (19) of the trilinear form K , ; By using property (31), Cauchy-Schwarz inequality and integrating by part, we obtain: By using Young's inequality (25) leads to,

Theorem 4 . 1 . 1 −
For any data f u at each time step k, problems (33) and (34) has a solution ω p