Spectral Element Discretization for the Vorticity, the Velocity and the Pressure Formulation of the Axisymmetric Navier-Stokes Problem

Abstract

We deal with the Navier-Stokes equations set in a three-dimensional axisymmetric bounded domain with non standard boundary conditions which involve the normal component of the velocity and tangential component of the vorticity. The axisymmetric property of the domain allows to reduce the three-dimensional problem into a two-dimensional one. We write a variational formulation with three independent unknowns: the vorticity, the velocity and the pressure. For the discretization, we use the spectral element methods, which are well-adapted here. We show the well-posedness of the obtained formulations and we establish error estimates for the three unknowns which proves the convergence of the method.

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1 Introduction

We consider, in this paper, the Navier-Stokes problem set in a three-dimensional axisymmetric bounded domain and provided with non standard boundary conditions, which are given on the normal component of the velocity and tangential component of the vorticity. This problem reads:

$$\displaystyle\begin{array}{rcl} \left \{\begin{array}{rlll} -\nu \varDelta \tilde{ u} + (\tilde{u}.\nabla )\tilde{u} + \nabla \tilde{P}& =\tilde{ f}&\mbox{ in} &\tilde{\varOmega }, \\ \mathrm{div}\tilde{u}& = 0 &\mbox{ in} &\tilde{\varOmega }, \\ \tilde{u}.\tilde{n}& = 0 &\mbox{ on}&\partial \tilde{\varOmega }, \\ \mathrm{curl}\tilde{u} \wedge \tilde{ n}& = 0 &\mbox{ on}&\partial \tilde{\varOmega }.\end{array} \right.& &{}\end{array}$$

(1)

where $\tilde{\varOmega }$ is a bounded connected three-dimensional axisymmetric domain, the generic point in $\tilde{\varOmega }$ is given by cylindrical components $(r,\theta,z) \in \mathbb{R}_{+}\times ]-\pi,\pi ] \times \mathbb{R}$.

ν is the viscosity of the fluid, $\tilde{u} = (u_{r},u_{\theta },u_{z})$ the velocity, $\tilde{P}$ the pressure and $\tilde{f}$ is the data, which represent the density of body forces. When the data is axisymmetric, problem (1) is equivalent to two decoupled systems [9]. In the first one, the unknowns are the components u _r and u _z of the velocity and pressure P, we will focus on. The second is a Laplace problem where the unknown is the velocity component u _θ .

At first, this problem was studied in [1] but in an unspecified bounded domain, then it was taken again by Azaiez et al. [10] in a bounded domain included in ${\mathbb{R}}^{2}$ or ${\mathbb{R}}^{3}$ in formulation (u, p), though the formulation that we consider here deals with three unknowns: vorticity, velocity and pressure. The first numerical analysis relying on this formulation has been realized in [13] and [8] for finite element methods and it has been extended to the case of spectral methods in [3] and [10], using analogues of Nédélec’s finite elements [6].

The discretization method which we use here is the spectral element methods, which are well adapted in domain decomposition. The main tool for the analysis of the nonlinear discrete problem is the theorem of Brezzi, Rappaz and Raviart [5]. We first prove the existence of a discrete solution. Then, by combining the results in [5, 11] and [7], we establish error estimates between the continuous solution and the discrete one, for the three unknowns.

The paper is organized as follows. In the next section, we introduce the variational formulation corresponding to the Navier-Stokes problem and we derive the existence of a solution. In Sect. 3, we study the discrete problem and we prove the well-posedness of this problem. We derive error estimates between the continuous solution and the discrete one in Sect. 4.

2 The Vorticity, Velocity and Pressure Formulation

The domain $\tilde{\varOmega }$ is obtained by rotating a two-dimensional domain Ω around the axis {r = 0}. We note by Γ ₀ the intersection of the boundary ∂ Ω with the axis r = 0, Γ = ∂ Ω∖Γ ₀ and by n the normal to Γ in the plane (r, z). We introduce the vorticity ω as a new unknown: ω = curlu. The bidimensional problem resulting from (1) reads:

$$\displaystyle\begin{array}{rcl} \left \{\begin{array}{rlll} \nu \mathrm{curl_{r}}\omega + \omega \times u + \nabla P & = f &\mbox{ in} &\varOmega, \\ \mathrm{div_{r}}u& = 0 &\mbox{ in} &\varOmega, \\ \omega & =\mathrm{ curl}u&\mbox{ in} &\varOmega, \\ u \cdot n& = 0 &\mbox{ on}&\varGamma, \\ \omega & = 0 &\mbox{ on}&\varGamma.\\ \end{array} \right.& &{}\end{array}$$

(2)

The operators div_r, curl and curl_r are given by: for $u = (u_{r},u_{z})$,

$\mathrm{div_{r}}u = \partial _{r}u_{r} + {r}^{-1}u_{r} + \partial _{z}u_{z}$ and $\mathrm{curl}u = \partial _{r}u_{z} - \partial _{z}u_{r}$. And for any scalar function $\varphi$, we define $\mathrm{curl_{r}}\varphi = \left (\partial _{z}\varphi,-{r}^{-1}\partial _{r}(r\varphi )\right )$. We refer to [11], for details.

In order to write the variational formulation of problem (2), we define the following weighted Sobolev spaces: For all s in $\mathbb{Z}$ and m in $\mathbb{N}$:

$$\displaystyle\begin{array}{rcl} & & L_{s}^{2}(\varOmega ) = \left \{v:\varOmega \rightarrow \mathbb{R}\quad \mathrm{measurable}\quad /\int _{\varOmega }\vert v(r,z){\vert }^{2}{r}^{s}\mathit{drdz} < \infty \right \} {}\\ & & H_{1}^{m}(\varOmega ) = \left \{v \in L_{ 1}^{2}(\varOmega )\quad /\quad \partial _{ r}^{l}\partial _{ z}^{m-l}v \in L_{ 1}^{2}(\varOmega )\quad \forall 0 \leq l \leq m\right \}, {}\\ & & H_{1}(\mathrm{curl},\varOmega ) = \left \{v \in L_{1}^{2}{(\varOmega )}^{2}\quad /\quad \mathrm{curl}v \in L_{ 1}^{2}(\varOmega )\right \}, {}\\ & & H_{1}(\mathrm{div}_{r},\varOmega ) = \left \{v \in L_{1}^{2}{(\varOmega )}^{2}\quad /\quad \mathrm{div}_{ r}v \in L_{1}^{2}(\varOmega )\right \}, {}\\ & & H_{1}^{\diamond}(\mathrm{div_{ r}},\varOmega ) = \left \{v \in H_{1}(\mathrm{div_{r}},\varOmega )/\quad v \cdot n = 0\quad \mathrm{on}\quad \varGamma \right \}, {}\\ & & H_{1}(\mathrm{curl}_{r},\varOmega ) = \left \{\varphi \in L_{1}^{2}(\varOmega )\quad /\quad \mathrm{curl}_{ r}\varphi \in L_{1}^{2}{(\varOmega )}^{2}\right \}, {}\\ & & V _{1}^{1}(\varOmega ) = H_{ 1}^{1}(\varOmega ) \cap L_{ -1}^{2}(\varOmega )\quad \mathrm{and}\quad V _{ 1\diamond}^{1}(\varOmega ) = \left \{v \in V _{ 1}^{1}(\varOmega )\ /\quad v = 0\quad \mathrm{on}\quad \varGamma \right \}. {}\\ \end{array}$$

The spaces $V _{1}^{1}(\varOmega )$, $H_{1}(\mathrm{div}_{r},\varOmega )$ and $H_{1}(\mathrm{curl}_{r},\varOmega )$ are respectively provided with:

$$\displaystyle\begin{array}{rcl} & & \left \|v\right \|_{V _{1}^{1}(\varOmega )} = {(\left \|\partial _{r}v\right \|_{L_{1}^{2}(\varOmega )}^{2} + \left \|\partial _{ z}v\right \|_{L_{1}^{2}(\varOmega )}^{2} + \left \|v\right \|_{ L_{-1}^{2}(\varOmega )}^{2})}^{\frac{1} {2} }, {}\\ & & \left \|v\right \|_{H_{1}(\mathrm{div}_{r},\varOmega )} ={ \left (\left \|v\right \|_{L_{1}^{2}(\varOmega )}^{2} + \left \|\mathrm{div}_{ r}v\right \|_{L_{1}^{2}(\varOmega )}^{2}\right )}^{\frac{1} {2} }, {}\\ & & \left \|\varphi \right \|_{H_{1}(\mathrm{curl}_{r},\varOmega )} ={ \left (\left \|\varphi \right \|_{L_{1}^{2}(\varOmega )}^{2} + \left \|\mathrm{curl}_{ r}\varphi \right \|_{L_{1}^{2}{(\varOmega )}^{2}}^{2}\right )}^{\frac{1} {2} }. {}\\ \end{array}$$

We note that the two norms $\left \|.\right \|_{H_{1}(\mathrm{curl}_{r},\varOmega )}$ and $\left \|.\right \|_{V _{1}^{1}(\varOmega )}$ are equivalent on $V _{1}^{1}(\varOmega )$.The variational problem reads:Find $(\omega,u,p) \in V _{1\diamond}^{1}(\varOmega ) \times H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega ) \times L_{1,0}^{2}(\varOmega )$ such that:

$$\displaystyle\begin{array}{rcl} \left \{\begin{array}{rlll} a(\omega,u;v) + K(\omega,u;v) + b(v,p)& = \left \langle f,v\right \rangle,&\forall v \in H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega ), \\ b(u,q)& = 0, &\forall q \in L_{1,0}^{2}(\varOmega ), \\ c(\omega,u,\varphi )& = 0, &\forall \varphi \in V _{1\diamond}^{1}(\varOmega ).\end{array} \right.& &{}\end{array}$$

(3)

where $\left \langle.,.\right \rangle$ is the duality pairing between $H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )$ and its dual space. The forms a(. , . ; . ), b(. , . ) and c(. , . ; . ) are defined by:

$$\displaystyle\begin{array}{rcl} & & a(\omega,u,\theta ) =\nu \int _{\varOmega }(\theta.\mathrm{curl_{r}}\omega )(r,z)\mathit{rdrdz},\quad b(v,q) = -\int _{\varOmega }(\mathrm{div_{r}}v)q(r,z)\mathit{rdrdz}, {}\\ & & c(\omega,u,\varphi ) =\int _{\varOmega }\omega (r,z)\varphi (r,z)\mathit{rdrdz} -\int _{\varOmega }(u.\mathrm{curl_{r}}\varphi )(r,z)\mathit{rdrdz}. {}\\ \end{array}$$

and K is the trilinear form given by: $K(\omega,u;v) =\int _{\varOmega }\left (\omega \times u\right ).v(r,z)\mathit{rdrdz}.$

Using density results, we first prove that problems (2) and (3) are equivalent. To prove the existence and the uniqueness of the solution of problem (3), we define the two following kernels V and W:

$$\displaystyle\begin{array}{rcl} & & V = \left \{v \in H_{1}^{\diamond}(\mathrm{div_{ r}},\varOmega ),\quad \forall q \in L_{1,0}^{2}(\varOmega )\quad /\quad b(v,q) = 0\right \}, {}\\ & & W = \left \{(\vartheta,v) \in V _{1\diamond}^{1}(\varOmega ) \times V \quad /\quad \forall \varphi \in V _{ 1\diamond}^{1}(\varOmega ),\quad \quad c(\vartheta,v;\varphi ) = 0\right \}, {}\\ \end{array}$$

and the reduced problem: Find (ω, u) in W such that:

$$\displaystyle{ \forall v \in V,\quad a(\omega,u;v) + K(\omega,u;v) = \left \langle f,v\right \rangle. }$$

(4)

By using standard arguments and properties on the linear forms, proven in [3] and [11], we can prove the existence and uniqueness of a solution for problem (4).So for any function f in $H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )^{\prime}$ such that

$$\displaystyle\begin{array}{rcl} \begin{array}{lll} c{_{\diamond}\nu }^{-2}\left \|f\right \|_{H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )^{\prime}} < 1,\end{array} & &{}\end{array}$$

(5)

Problem (3) admits a unique solution (ω, u; p) in $V _{1\diamond}^{1}(\varOmega ) \times H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega ) \times L_{1,0}^{2}(\varOmega )$, such that

$$\displaystyle\begin{array}{rcl} & & \left \|\omega \right \|_{V _{1}^{1}(\varOmega )} + \left \|u\right \|_{H_{1}(\mathrm{div_{r}},\varOmega )} {+\nu }^{-1}\left \|p\right \|_{ L_{1}^{2}(\varOmega )} \\ & & \qquad \qquad \qquad \leq {c\nu }^{-1}\left \|f\right \|_{ H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )^{\prime}}\left (1 {+\nu }^{-2}\left \|f\right \|_{ H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )^{\prime}}\right ).{}\end{array}$$

(6)

3 Discrete Navier-Stokes Problem

From now on, we assume that Ω is the rectangle ]0, 1[×] − 1, 1[ and admits a partition without overlap into a finite number of subdomains:

$$\displaystyle{\overline{\varOmega } =\bigcup _{ k=1}^{K}\varOmega _{ k}\quad \mathrm{and}\quad \varOmega _{k} \cap \varOmega _{k^{\prime}} = \emptyset \quad,\quad 1 \leq k < k^{\prime} \leq K,\mbox{ such that:}}$$

• 1. Each Ω _k , 1 ≤ k ≤ K is a rectangle.

• 2. The intersection between two subdomains $\overline{\varOmega }_{k}$ and $\overline{\varOmega }_{k^{\prime}}$, 1 ≤ k < k′ ≤ K, if not empty, is either a vertex or a whole edge of both Ω _k and Ω _k′.

The discrete spaces $\mathbb{D}_{N}$, $\mathbb{C}_{N}$ and $\mathbb{M}_{N}$ which approximate, respectively, $H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )$, $V _{1\diamond}^{1}(\varOmega )$ and $L_{1,0}^{2}(\varOmega )$ are defined from local discrete ones, for an integer N ≥ 2 and 1 ≤ k ≤ K, by:$\quad \mathbb{D}_{N} = \left \{v_{N} \in H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega );\ v_{N}\vert _{\varOmega _{k}} \in \mathbb{P}_{N,N-1}(\varOmega _{k}) \times \mathbb{P}_{N-1,N}(\varOmega _{k}),\ 1 \leq k \leq K\right \}$, $\quad \quad \mathbb{C}_{N} = \left \{\varphi _{N} \in V _{1\diamond}^{1}(\varOmega );\ \varphi _{N}\vert _{\varOmega _{k}} \in \mathbb{P}_{N}(\varOmega _{k}),\,1 \leq k \leq K\right \}$ and $\quad \quad \quad \mathbb{M}_{N} = \left \{q_{N} \in L_{1,0}^{2}(\varOmega );\ q_{N}\vert _{\varOmega _{k}} \in \mathbb{P}_{N-1}(\varOmega _{k}),\ 1 \leq k \leq K\right \}.$ where $\mathbb{P}_{n,m}\left (\varOmega _{k}\right )$ is the space of restrictions to Ω _k of polynomials with degree ≤ n with respect to r and ≤ m with respect to z, for any nonnegative integers n and m.

To calculate the integrals involved in the discrete forms, we define $(\xi _{i},\rho _{i})$, 0 ≤ i ≤ N the nodes and weights of the Gauss-Lobatto quadrature formula on [−1, 1] for the measure d ζ and $(\zeta _{j},\omega _{j})$, 1 ≤ j ≤ N + 1 their analogues for the measure (1 +ζ)d ζ, see [9] for a more explicit definition, we need two different quadrature formulas. The quadrature formula on [−1, 1] is given by:

$$\displaystyle{ \forall \phi \in \mathbb{P}_{2N-1}([-1,1]),\quad \int _{-1}^{1}\phi (\xi )d\xi =\sum _{ i=0}^{N}\phi (\xi _{ i})\rho _{i}, }$$

(7)

and by setting $r = \frac{1} {2}(1+\zeta )$, we define the quadrature formula with the measure rdr:

$$\displaystyle{ \forall \phi \in \mathbb{P}_{2N-1}([0,1]),\quad \int _{0}^{1}\phi (r)\mathit{rdr} = \frac{1} {4}\sum _{j=1}^{N+1}\phi (r_{ j})\omega _{j}. }$$

(8)

We denote by $(\varOmega _{k})_{1\leq k\leq K_{0}}$ the rectangles such that $\partial \overline{\varOmega }_{k} \cap \varGamma _{0}\neq \emptyset$ and by $(\varOmega _{k})_{K_{0}+1\leq k\leq K}$ those such that $\partial \overline{\varOmega }_{k} \cap \varGamma _{0} = \emptyset$. Denoting by F _k the affine mapping that sends ]0, 1[×] − 1, 1[ onto Ω _k , 1 ≤ k ≤ K ₀ and sends ] − 1, 1[² onto Ω _k , K ₀ + 1 ≤ k ≤ K. We define the discrete scalar product: For all functions u and v such that $u_{k} = u\vert _{\varOmega _{k}}$ and $v_{k} = v\vert _{\varOmega _{k}}$ are continuous on $\overline{\varOmega }_{k}$, 1 ≤ k ≤ K, by:

$$\displaystyle\begin{array}{rcl} & & ((u,v))_{N} =\sum _{ k=1}^{K_{0} } \frac{\mathit{mes}(\varOmega _{k})} {4} \sum _{i=0}^{N}\sum _{ j=1}^{N+1}u \circ F_{ k}(r_{j},\xi _{i}).v \circ F_{k}(r_{j},\xi _{i})\rho _{i}\omega _{j}. {}\\ & & \qquad \qquad +\sum _{ k=K_{0}+1}^{K}\frac{\mathit{mes}(\varOmega _{k})} {4} \sum _{i=0}^{N}\sum _{ j=0}^{N}u \circ F_{ k}(\xi _{j},\xi _{i}).v \circ F_{k}(\xi _{j},\xi _{i})\rho _{i}\rho _{j}. {}\\ \end{array}$$

We denote by $I_{N}^{k}$, 1 ≤ k ≤ K, the Lagrange interpolation operators associated with the nodes $F_{k}(r_{j},\xi _{i})_{1\leq j\leq N+1,0\leq i\leq N}$ for 1 ≤ k ≤ K ₀ and with $F_{k}(\xi _{j},\xi _{i})_{0\leq j,i\leq N}$ for K ₀ + 1 ≤ k ≤ K, with values in $\mathbb{P}_{N}(\varOmega _{k})$, 1 ≤ k ≤ K. For each function ϕ continuous on $\bar{\varOmega }$, I _N ϕ denotes the function such that $I_{N}\phi \vert _{\varOmega _{k}} = I_{N}^{k}\phi$, 1 ≤ k ≤ K. Using the Galerkin method with numerical integration, we build from the continuous problem (3) the following discrete problem:

Find $(\omega _{N},u_{N};p_{N})$ in $\mathbb{C}_{N} \times \mathbb{D}_{N} \times \mathbb{M}_{N}$ such that

$$\displaystyle{ \left \{\begin{array}{rlll} a_{N}(\omega _{N},u_{N},v_{N}) + K_{N}(\omega _{N},&u_{N},v_{N}), \\ + b_{N}(v_{N},p_{N})& = \left (\left (f,v_{N}\right )\right )_{N},&\forall v_{N} \in \mathbb{D}_{N}, \\ b_{N}(u_{N},q_{N})& = 0, &\forall q_{N} \in \mathbb{M}_{N}, \\ c_{N}(\omega _{N},u_{N},\varphi _{N})& = 0, &\forall \varphi _{N} \in \mathbb{C}_{N}. \end{array} \right. }$$

(9)

where the bilinear forms a _N (. , . ; . ), b _N (. , . ) and c _N (. , . ; . ) are defined by:$a_{N}(\omega _{N},u_{N};v_{N}) =\nu ((\mathrm{curl_{r}}\omega _{N},v_{N}))_{N}$, $b_{N}(v_{N},q_{N}) = -((\mathrm{div_{r}}v_{N},q_{N}))_{N}$,$c_{N}(\omega _{N},u_{N},\varphi _{N}) = ((\omega _{N},\varphi _{N}))_{N} - ((u_{N},\mathrm{curl_{r}}\varphi _{N}))_{N}$, while the trilinear form K _N (. , . ; . ) is given by: $K_{N}(\omega _{N},u_{N};v_{N}) = \left (\left (\omega _{N} \times u_{N},v_{N}\right )\right )_{N}$. In order to prove the well-posedness of the discrete problem, we need to introduce the kernels:

$$\displaystyle\begin{array}{rcl} & & V _{N} = \left \{v_{N} \in \mathbb{D}_{N}/\forall q_{N} \in \mathbb{M}_{N}\quad,\quad b_{N}(vN,q_{N}) = 0\right \}, {}\\ & & {}\\ \end{array}$$

$$\displaystyle\begin{array}{rcl} & & W_{N} = \left \{(\omega _{N},u_{N}) \in \mathbb{C}_{N} \times V _{N}/\forall \theta _{N} \in \mathbb{C}_{N},c_{N}(\omega _{N},u_{N},\theta _{N}) = 0\right \}. {}\\ \end{array}$$

We observe that, for any solution $(\omega _{N},u_{N},p_{N})$ of problem (9), the pair $(\omega _{N},u_{N})$ is a solution of the reduced problem: Find $(\omega _{N},u_{N}) \in W_{N}$ such that:

$$\displaystyle{ \forall v_{N} \in V _{N}\quad,\quad a_{N}(\omega _{N},u_{N};v_{N}) + K_{N}(\omega _{N},u_{N};v_{N}) = \left (\left (f,v_{N}\right )\right )_{N}. }$$

(10)

We recall from [4] and [7] that the bilinear form a _N (. , . ; . ) satisfies, on the discrete spaces, a positivity property and an inf − sup condition with constants independent of N. We also refer to [4], for a discrete inf − sup condition on the form b _N (. , . ). Using the fixed point theorem of Brower, we can prove the wellposedness of problem (10) and then derive the:

Theorem 1.

For any data f continuous on $\overline{\varOmega }$, the discrete problem(9)admits a solution $(\omega _{N},u_{N};p_{N})$ in $\mathbb{C}_{N} \times \mathbb{D}_{N} \times \mathbb{M}_{N}$. Moreover, $(\omega _{N},u_{N})$ satisfies:

$$\displaystyle{ \left \|\omega _{N}\right \|_{L_{1}^{2}(\varOmega )} + \left \|u_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}} \leq {c\nu }^{-1}\left \|\mathit{I}_{ N}f\right \|_{L_{1}^{2}{(\varOmega )}^{2}}. }$$

(11)

4 Error Estimates

We now intend to prove an error estimate between the solutions of problems (3) and (9). Since the error analysis of the discrete problem relies on the theory of Brezzi, Rappaz and Raviart [5], we express both problems (4) and (10) in a different form and we set $X = V _{1\diamond}^{1}(\varOmega ) \times \left (V \cap H_{1}(\mathrm{curl},\varOmega )\right )$. We denote by S the linear operator of Stokes which for any f in the dual space of $H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )$, associates the solution (ω, u) of the following reduced problem:

$\qquad \qquad \mbox{ Find }(\omega,u) \in W\mbox{ such that }\quad \forall v \in V,\quad a(\omega,u;v) = \left \langle f,v\right \rangle$.

We introduce the mapping G defined from X into the dual space of $H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )$ by:$\forall (\omega,u) \in X,\quad \forall v \in H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega ),$ $\left \langle G(\omega,u),v\right \rangle = K(\omega,u;v) -\left \langle f,v\right \rangle$.

Then, problem (4) can be equivalently written as: Find (ω, u) ∈ X such that

$$\displaystyle{ (\omega,u) + SG(\omega,u) = 0. }$$

(12)

Similarly, we define the discrete space $X_{N} = \mathbb{C}_{N} \times (V _{N} \cap H_{1}(\mathrm{curl},\varOmega ))$. We thus define the discrete Stokes operator S _N : for any f in the dual space of $H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )$, S _N f denotes the solution $(\omega _{N},u_{N})$ of problem: Find $(\omega _{N},u_{N}) \in W_{N}$ such that

$$\displaystyle{ \forall v_{N} \in V _{N},\quad a_{N}(\omega _{N},u_{N};v_{N}) = \left \langle f,v_{N}\right \rangle. }$$

(13)

The well-posedness of problem (13) is proven in [4], for a slightly different right-hand side. Finally, we consider the mapping G _N defined from X _N in the dual space of $\mathbb{D}_{N}$ by $\forall (\omega _{N},u_{N}) \in X_{N},\quad \forall v_{N} \in \mathbb{D}_{N}$

$$\displaystyle{ \left \langle G_{N}(\omega _{N},u_{N}),v_{N}\right \rangle = K_{N}(\omega _{N},u_{N};v_{N}) - ((f,v_{N}))_{N}. }$$

(14)

Problem (10) can equivalently be written as: Find $(\omega _{N},u_{N}) \in X_{N}$ such that

$$\displaystyle{ \left (\omega _{N},u_{N}\right ) + S_{N}G_{N}\left (\omega _{N},u_{N}\right ) = 0. }$$

(15)

Using analogous arguments to those in [4], we easily derive that the operator S _N satisfies a stability property, with a constant independent of N and that, the following error estimate holds for all f in $H_{1}^{s+1}(\varOmega ) \times H_{1}^{s}{(\varOmega )}^{2}$, s > 1,

$$\displaystyle{ \left \|(S - S_{N})f\right \|_{X} \leq {\mathit{cN}}^{-s}\left \|Sf\right \|_{ H_{1}^{s+1}(\varOmega )\times H_{1}^{s}{(\varOmega )}^{2}}. }$$

(16)

We are led to make the following assumptions. Here, D is the differential operator.

Assumption 1.

The triplet (ω, u, p) is a solution of the problem (3) such that the operator Id + SDG(ω, u) is an isomorphism of X.

This assumption can equivalently be written as: For any data g in $H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )^{\prime}$, the linearized problem Find $(\vartheta,w,r)$ in $V _{1\diamond}^{1}(\varOmega ) \times \left (H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega ) \cap H_{1}(\mathrm{curl},\varOmega )\right ) \times L_{1,0}^{2}(\varOmega )$ such that:

$$\displaystyle{ \left \{\begin{array}{rlll} a(\vartheta,w,v) + K(\omega,w,v)+&K(\vartheta,u;v)& \\ + b(v,r)& = \left \langle g,v\right \rangle,\quad &\forall v \in H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega ) \cap H_{1}(\mathrm{curl},\varOmega ), \\ b(w,q)& = 0,\quad &\forall q \in L_{1,0}^{2}(\varOmega ), \\ c(\vartheta,w,\varphi )& = 0,\quad &\forall \varphi \in V _{1\diamond}^{1}(\varOmega ). \end{array} \right. }$$

(17)

has a unique solution with norm bounded by a constant times $\Vert g\Vert _{H_{1}^{\diamond}(\mathrm{div_{r}},\varOmega )}$. It yields the local uniqueness of the solution (ω, u, p) but is much less restrictive than the global uniqueness condition. We need to prove a few technical results in order to derive the error estimate. For this, we make the:

Assumption 2.

The solution (ω, u, p) of problem (3) satisfying Assumption 1, belongs to $H_{1}^{s+1}(\varOmega ) \times H_{1}^{s}{(\varOmega )}^{2} \times H_{1}^{s}(\varOmega )$, s > 1.

Then, we prove:

Lemma 1.

For any $(\omega _{N},u_{N};v_{N})$ in $\mathbb{C}_{N} \times \mathbb{D}_{N} \times \mathbb{D}_{N},$

$$\displaystyle\begin{array}{rcl} \left \vert K(\omega _{N},u_{N};v_{N})\right \vert \leq c_{1}N\left \|\omega _{N}\right \|_{V _{1}^{1}(\varOmega )}\left \|u_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}}\left \|v_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}},& &{}\end{array}$$

(18)

$$\displaystyle{ \vert K_{N}(\omega _{N},u_{N};v_{N})\vert \leq c_{2}N\left \|\omega _{N}\right \|_{V _{1}^{1}(\varOmega )}\left \|u_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}}\left \|v_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}}. }$$

(19)

the constants c ₁ and c ₂ are independent of N.

Proof.

According to the Cauchy-Schwarz inequality we have:

$$\displaystyle{ \left \vert K(\omega _{N},u_{N};v_{N})\right \vert = \left \vert \int _{\varOmega }(\omega _{N} \times u_{N}).v_{N}\mathit{rdrdz}\right \vert \leq \left \|\omega _{N}\right \|_{L_{1}^{4}(\varOmega )}\left \|u_{N}\right \|_{L_{1}^{4}{(\varOmega )}^{2}}\left \|v_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}}. }$$

Using the inclusion of $V _{1}^{1}(\varOmega )$ in $L_{1}^{4}(\varOmega )$ and inequality:

$$\displaystyle{ \forall z_{N} \in \mathbb{P}_{N}(\varOmega ),\qquad \left \|z_{N}\right \|_{L_{1}^{4}(\varOmega )} \leq \mathit{cN}\left \|z_{N}\right \|_{L_{1}^{2}(\varOmega )}, }$$

(20)

see [2], we have the first previous result. For the second one, we have with obvious notation,

$$\displaystyle\begin{array}{rcl} K_{N}(\omega _{N},u_{N};v_{N})& =& ((\omega _{N}u_{Nr},v_{Nz}))_{N} - ((\omega _{N}u_{Nz},v_{Nr}))_{N} {}\\ & =& ((I_{N}(\omega _{N}u_{Nr}),v_{Nz}))_{N} - ((I_{N}(\omega _{N}u_{Nz}),v_{Nr}))_{N}. {}\\ \end{array}$$

By combining the Cauchy-Schwarz inequalities with inequality (3. 7) in [7], we obtain

$$\displaystyle{\left \vert K_{N}(\omega _{N},u_{N};v_{N})\right \vert \leq \left \|I_{N}(\omega _{N}u_{N})\right \|_{L_{1}^{2}{(\varOmega )}^{2}}\left \|v_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}}.}$$

Then, we use the following result which can be derived from its one-dimensional analogue [7],

$$\displaystyle{\forall \varphi _{M} \in \mathbb{P}_{M}(\varOmega _{k}),\quad \left \|I_{N}^{k}\varphi _{ M}\right \|_{L_{1}^{2}(\varOmega _{k})} \leq c{(1 + \frac{M} {m(N)})}^{2}\left \|\varphi _{ M}\right \|_{L_{1}^{2}(\varOmega _{k})},}$$

with m(N) = E((1 +δ)N) and δ a real number between 0 and 1. We conclude, by using the inequalities (20) and $\left \|\omega _{N}u_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}} \leq \left \|\omega _{N}\right \|_{L_{1}^{4}(\varOmega )}\left \|u_{N}\right \|_{L_{1}^{4}{(\varOmega )}^{2}}$, together with the continuous inclusion of $V _{1}^{1}(\varOmega )$ in $L_{1}^{4}(\varOmega )$, that

$$\displaystyle{\left \|\omega _{N}u_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}} \leq \mathit{cN}\left \|\omega _{N}\right \|_{V _{1}^{1}(\varOmega )}\left \|u_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}}.}$$

Remark 1.

Similar arguments lead to estimate (18), if at most two of the three functions ω _N , u _N and v _N are replaced by their analogues ω in $V _{1\diamond}^{1}(\varOmega )$, u and v in $\mathbb{D}(\varOmega )$.

Remark 2.

Under Assumption 2 and taking $\tilde{N} = E(2\delta N - 1)$, we can find $(\tilde{\omega }_{N},\tilde{u}_{N})$ in $\mathbb{C}_{\tilde{N}} \times V _{\tilde{N}}$ such that:

$$\displaystyle{ \left \|\left (\omega -\tilde{\omega }_{N},u -\tilde{ u}_{N}\right )\right \|_{X} \leq c\tilde{{N}}^{-s}\left \|\left (\omega,u\right )\right \|_{ H_{1}^{s+1}(\varOmega )\times H_{1}^{s}{(\varOmega )}^{2}},s > 1. }$$

(21)

Note that estimate (21) makes sense only when $\tilde{N} \geq 2$.

Lemma 2.

If Assumptions  1 and  2 hold, there exists an integer N ₀ such that, for all N ≥ N ₀, the operator $\mathit{Id} + S_{N}DG_{N}(\tilde{\omega }_{N},\tilde{u}_{N})$ is an isomorphism of X _N. Moreover, the norm of its inverse operator is bounded independently of N.

Proof.

We can write that:

$$\displaystyle\begin{array}{rcl} \mathit{Id} + S_{N}\mathit{DG}_{N}(\tilde{\omega }_{N},\tilde{u}_{N}) = \mathit{Id} + S\mathit{DG}(\omega,u) - (S - S_{N})\mathit{DG}(\omega,u)& & \\ -S_{N}(\mathit{DG}(\omega,u) -\mathit{DG}(\tilde{\omega }_{N},\tilde{u}_{N})) - S_{N}(\mathit{DG}(\tilde{\omega }_{N},\tilde{u}_{N}) -\mathit{DG}_{N}(\tilde{\omega }_{N},\tilde{u}_{N})).& &{}\end{array}$$

(22)

It follows from the definition of G and G _N that, for all $(\theta _{N},\omega _{N})$ in X _N and v _N in V _N :

$\left \langle \mathit{DG}(\tilde{\omega }_{N},\tilde{u}_{N}).(\theta _{N},w_{N}),v_{N}\right \rangle = K(\tilde{\omega }_{N},w_{N};v_{N}) + K(\theta _{N},\tilde{u}_{N};v_{N}),$ and $\left \langle \mathit{DG}_{N}(\tilde{\omega }_{N},\tilde{u}_{N}).(\theta _{N},w_{N}),v_{N}\right \rangle = K_{N}(\tilde{\omega }_{N},w_{N};v_{N}) + K_{N}(\theta _{N},\tilde{u}_{N};v_{N}).$ Thanks to the choice of $(\tilde{\omega }_{N},\tilde{u}_{N})$, the term $S_{N}(\mathit{DG}(\tilde{\omega }_{N},\tilde{u}_{N}) -\mathit{DG}_{N}(\tilde{\omega }_{N},\tilde{u}_{N}))$ vanishes. Then, using the stability of S _N , we can derive that:

$$\displaystyle\begin{array}{rcl} & & \left \|S_{N}(\mathit{DG}(\omega,u) -\mathit{DG}(\tilde{\omega }_{N}.\tilde{u})).(\theta _{N},w_{N})\right \|_{X} {}\\ & & \leq c\sup _{v_{N}\in V _{N}}\frac{K(\omega -\tilde{\omega }_{N},w_{N},v_{N}) + K(\theta _{N},u -\tilde{ u}_{N},v_{N})} {\Vert v_{N}\Vert _{L_{1}^{2}{(\varOmega )}^{2}}}. {}\\ \end{array}$$

By Lemma 1, we have:

$$\displaystyle\begin{array}{rcl} & & \left \|S_{N}(\mathit{DG}(\omega,u) -\mathit{DG}(\tilde{\omega }_{N}.\tilde{u}_{N})).(\theta _{N},w_{N})\right \|_{X} \\ & & \qquad \leq \mathit{cN}\left (\left \|\omega -\tilde{\omega }_{N}\right \|_{V _{1}^{1}(\varOmega )}\left \|w_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}} + \left \|\theta _{N}\right \|_{V _{1}^{1}(\varOmega )}\left \|u -\tilde{ u}_{N}\right \|_{L_{1}^{2}{(\varOmega )}^{2}}\right ).{}\end{array}$$

(23)

Estimate (21) leads to

$$\displaystyle{ \lim _{N\rightarrow +\infty }\left \|S_{N}(\mathit{DG}(\omega,u) -\mathit{DG}(\tilde{\omega }_{N}.\tilde{u}_{N}))\right \|_{L(X_{N})} = 0. }$$

(24)

Finally, it follows from Assumption 2 that, when (θ, w) runs through the unit ball of X, DG(ω, u)(θ, w) belongs to a compact subset of $L_{1}^{2}{(\varOmega )}^{2}$. So, the next property is derived from the stability of S _N and from inequality (16) by standard arguments:

$$\displaystyle{ \lim _{N\rightarrow +\infty }\left \|\left (S - S_{N}\right )\mathit{DG}(\omega,u)\right \|_{L(X_{N})} = 0. }$$

(25)

Thanks to Assumption 1, for $\gamma = \left \|{\left (\mathit{Id} + S\mathit{DG}(\omega,u)\right )}^{-1}\right \|_{L(X)}$, and by choosing N large enough so that the quantities in (24) and (25) are smaller than $\frac{1} {4\gamma }$, we obtain the desired property with $\left \|{\left (\mathit{Id} + S_{N}\mathit{DG}_{N}(\tilde{\omega }_{N},\tilde{u}_{N})\right )}^{-1}\right \|_{L(X_{N})} < 2\gamma.$

Lemma 3.

The following Lipschitz property holds: $\forall (\omega _{N}^{{\ast}},u_{N}^{{\ast}}) \in X_{N}$ ,

$$\displaystyle{ \left \|S_{N}\left (\mathit{DG}_{N}(\tilde{\omega }_{N},\tilde{u}_{N}) -\mathit{DG}_{N}(\omega _{N}^{{\ast}},u_{ N}^{{\ast}})\right )\right \|_{ L(X_{N})} \leq \mathit{cN}\left \|\left (\tilde{\omega }_{N} -\omega _{N}^{{\ast}},\tilde{u}_{ N} - u_{N}^{{\ast}}\right )\right \|_{ X}. }$$

(26)

Proof.

We just note that

$$\displaystyle\begin{array}{rcl} & & \left \langle \left (\mathit{DG}_{N}(\tilde{\omega }_{N},\tilde{u}_{N}) -\mathit{DG}_{N}(\omega _{N}^{{\ast}},u_{ N}^{{\ast}})\right ).\left (\theta _{ N},w_{N}\right ),v_{N}\right \rangle {}\\ & & \qquad \qquad \qquad \qquad = K_{N}(\tilde{\omega }_{N} -\omega _{N}^{{\ast}},w_{ N};v_{N}) + K_{N}(\theta _{N},\tilde{u}_{N} - u_{N}^{{\ast}};v_{ N}). {}\\ \end{array}$$

Lemma 1 leads to the desired property.

Lemma 4.

Assume that the data $f \in H_{1}^{\sigma }{(\varOmega )}^{2}$ , $\sigma > \frac{3} {2}$. Under Assumption 2,

$$\displaystyle\begin{array}{rcl} & & \left \|(\tilde{\omega }_{N},\tilde{u}_{N}) + S_{N}G_{N}(\tilde{\omega }_{N},\tilde{u}_{N})\right \|_{X} {}\\ & & \qquad \leq c(\omega,u)\left ({N}^{-s}\left \|(\omega,u)\right \|_{ H_{1}^{s+1}(\varOmega )\times H_{1}^{s}{(\varOmega )}^{2}} + {N}^{-\sigma }\left \|f\right \|_{ H_{1}^{\sigma }{(\varOmega )}^{2}}\right ), {}\\ \end{array}$$

for a constant c(ω,u) only depending on the solution (ω,u).

Proof.

From (12), we derive

$$\displaystyle\begin{array}{rcl} & & \left \|\left (\tilde{\omega }_{N},\tilde{u}_{N}\right ) + S_{N}G_{N}\left (\tilde{\omega }_{N},\tilde{u}_{N}\right )\right \|_{X} \leq \left \|\left (\omega -\tilde{\omega }_{N},u -\tilde{ u}_{N}\right )\right \|_{X} + \left \|\left (S - S_{N}\right )G\left (\omega,u\right )\right \|_{X} {}\\ & & \quad + \left \|S_{N}\left (G\left (\omega,u\right ) - G\left (\tilde{\omega }_{N},\tilde{u}_{N}\right )\right )\right \|_{X} + \left \|S_{N}\left (G\left (\tilde{\omega }_{N},\tilde{u}_{N}\right ) - G_{N}\left (\tilde{\omega }_{N},\tilde{u}_{N}\right )\right )\right \|_{X} {}\\ \end{array}$$

The bound for the first term in the right-hand side obviously follows from (21). From estimate (16) with Assumption 2, we also derive

$$\displaystyle{\left \|\left (S - S_{N}\right )G\left (\omega,u\right )\right \|_{X} \leq {\mathit{cN}}^{-s}\left \|(\omega,u)\right \|_{ H_{1}^{s+1}(\varOmega )\times H_{1}^{s}{(\varOmega )}^{2}}.}$$

On the other hand,

$$\displaystyle\begin{array}{rcl} K(\omega,u;v_{N}) - K(\tilde{\omega }_{N},\tilde{u}_{N};v_{N})& = K(\omega -\tilde{\omega }_{N},u;v_{N}) + K(\omega,u -\tilde{ u}_{N};v_{N})& {}\\ & & -K(\omega -\tilde{\omega }_{N},u -\tilde{ u}_{N};v_{N}).{}\\ \end{array}$$

So, we have from the stability property on S _N

$$\displaystyle\begin{array}{rcl} \Vert S_{N}\left (G\left (\omega,u\right ) - G\left (\tilde{\omega }_{N},\tilde{u}_{N}\right )\right )\Vert _{X} \leq c\sup _{v_{N}\in V _{N}}\frac{\langle K(\omega,u;v_{N}) - K(\tilde{\omega }_{N},\tilde{u}_{N};v_{N}),v_{N}\rangle } {\Vert v_{N}\Vert _{L_{1}^{2}{(\varOmega )}^{2}}},& & {}\\ \end{array}$$

From (21), Remarks 1 and 2, we have

$$\displaystyle{\left \|S_{N}\left (G\left (\omega,u\right ) - G\left (\tilde{\omega }_{N},\tilde{u}_{N}\right )\right )\right \|_{X} \leq c(\omega,u){N}^{-s}\left \|\left (\omega,u\right )\right \|_{ H_{1}^{s+1}(\varOmega )\times H_{1}^{s}{(\varOmega )}^{2}}.}$$

We note that $\forall v_{N} \in \mathbb{D}_{N}$, the quantities $K(\tilde{\omega }_{N},\tilde{u}_{N};v_{N})$ and $K_{N}(\tilde{\omega }_{N},\tilde{u}_{N};v_{N})$ coincide. Then, if Π _N−1 denotes the orthogonal projection operator from $L_{1}^{2}(\varOmega )$ onto the space of functions such that their restrictions to all Ω _k , 1 ≤ k ≤ K, belong to $\mathbb{P}_{N-1}(\varOmega _{k})$, and by adding and subtracting the quantity Π _N−1 f in

$\left \|S_{N}\left (G\left (\tilde{\omega }_{N},\tilde{u}_{N}\right ) - G_{N}\left (\tilde{\omega }_{N},\tilde{u}_{N}\right )\right )\right \|_{X}$, we can prove that$\left \|S_{N}\left (G\left (\tilde{\omega }_{N},\tilde{u}_{N}\right ) - G_{N}\left (\tilde{\omega }_{N},\tilde{u}_{N}\right )\right )\right \|_{X} \leq c\left (\left \|f -\varPi _{N-1}f\right \|_{L_{1}^{2}{(\varOmega )}^{2}} + \left \|f\right.\right.$ $\left.\left.-I_{N}f\right \|_{L_{1}^{2}{(\varOmega )}^{2}}\right ).$ Finally, the standard approximation properties of the operators Π _N−1 and I _N , lead to

$$\displaystyle{\left \|S_{N}\left (G\left (\tilde{\omega }_{N},\tilde{u}_{N}\right ) - G_{N}\left (\tilde{\omega }_{N},\tilde{u}_{N}\right )\right )\right \|_{X} \leq {\mathit{cN}}^{-\sigma }\left \|f\right \|_{{ H}^{\sigma }{(\varOmega )}^{2}}.}$$

The desired bound is then derived by combining the previous estimates.

We are now in a position to prove the error estimate.

Theorem 2.

We assume that the data f is in $H_{1}^{\sigma }{(\varOmega )}^{2},\quad \sigma > \frac{3} {2}$, and that the solution (ω,u,p), of problem (3) satisfies Assumptions  1 and  2 .

Then, there exists an integer $N_{\diamond}$ and a constant $c_{\diamond}$ such that for any $N \geq N_{\diamond}$, the problem (9) has a unique solution $(\omega _{N},u_{N},p_{N})$ satisfying the following estimate:

$$\displaystyle\begin{array}{rcl} & & \left \|\omega -\omega _{N}\right \|_{V _{1}^{1}(\varOmega )} + \left \|u - u_{N}\right \|_{H_{1}(\mathrm{div_{r}},\varOmega )} + \left \|p - p_{N}\right \|_{L_{1}^{2}(\varOmega )} \\ & & \leq c(\omega,u)\Big[{N}^{1-s}\Big(\left \|\omega \right \|_{ H_{1}^{s+1}(\varOmega )} + \left \|u\right \|_{H_{1}^{s}{(\varOmega )}^{2}} + \left \|p\right \|_{H_{1}^{s}(\varOmega )}\Big) + {N}^{-\sigma }\left \|f\right \|_{ H_{1}^{\sigma }{(\varOmega )}^{2}}\Big].{}\end{array}$$

(27)

Proof.

Combining Lemmas 2–4 with the Brezzi-Rappaz-Raviart theorem [5], yields that, for N sufficiently large, problem (10) has a unique solution (ω _N , u _N ). Moreover, thanks to the discrete inf-sup condition of b _N (. , . ), there exists a unique p _N in $\mathbb{M}_{N}$ such that

$$\displaystyle{\forall v_{N} \in \mathbb{D}_{N},\quad b_{N}(v_{N},p_{N}) = ((f,v_{N}))_{N} - a_{N}(\omega _{N},u_{N};v_{N}) - K_{N}(\omega _{N},u_{N};v_{N}).}$$

Hence, the existence and local uniqueness result follows. Moreover,

$$\displaystyle\begin{array}{rcl} & & \forall q_{N}\;\mathrm{in}\;\mathbb{M}_{N},\;b_{N}(v_{N},p_{N} - q_{N}) = b(v_{N},p - q_{N}) -\left \langle f,v_{N}\right \rangle + ((f,v_{N}))_{N} \\ & & +a(\omega -\omega _{N},u - u_{N};v_{N}) + (a - a_{N})(\omega _{N},u_{N};v_{N}) \\ & & +K(\omega,u;v_{N}) - K_{N}(\omega _{N},u_{N};v_{N}). {}\end{array}$$

(28)

so that the estimate for $\left \|p - p_{N}\right \|_{L_{1}^{2}(\varOmega )}$ follows from the discrete inf-sup condition of b _N (. , . ), a triangle inequality and the same arguments as in the proof of Lemma 4.

To conclude, the vorticity-velocity and pressure formulation allows to decouple the calculus of the velocity and the pressure, to handle easily non standard boundary conditions and leads to a more accurate approximation of the pressure. The axisymmetric property of domain allows to move from a three-dimensional problem to a two-dimensional one, which reduces the cost of the resolution. In addition, the tensorization properties of the polynomial spaces, which characterize the spectral methods, enable to inverse the obtained system matrix with a raisonable cost.