Spectral element discretization for the stream-function and vorticity formulation of the axisymmetric Stokes problem

Abstract

We deal in this work with the Stokes equations set in a three-dimensional axisymmetric bounded domain. The boundary conditions that we consider are given on the normal component of the velocity and the tangential components of the vorticity. Under assumptions on the data of the problem, the three-dimensional problem is reduced to a two-dimensional one. We write a stream function–vorticity formulation for this problem with two scalar unknowns. For the discretization, we use a domain decomposition method: the spectral element method which is well-adapted here. We prove the well-posedness of the obtained formulations and we derive error estimates between the exact solution and the discrete one.

Appendix 1

We give the proof of the approximation properties of the discrete operators $\mathscr {R}_N^0, \mathscr {R}_N^1$ and $\mathscr {R}_N^{1,0}$.

Lemma 5.1

For any function $\theta \in L^2_1(\Omega )$,

$$\begin{aligned} \Vert \theta -\mathscr {R}_N^0\theta \Vert _{L^2_1(\Omega )}\le 2 \Vert \theta -\theta _{N-1}\Vert _{L^2_1(\Omega )},\forall \theta _{N-1}\in \mathbb {P}_{N-1}(\Omega ). \end{aligned}$$

Proof

Let $\theta _{N-1}\in \mathbb {P}_{N-1}(\Omega ),$ we have

$$\begin{aligned} \Vert \theta - \mathscr {R}_N^0\theta \Vert _{L_1^2(\Omega )}\le & {} \Vert \theta - \theta _{N-1}\Vert _{L_1^2(\Omega )} +\Vert \theta _{N-1} - \mathscr {R}_N^0\theta \Vert _{L_1^2(\Omega )} \nonumber \\&\Vert \theta _{N-1} - \mathscr {R}_N^0\theta \Vert _N +\Vert \mathscr {R}_N^0(\theta _{N-1} - \theta )\Vert _N \end{aligned}$$

(5.1)

Note that

$$\begin{aligned} \forall \theta \in {L_1^2(\Omega )}, {\Vert \mathscr {R}_N^0\theta \Vert _N}^2=(\mathscr {R}_N^0\theta ,\mathscr {R}_N^0\theta )_N =(\theta ,\mathscr {R}_N^0\theta ). \end{aligned}$$

We can deduce that

$$\begin{aligned} {\Vert \mathscr {R}_N^0\theta \Vert _N}\le {\Vert \theta \Vert _{L_1^2(\Omega )}}. \end{aligned}$$

Applying this last inequality for $\theta -\theta _{N-1}$ in (5.1) gives the result. $\square$

Lemma 5.2

For any function $\chi \in V^1_1(\Omega )$, there exists a positive constant c such that:

$$\begin{aligned} \Vert \chi -\mathscr {R}_N^1\chi \Vert _{V^1_1(\Omega )}\le c \Vert \chi -\chi _{N-1}\Vert _{V^1_1(\Omega )},\forall \chi _{N-1}\in \mathbb {P}_{N-1}(\Omega ). \end{aligned}$$

Proof

Let $\chi _{N-1} \in \mathbb {P}_{N-1}(\Omega )$

$$\begin{aligned} \Vert \chi -\mathscr {R}_N^1\chi \Vert _{V^1_1(\Omega )}\le \Vert \chi -\chi _{N-1}\Vert _{V^1_1(\Omega )}+\Vert \chi _{N-1}-\mathscr {R}_N^1\chi \Vert _{V^1_1(\Omega )} \end{aligned}$$

First, we have

$$\begin{aligned} \Vert \mathscr {R}_N^1\chi -\chi _{N-1}\Vert _{V^1_1(\Omega )}\le & {} \Vert \mathbf {curl}_{r} (\mathscr {R}_N^1\chi -\chi _{N-1})\Vert _{L_1^2(\Omega )}\nonumber \\\le & {} \Vert \mathbf {curl}_{r} (\mathscr {R}_N^1\chi -\chi _{N-1})\Vert _N \nonumber \\\le & {} \Vert \mathbf {curl}_{r} (\mathscr {R}_N^1(\chi -\chi _{N-1}))\Vert _N \end{aligned}$$

(5.2)

Next, for any $\chi \in V^1_1(\Omega )$,

$$\begin{aligned} {\Vert \mathbf {curl}_{r} (\mathscr {R}_N^1\chi )\Vert _N}^2= & {} (\mathbf {curl}_{r} (\mathscr {R}_N^1\chi ),\mathbf {curl}_{r} (\mathscr {R}_N^1\chi ))_N \nonumber \\= & {} (\mathbf {curl}_{r} (\chi ),\mathbf {curl}_{r} (\mathscr {R}_N^1\chi ))_N \end{aligned}$$

(5.3)

This leads to

$$\begin{aligned} \Vert \mathbf {curl}_{r} (\mathscr {R}_N^1\chi )\Vert _N \le \Vert \mathbf {curl}_{r} \chi \Vert _{L_1^2(\Omega )}\le c \Vert \chi \Vert _{V^1_1(\Omega )}. \end{aligned}$$

Applying this last result to $\chi -\chi _{N-1}$ together with (5.2) give

$$\begin{aligned} \Vert \mathscr {R}_N^1\chi -\chi _{N-1}\Vert _{V^1_1(\Omega )} \le c \Vert \chi -\chi _{N-1}\Vert _{V^1_1(\Omega )}. \end{aligned}$$

$\square$

Lemma 5.3. can be proven by using the same arguments as previously, we just state it.

Lemma 5.3

For any function $\chi \in V^1_{1\diamond }(\Omega )$, there exists a positive constant c such that:

$$\begin{aligned} \Vert \chi -\mathscr {R}_N^{1,0}\chi \Vert _{V^1_1(\Omega )}\le c \Vert \chi -\chi _{N-1}\Vert _{V^1_1(\Omega )},\forall \chi _{N-1}\in \mathbb {P}_{N-1}^0(\Omega ). \end{aligned}$$