For the time discretization of the nonstationnary Navier–Stokes problem, we use the backward Euler method. We start by doing a partition of the interval \([0,T]\) in subintervals \([t_{i-1},t_{i}]\) for \(1\leq i\leq I\), where I is a positive integer such that \(0=t_{0}< t_{1}<\cdots<t_{I}=T\). Let \(h_{i}=t_{i}-t_{i-1}\), \(h=(h_{1},h_{2},\dotsc ,h_{I})\) and \(|h|=\max_{1\leq i\leq I}h_{i}\).
If the data f belongs to the space \(L^{2}(0,T;(H_{0}(\operatorname{div},\Omega ))^{\prime })\) and \((\theta _{0},{\mathbf{v}}_{0})\in H_{0}(\operatorname{curl},\Omega )\times H_{0}( \operatorname{div},\Omega )\) satisfy condition (3), then the time semidiscrete problem is:
Find \((\theta ^{i})_{0\leq i\leq I}\in (H_{0}(\operatorname{curl},\Omega ))^{I+1}\), \(({ \mathbf{v}}^{i})_{0\leq i\leq I}\in (H_{0}(\operatorname{div},\Omega ))^{I+1}\), and \((p^{i})_{1\leq i\leq I}\in (L^{2}_{0}(\Omega ))^{I}\) such that
$$ \theta ^{0}=\theta _{0}\quad \mbox{and}\quad {\mathbf{v}}^{0}={\mathbf{v}}_{0} \quad \mbox{in } {\Omega } $$
(7)
and for all \(1 \leq i \leq I\),
$$ \begin{aligned} &\forall {\mathbf{w}}\in H_{0}(\operatorname{div},\Omega ), \hat{l}\bigl( \theta ^{i},{ \mathbf{v}}^{i};{\mathbf{w}}\bigr)+h_{i}Z\bigl(\theta ^{i},{\mathbf{v}}^{i};{\mathbf{w}}\bigr)+ h_{i} b\bigl({\mathbf{w}},p^{i}\bigr)=L({\mathbf{w}}), \\ &\forall q\in L_{0}^{2}(\Omega ),\quad b\bigl({ \mathbf{v}}^{i},q\bigr)=0, \\ &\forall \boldsymbol{\vartheta }\in H_{0}(\operatorname{curl},\Omega ),\quad t \bigl( \theta ^{i},{\mathbf{v}}^{i};\boldsymbol{ \vartheta }\bigr)=0, \end{aligned} $$
(8)
where \({\mathbf{f}}^{i}={\mathbf{f}}(\cdot ,t_{i})\),
$$ \hat{l}\bigl(\theta ^{i},{\mathbf{v}}^{i};{\mathbf{w}} \bigr)=\bigl({\mathbf{v}}^{i} ,{\mathbf{w}}\bigr) + h_{i} l\bigl(\theta ^{i},{\mathbf{v}}^{i};{ \mathbf{w}}\bigr) , $$
and
$$ L({\mathbf{w}})=\bigl({\mathbf{v}}^{i-1},{\mathbf{w}} \bigr)+h_{i}\prec {\mathbf{f}}^{i},{\mathbf{w}} \succ . $$
So when \((\theta ^{i},{\mathbf{v}}^{i},p^{i})\) is a solution of problem (7)–(8), the couple \((\theta ^{i},{\mathbf{v}}^{i})\in \mathrm{W}\) is a solution of the problem
$$ \forall {\mathbf{w}}\in \mathrm{K},\quad \hat{l}\bigl(\theta ^{i},{\mathbf{v}}^{i};{ \mathbf{w}} \bigr)+h_{i} Z\bigl(\theta ^{i},{\mathbf{v}}^{i};{ \mathbf{w}}\bigr)=L({\mathbf{w}}). $$
(9)
The existence of a solution for problems (9) and (8) is deduced from the properties (positivity and inf-sup conditions) of the bilinear form \(\hat{l}(\cdot ,\cdot \,;\cdot )\) proved in [11, Lemma 1], the properties of the trilinear form \(Z(\cdot ,\cdot \,;\cdot )\) (continuity and antisymmetry) proved in [12, Lemma 1], and the inf-sup condition (6). We summarize this result of the existence in dimensions two and three in the following theorem; see [12, Sect. 3], for its proof.
Theorem 1
Suppose the data \({\mathbf{f}}\in L^{2}(0,t;(H_{0}(\operatorname{div},\Omega ))^{\prime })\) and the initial vorticity–velocity \((\theta _{0},{\mathbf{v}}_{0})\) both belong to \(H_{0}(\operatorname{curl},\Omega )\times H_{0}(\operatorname{div},\Omega )\) and satisfy condition (3). In dimension \(d=2\), for any \(i, 1 \leq i \leq I\), problem (7)–(8) has a solution \((\theta ^{i},{\mathbf{v}}^{i},p^{i})\) in \(H_{0}(\operatorname{curl},\Omega )\times H_{0}(\operatorname{div},\Omega )\times L^{2}( \Omega )\). In dimension \(d=3\), if Assumption 1holds and there exists a constant ϱ such that
$$ \varrho \nu ^{-2} \Vert L \Vert _{\mathcal{L}(H_{0}(\operatorname{div}, \Omega ))}< 1, $$
then problem (7)–(8) has a solution \((\theta ^{i}, {\mathbf{v}}^{i}, p^{i})\) in \(H_{0}(\operatorname{curl},\Omega )\times H_{0}(\operatorname{div},\Omega )\times L^{2}( \Omega ) \). The pair \((\theta ^{i}, {\mathbf{v}}^{i})\) satisfies the following stability condition:
$$ \sum_{j=1}^{i} \bigl\Vert \theta ^{j} \bigr\Vert ^{2}_{H({{\mathbf{curl}}}, \Omega )}+ \bigl\Vert { \mathbf{v}}^{i} \bigr\Vert ^{2}_{L^{2}(\Omega )^{d}} \leq \frac{c }{\nu } \Biggl( \Vert {\mathbf{v}}_{0} \Vert ^{2}_{L^{2}( \Omega )^{d}} + \sum_{j=1}^{i} h_{j} \bigl\Vert {\mathbf{f}}^{j} \bigr\Vert ^{2}_{H_{0}( \operatorname{div},\Omega )^{\prime }} \Biggr), $$
where c is a positive constant independent of i.
Hereinafter, for the spectral discretization, we assume that Ω is a square or cube. Using the same idea of Nédélec’s finite elements (see [27, Sect. 2]), we introduce our discrete spaces.
Let \(N\ge 2\) be an integer. The velocity discrete space \({\mathbb{V}}_{N}\) is defined as
$$ {\mathbb{V}}_{N} = H_{0}({\mathrm{div}},\Omega ) \, \cap \, \textstyle\begin{cases} {\mathbb{P}}_{N,N-1}(\Omega ) \times {\mathbb{P}}_{N-1,N}(\Omega ) & \text{if }d = 2, \\ {\mathbb{P}}_{N,N-1,N-1}(\Omega ) \times {\mathbb{P}}_{N-1,N,N-1}( \Omega ) \times {\mathbb{P}}_{N-1,N-1,N}(\Omega ) & \text{if }d = 3. \end{cases} $$
The vorticity discrete space \({\mathbb{T}}_{N}\) is defined as
$$ {\mathbb{T}}_{N} = \textstyle\begin{cases} H^{1}_{0}(\Omega ) \cap {\mathbb{P}}_{N}(\Omega ) &\text{if }d = 2, \\ H_{0}({\mathbf{curl}},\Omega ) \cap ({\mathbb{P}}_{N-1,N,N}(\Omega ) \times {\mathbb{P}}_{N,N-1,N}(\Omega ) \times {\mathbb{P}}_{N,N,N-1}( \Omega ) ) &\text{if }d = 3. \end{cases} $$
Finally, the pressure discrete spaces \({\mathbb{M}}_{N}\) are defined as
$$ {\mathbb{M}}_{N} = L^{2}_{0}(\Omega ) \cap {\mathbb{P}}_{N-1}(\Omega ). $$
Let the nodes \(\epsilon _{i}\), \(0 \le i \le N\), be the zeros of the polynomial \((1-x^{2})L_{N}^{\prime }\), where \(L_{N}\) is the Legendre polynomial of degree N on the interval \([-1,1]\), and let \(\rho _{i}\), \(0 \le i \le N\), be the set of weights for the Gauss–Lobatto quadrature formula. Then
$$ \forall \varphi \in {\mathbb{P}}_{2N-1}(-1,1), \quad \int _{-1}^{1} \varphi (\zeta )\,d\zeta = \sum _{j = 0}^{N} \varphi (\epsilon _{j}) \,\rho _{j}. $$
(10)
We have the following inequality [28]:
$$ \forall u_{N} \in {\mathbb{P}}_{N}(-1,1), \quad \Vert u_{N} \Vert _{L^{2}(-1,1)}^{2} \le \sum_{k = 0}^{N} u_{N}^{2}( \epsilon _{k})\,\rho _{k} \le 3\, \Vert u_{N} \Vert _{L^{2}(-1,1)}^{2}. $$
(11)
For continuous functions φ and ψ on Ω̄, we define the discrete scalar product
$$ (\varphi ,\psi )_{N} = \textstyle\begin{cases} \sum_{k = 0}^{N} \sum_{l = 0}^{N} \varphi (\epsilon _{k},\epsilon _{l}) \psi (\epsilon _{k},\epsilon _{l})\,\rho _{k}\rho _{l} \quad &\text{if $d = 2$,} \\ \sum_{k = 0}^{N} \sum_{l = 0}^{N} \sum_{r = 0}^{N} \varphi ( \epsilon _{k},\epsilon _{l},\epsilon _{r})\psi (\epsilon _{k}, \epsilon _{l},\epsilon _{r})\,\rho _{k}\rho _{l}\rho _{r} \quad &\text{if $d = 3$.} \end{cases} $$
Hereinafter, we suppose that f is continuous on \(\overline{\Omega }\times [0,T]\). The full discrete problem is constructed from problem (7)–(8) by using the Galerkin method combined with numerical integration.
If \({\mathbf{v}}_{N}^{0}={\mathrm{I}}_{N}({\mathbf{v}}_{0})\), then knowing \({\mathbf{v}}_{N}^{i-1}\), we find \(({\boldsymbol{{\tau }}}^{i}_{N},{\mathbf{v}}^{i}_{N},p^{i}_{N})\) in \({\mathbb{T}_{N}}\times {\mathbb{V}}_{N} \times {\mathbb{M}}_{N}\) such that for \(1\leq i\leq I\),
$$ \begin{aligned} &\forall {\mathbf{w}}_{N} \in {\mathbb{V}}_{N}, \quad \hat{l}_{N}\bigl( \theta ^{i}_{N},{\mathbf{v}}^{i}_{N}; {\mathbf{w}}_{N}\bigr) +h_{i} Z_{N}\bigl( \theta ^{i}_{N},{ \mathbf{v}}^{i}_{N}; {\mathbf{w}}_{N}\bigr) + h_{i} b_{N} \bigl({\mathbf{w}}_{N},p^{i}_{N}\bigr) = L_{N}({ \mathbf{w}}_{N}) , \\ &\forall q_{N} \in {\mathbb{M}}_{N}, \quad b_{N}\bigl({\mathbf{v}}^{i}_{N},q_{N} \bigr) = 0, \\ &\forall \boldsymbol{\vartheta }_{N} \in {\mathbb{T}}_{N}, \quad t_{N}\bigl( \theta ^{i}_{N},{ \mathbf{v}}^{i}_{N};\boldsymbol{\vartheta }_{N}\bigr) = 0. \end{aligned} $$
(12)
The bilinear forms \(\hat{l}_{N}(\cdot ,\cdot \,;\cdot )\), \(b_{N}(\cdot ,\cdot )\), and \(t_{N}(\cdot ,\cdot ;\cdot )\) are defined as follows:
$$ \begin{aligned} & \hat{l}_{N}\bigl(\theta ^{i}_{N},{\mathbf{v}}^{i}_{N}; {\mathbf{w}}_{N}\bigr) = \bigl({\mathbf{v}}_{N}^{i},{ \mathbf{w}}_{N}\bigr)_{N} +h_{i}\nu \, \bigl({\mathbf{curl}}\,\theta ^{i}_{N} ,{ \mathbf{w}}_{N}\bigr)_{N},\qquad b_{N}({ \mathbf{w}}_{N},q_{N}) = - ({\mathrm{div}}\,{ \mathbf{w}}_{N}, q_{N})_{N}, \\ & \quad \text{and}\quad t_{N}\bigl(\theta ^{i}_{N},{ \mathbf{v}}^{i}_{N}; \boldsymbol{\varphi }_{N} \bigr) = \bigl(\theta ^{i}_{N}, \boldsymbol{\varphi }_{N}\bigr)_{N} - \bigl({ \mathbf{v}}^{i}_{N}, {\mathbf{curl}}\, \boldsymbol{ \varphi }_{N}\bigr)_{N}. \end{aligned} $$
From (11) combined with the Cauchy–Schwarz inequality it follows that the bilinear forms\(\hat{l}_{N}(\cdot ,\cdot ;\cdot )\), \(b_{N}(\cdot ,\cdot )\). and \(t_{N}(\cdot ,\cdot ;\cdot )\) are continuous respectively on \(({\mathbb{T}}_{N} \times {\mathbb{V}}_{N} ) \times {\mathbb{V}}_{N}\), \({\mathbb{V}}_{N} \times {\mathbb{M}}_{N}\), and \(({\mathbb{T}}_{N} \times {\mathbb{V}}_{N} ) \times {\mathbb{T}}_{N}\) with norms bounded independently of N. The functional \(L_{N}({\mathbf{w}}_{N})=({\mathbf{v}}_{N}^{i-1},{\mathbf{w}}_{N})_{N}+h_{i} ({ \mathrm{I}}_{N}({\mathbf{f}}^{i}),{\mathbf{w}}_{N})_{N}\) is linear and continuous on \({\mathbb{V}}_{N}\). As a consequence of the exactness property (10), the bilinear forms \(b(\cdot ,\cdot )\) and \(b_{N}(\cdot ,\cdot )\) coincide on \({\mathbb{V}}_{N} \times {\mathbb{M}}_{N}\). The discrete nonlinear term \(Z_{N}(\cdot ,\cdot ;\cdot )\) is defined as
$$ Z_{N}\bigl(\theta ^{i}_{N},{ \mathbf{v}}^{i}_{N}; {\mathbf{w}}_{N} \bigr)=\bigl(\theta ^{i}_{N} \times { \mathbf{v}}^{i}_{N},{\mathbf{w}}_{N} \bigr)_{N}. $$
We introduce the kernel of the discrete bilinear form \(b_{N}(\cdot ,\cdot )\)
$$ \mathrm{K}_{N} = \bigl\{ {\mathbf{w}}_{N} \in { \mathbb{V}}_{N}; \forall q_{N} \in { \mathbb{M}}_{N},\,b_{N}({\mathbf{w}}_{N},q_{N}) = 0 \bigr\} , $$
which is equal to the space of divergence-free polynomials in \({\mathbb{D}}_{N}\).
We also define the discrete kernel of the bilinear form \(t_{N}(\cdot ,\cdot \,;\cdot )\)
$$ \mathrm{W}_{N} = \bigl\{ (\boldsymbol{\vartheta }_{N},{ \mathbf{v}}_{N}) \in {\mathbb{T}}_{N} \times \mathrm{K}_{N}; \forall \boldsymbol{\varphi }_{N} \in { \mathbb{T}}_{N}, \, t_{N}( \boldsymbol{\vartheta }_{N},{\mathbf{v}}_{N};\boldsymbol{\varphi }_{N}) = 0 \bigr\} . $$
We remark that the discrete kernel \(\mathrm{W}_{N}\) is not included in the continuous kernel W; see [10, Cor 3.2],. So the full discrete problem (12) is reduced as follows:
If \({\mathbf{v}}_{N}^{0}={\mathrm{I}}_{N}({\mathbf{v}}_{0})\), then knowing \({\mathbf{v}}_{N}^{i-1}\), find \((\theta ^{i}_{N},{\mathbf{v}}^{i}_{N}) \in \mathrm{W}_{N}\) such that for \(1\leq i\leq I\),
$$ \forall {\mathbf{w}}_{N} \in \mathrm{K}_{N}, \hat{l}_{N}\bigl(\theta ^{i}_{N},{ \mathbf{v}}^{i}_{N}; {\mathbf{w}}_{N}\bigr) +h_{i} Z_{N}\bigl(\theta ^{i}_{N},{ \mathbf{v}}^{i}_{N}; {\mathbf{w}}_{N}\bigr) = L_{N}({\mathbf{w}}_{N}). $$
(13)
We consider the inf-sup condition proved in [10, Lemma 3.9]. There exists a positive constant β independent of N such that the discrete bilinear form \(b_{N} (\cdot ,\cdot )\) satisfies
$$ \forall q_{N} \in {\mathbb{M}}_{N}, \quad \sup_{{\mathbf{v}}_{N} \in { \mathbb{V}}_{N}} \,\frac{b_{N}({\mathbf{v}}_{N}, q_{N})}{ \Vert {\mathbf{v}}_{N} \Vert _{H({\mathrm{div}},\Omega )}} \ge \beta \, \Vert q_{N} \Vert _{L^{2}( \Omega )}. $$
(14)
The arguments used to prove the existence of a solution of problems (13) and (12) are exactly the same as those for the continuous problems (9) and (8). These arguments are based on Brouwer’s fixed point theorem [20, Chap. IV, Cor. 1.1] and the inf-sup condition (14). We summarize this result on the existence in the following theorem proved in [12, Sect. 4]
Theorem 3.1
Assume that the data f is continuous on \(\bar{\Omega }\times [0,T] \). Then, knowing \({\mathbf{v}}^{i-1}_{N}\) at each time step i, problem (12) has a solution \((\theta ^{i}_{N},{\mathbf{v}}^{i}_{N},p^{i}_{N})\) in \({\mathbb{T}}_{N} \times {\mathbb{V}}_{N} \times {\mathbb{M}}_{N}\). Moreover, the pair \((\theta ^{i}_{N},{\mathbf{v}}^{i}_{N})\) of this solution satisfies
$$ \sum_{j=1}^{i} \bigl\Vert \theta _{N}^{j} \bigr\Vert ^{2}_{H({{\mathbf{curl}}}, \Omega )}+ \bigl\Vert {\mathbf{v}}^{i}_{N} \bigr\Vert ^{2}_{L^{2}(\Omega )^{d}}\leq \frac{{3^{d}c} }{{2\nu }} \Biggl( \bigl\Vert {\mathbf{v}}_{N}^{0} \bigr\Vert ^{2}_{L^{2}(\Omega )^{d}} + \sum_{j=1}^{i} h_{j} \bigl\Vert {\mathrm{I}}_{N}\bigl({\mathbf{f}}^{j} \bigr) \bigr\Vert ^{2}_{L^{2}( \Omega )^{d}} \Biggr), $$
where c is a positive constant independent of N and i..
Remark 1
Note that the previous existence result still holds when \(Z_{N}(\cdot ,\cdot \,;\cdot )\) is replaced by \(Z(\cdot ,\cdot \,;\cdot )\) in problem (12). In practice, this means that a more precise quadrature formula, exact on \({\mathbb{P}}_{3N-1}(\Omega )\), is used to evaluate the integrals that appear in the treatment of the nonlinear term. The corresponding discrete problem reads:
If \({\mathbf{v}}_{N}^{0}={\mathrm{I}}_{N}({\mathbf{v}}_{0})\), then knowing \({\mathbf{v}}_{N}^{i-1}\), find \((\theta ^{i}_{N},{\mathbf{v}}^{i}_{N},p^{i}_{N})\) in \({\mathbb{T}_{N}}\times {\mathbb{V}}_{N} \times {\mathbb{M}}_{N}\) such that for \(1\leq i\leq I\),
$$ \begin{aligned} &\forall {\mathbf{w}}_{N} \in {\mathbb{V}}_{N}, \quad \hat{l}_{N}\bigl( \theta ^{i}_{N},{\mathbf{v}}^{i}_{N}; {\mathbf{w}}_{N}\bigr) +h_{i} Z\bigl(\theta ^{i}_{N},{ \mathbf{v}}^{i}_{N}; {\mathbf{w}}_{N}\bigr) + h_{i} b_{N} \bigl({\mathbf{w}}_{N},p^{i}_{N}\bigr) = L_{N}({ \mathbf{w}}_{N}) , \\ &\forall q_{N} \in {\mathbb{M}}_{N}, \quad b_{N}\bigl({\mathbf{v}}^{i}_{N},q_{N} \bigr) = 0, \\ &\forall \boldsymbol{\vartheta }_{N} \in {\mathbb{T}}_{N}, \quad t_{N}\bigl( \theta ^{i}_{N},{ \mathbf{v}}^{i}_{N};\boldsymbol{\vartheta }_{N}\bigr) = 0. \end{aligned} $$
(15)
In the same way the discrete reduced problem (13) is written as:
If \({\mathbf{v}}_{N}^{0}={\mathrm{I}}_{N}({\mathbf{v}}_{0})\), then knowing \({\mathbf{v}}_{N}^{i-1}\), find \((\theta ^{i}_{N},{\mathbf{v}}^{i}_{N}) \in \mathrm{W}_{N}\), such that for \(1\leq i\leq I\),
$$ \forall {\mathbf{w}}_{N} \in \mathrm{K}_{N}, \hat{l}_{N}\bigl(\theta ^{i}_{N},{ \mathbf{v}}^{i}_{N}; {\mathbf{w}}_{N}\bigr) +h_{i} Z\bigl(\theta ^{i}_{N},{ \mathbf{v}}^{i}_{N}; {\mathbf{w}}_{N}\bigr) = L_{N}({\mathbf{w}}_{N}). $$
(16)