2.1 Stabilized finite element method
Throughout this paper, we focus on the following finite element subspaces:
$$\begin{aligned}& X_{H}=\bigl\{ \mathbf{v}\in C^{0}(\overline{\Omega })^{2}\cap X:\mathbf{v}|_{K} \in P_{1}(K)^{2}, \forall K\in \mathcal{T}_{H}\bigr\} , \\& M_{H}=\bigl\{ q\in C^{0}(\overline{\Omega })\cap M:q|_{K}\in P_{1}(K), \forall K\in \mathcal{T}_{H} \bigr\} , \end{aligned}$$
where \(P_{1}(K)\) is the space of affine polynomials on the element K.
For the solenoidal vector a, we make the following assumption (see [3, 6, 11]).
Assumption A1
There is a constant \(C_{D}\) such that the m derivatives of a within the element K are bounded above by \(C_{D}|\mathbf{a}|_{\infty,K}\), \(\forall K\in \mathcal{T}_{h}\).
Under the assumption of weakly divergence free of a, for all \(\mathbf{v}\in X\), we have
$$\begin{aligned} (\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{v},\mathbf{v})= \frac{1}{2} \bigl(\mathbf{a}\cdot \boldsymbol{\nabla} (\mathbf{v}\cdot \mathbf{v}),1\bigr)=- \frac{1}{2}(\boldsymbol{\nabla} \cdot \mathbf{a}, \mathbf{v}\cdot \mathbf{v})=0. \end{aligned}$$
(2.1)
It is well known that the above chosen finite element spaces \(X_{H}\) and \(M_{H}\) do not satisfy the discrete inf-sup condition (1.5), but they are of practical importance in real applications. A recently popular stabilized approach, called local pressure projection method, is used in [2, 17, 21, 22] to stabilize the lower order finite element for incompressible flow.
The stabilized finite element method for problem (1.2) is to find \((\mathbf{u}_{H},p_{H})\in X_{H}\times M_{H}\) satisfying
$$\begin{aligned} \textstyle\begin{cases} \nu (\nabla \mathbf{u}_{H},\nabla \mathbf{v}_{H})-(\nabla \cdot \mathbf{v}_{H},p _{H}) +(\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u}_{H},\mathbf{v}_{H})=(\mathbf{f},\mathbf{v}_{H}), & \forall \mathbf{v}_{H}\in X_{H}, \\ (\boldsymbol{\nabla}\cdot \mathbf{u}_{H},q_{H})+G_{h}(p_{H},q _{H})=0, & \forall q_{H}\in M_{H}. \end{cases}\displaystyle \end{aligned}$$
(2.2)
Here, the stabilized term \(G(\cdot,\cdot)\) is defined by
$$\begin{aligned} G_{h}(p_{H},q_{H})=(p_{H}- \Pi_{H}p_{H},q_{H}-\Pi_{H}q_{H}) \quad \forall p_{H},q_{H}\in M_{H}, \end{aligned}$$
(2.3)
and the local projection \(\Pi_{H}:L^{2}(\Omega)\rightarrow P_{0}(K)\) satisfies
$$\begin{aligned}& \begin{aligned}&(p,q_{H})=(\Pi_{H}p,q_{H}), \Vert \Pi_{H}p \Vert _{0}\leq C \Vert p \Vert _{0}, \quad \forall p \in M, q_{H}\in P_{0}(K), \\ & \Vert p-\Pi_{H}p \Vert _{0}\leq CH \Vert p \Vert _{1}, \quad \forall p\in H^{1}(\Omega) \cap M, \end{aligned} \end{aligned}$$
(2.4)
where \(P_{0}(K)\) denotes a piecewise constant on each element K.
We first present the Scott–Zhang [32] interpolating property as the following lemma.
Lemma 2.1
Let
\(I_{H}\)
be the interpolation operator from
\(X\cap C^{0}(\overline{\Omega })^{2}\)
into
\(X_{h}\). It holds
$$\begin{aligned}& \Vert \mathbf{w}-I_{H}\mathbf{w} \Vert _{0,K}\leq C_{1}H_{K} \Vert \mathbf{w} \Vert _{1,\omega _{K}}, \\& \Vert \mathbf{w}-I_{H}\mathbf{w} \Vert _{0,E}\leq C_{1}H_{E}^{1/2} \Vert \mathbf{w} \Vert _{1, \omega_{E}}, \\& \Vert I_{H}\mathbf{w} \Vert _{1}\leq \Vert \mathbf{w} \Vert _{1}, \end{aligned}$$
where
\(\omega_{K}=\bigcup_{K'\cap K\neq \emptyset }K'\) (\(K'\in \mathcal{T}_{h}\)) and
\(\omega_{E}=\bigcup_{E\cap K\neq \emptyset }K\).
Due to the quasi-uniformness of the triangulation \(\mathcal{T}_{H}\), the inverse inequality holds
$$\begin{aligned} \Vert \mathbf{w}_{H} \Vert _{1}\leq C_{2}H^{-1} \Vert \mathbf{w}_{H} \Vert _{0}, \quad \forall \mathbf{w}_{H}\in X_{H}. \end{aligned}$$
(2.5)
Denote
$$\begin{aligned}& \mathscr{B}\bigl((\mathbf{u}_{H},p_{H}),( \mathbf{v}_{H},q_{H})\bigr) \\& \quad \equiv \nu ( \nabla \mathbf{u}_{H},\nabla \mathbf{v}_{H})-( \nabla \cdot \mathbf{v}_{H},p _{H}) +(\mathbf{a}\cdot \boldsymbol{\nabla} \mathbf{u}_{H},\mathbf{v}_{H})+( \boldsymbol{\nabla}\cdot \mathbf{u} _{H},q_{H})+G_{h}(p_{H},q_{H}). \end{aligned}$$
Then the well-posedness of problem (2.2) can be obtained from the following theorem.
Theorem 2.2
For all
\((\mathbf{u}_{H},p_{H}),(\mathbf{v}_{H},q_{H}) \in X_{H}\times M_{H}\), it holds
$$\begin{aligned} \mathscr{B}\bigl((\mathbf{u}_{H},p_{H}),( \mathbf{v}_{H},q_{H})\bigr)\leq C\bigl( \Vert \mathbf{u} _{H} \Vert _{1}+ \Vert p_{H} \Vert _{0}\bigr) \bigl( \Vert \mathbf{v}_{H} \Vert _{1}+ \Vert q_{H} \Vert _{0}\bigr). \end{aligned}$$
(2.6)
Furthermore, there exists a constant
\(\beta^{*}\)
such that, for all
\((\mathbf{u}_{H},p_{H})\in X_{H}\times M_{H}\),
$$\begin{aligned} \beta^{*}\bigl( \Vert \nabla \mathbf{u}_{H} \Vert _{0}+ \Vert p_{H} \Vert _{0}\bigr) \leq \sup_{0\neq (\mathbf{v}_{H},q_{H})\in X_{H}\times M_{H}} \frac{ \vert \mathscr{B}((\mathbf{u}_{H},p_{H}),(\mathbf{v}_{H},q_{H})) \vert }{ \Vert \nabla \mathbf{v} _{H} \Vert _{0}+ \Vert q_{H} \Vert _{0}}. \end{aligned}$$
(2.7)
Proof
The continuity (2.6) holds by using the Cauchy inequality and Assumption A1.
Now, we present the proof of (2.7). For each \(p_{H}\in M _{H}\subset M\), there exists \(\mathbf{w}\in X\) [10, 16] such that
$$\begin{aligned} \nabla \cdot \mathbf{w}=p_{H} \end{aligned}$$
(2.8)
and
$$\begin{aligned} \Vert \mathbf{w} \Vert _{1}\leq C_{3} \Vert p_{H} \Vert _{0},\qquad (\nabla \cdot \mathbf{w},p_{H})\geq C_{4} \Vert \mathbf{w} \Vert _{1} \Vert p_{H} \Vert _{0}. \end{aligned}$$
(2.9)
Let \(\mathbf{w}_{H}=I_{H}\mathbf{w}\in X_{H}\), which satisfies Lemma 2.1. Then it follows from the Green’s formula, Poincare’s inequality, the inverse inequality (2.5), and (2.8)–(2.9) that
$$\begin{aligned}& \bigl\vert \nu (\boldsymbol{\nabla}\mathbf{u}_{H},\boldsymbol{ \nabla}\mathbf{w}_{H}) \bigr\vert \leq \nu \Vert \mathbf{u}_{H} \Vert _{1} \Vert \mathbf{w}_{H} \Vert _{1} \\& \hphantom{ \vert \nu (\boldsymbol{\nabla}\mathbf{u}_{H},\boldsymbol{\nabla}\mathbf{w}_{H}) \vert } \leq \nu \Vert \mathbf{u}_{H} \Vert _{1} \Vert \mathbf{w} \Vert _{1} \leq C_{3}\nu \Vert \mathbf{u}_{H} \Vert _{1} \Vert p_{H} \Vert _{0}\leq \frac{1}{4} \Vert p_{H} \Vert ^{2}_{0}+C_{3}^{2}\nu^{2} \Vert \mathbf{u}_{H} \Vert ^{2}_{1}, \\& \bigl\vert (\nabla \cdot \mathbf{w}_{H},p_{H}) \bigr\vert =(\nabla \cdot \mathbf{w},p_{H})-\bigl( \nabla \cdot ( \mathbf{w}-\mathbf{w}_{H}),p_{H}\bigr) \geq \Vert p_{H} \Vert _{0}^{2}-C_{1}C_{4}H \Vert \mathbf{w} \Vert _{1} \Vert \nabla p_{H} \Vert _{0} \\& \hphantom{ \vert (\nabla \cdot \mathbf{w}_{H},p_{H}) \vert }= \Vert p_{H} \Vert _{0}^{2}-C_{1}C_{4}H \Vert \mathbf{w} \Vert _{1}\sum_{K\in \mathcal{T} _{H}} \bigl\Vert \nabla (p_{H}-\Pi_{H}p_{H}) \bigr\Vert _{0} \\& \hphantom{ \vert (\nabla \cdot \mathbf{w}_{H},p_{H}) \vert }\geq \Vert p_{H} \Vert _{0}^{2}-C_{1}C_{2}C_{3}C_{4} \Vert p_{H} \Vert _{0} \sum _{K\in \mathcal{T}_{H}} \Vert p_{H}-\Pi_{H}p_{H} \Vert _{0} \\& \hphantom{ \vert (\nabla \cdot \mathbf{w}_{H},p_{H}) \vert }\geq \frac{3}{4} \Vert p_{H} \Vert _{0}^{2}-C^{2}_{1}C^{2}_{2}C^{2}_{3}C^{2} _{4} \Vert p_{H}-\Pi_{H}p_{H} \Vert ^{2}_{0}, \\& \bigl\vert (\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u}_{H}, \mathbf{w}_{H}) \bigr\vert \leq C_{D} \vert \mathbf{a} \vert _{ \infty } \Vert \boldsymbol{\nabla}\mathbf{u}_{H} \Vert _{0} \Vert \mathbf{w}_{H} \Vert _{0} \\& \hphantom{ \vert (\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u}_{H},\mathbf{w}_{H}) \vert }\leq C_{D} \vert \mathbf{a} \vert _{\infty } \Vert \boldsymbol{\nabla}\mathbf{u}_{H} \Vert _{0} \Vert \mathbf{w}_{H} \Vert _{1} \leq \frac{1}{4} \Vert p_{H} \Vert ^{2}_{0}+C_{3}^{2}C^{2}_{D} \vert \mathbf{a} \vert _{\infty }^{2} \Vert \mathbf{u}_{H} \Vert ^{2}_{1}. \end{aligned}$$
Set \(\delta >0\) and take
$$\begin{aligned} \mathbf{v}_{H}=\mathbf{u}_{H}-\delta \mathbf{w}_{H} \quad \text{and}\quad q_{H}=p_{H}. \end{aligned}$$
Thanks to (2.1) and the above inequalities, one finds
$$\begin{aligned}& \bigl\vert \mathscr{B}\bigl((\mathbf{u}_{H},p_{H}),( \mathbf{u}_{H}-\delta \mathbf{w}_{H},p _{H}) \bigr) \bigr\vert \\& \quad = \bigl\vert \nu (\boldsymbol{\nabla}\mathbf{u}_{H}, \boldsymbol{\nabla}\mathbf{u}_{H})+G_{h}(p_{H},p_{H})+( \mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u}_{H}, \mathbf{u}_{H}) \\& \qquad {}-\delta \bigl[\nu (\boldsymbol{\nabla}\mathbf{u}_{H}, \boldsymbol{\nabla}\mathbf{w}_{H})- ( \boldsymbol{\nabla}\cdot \mathbf{w}_{H},p_{H}) +(\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u}_{H},\mathbf{w}_{H})\bigr] \bigr\vert \\& \quad \geq \nu \Vert \boldsymbol{\nabla}\mathbf{u}_{H} \Vert ^{2}_{0}+ \bigl\Vert (I-\Pi_{H})p_{H} \bigr\Vert _{0} ^{2} \\& \qquad {}-\delta \biggl(\bigl(C_{3}^{2} \nu^{2}+C_{3}^{2}C_{D}^{2} \vert \mathbf{a} \vert _{\infty }^{2}\bigr) \Vert \mathbf{u}_{H} \Vert ^{2}_{1}-\frac{1}{4} \Vert p_{H} \Vert _{0}^{2}+C^{2}_{1}C ^{2}_{2}C^{2}_{3}C^{2}_{4} \Vert p_{H}-\Pi_{H}p_{H} \Vert ^{2}_{0} \biggr) \\& \quad \geq \bigl(\nu -\delta \bigl(C_{3}^{2} \nu^{2}+C_{3}^{2}C_{D}^{2} \vert \mathbf{a} \vert _{ \infty }^{2}\bigr) \bigr) \Vert \mathbf{u}_{H} \Vert _{1}^{2}+\frac{\delta }{4} \Vert p_{H} \Vert _{0}^{2} +\bigl(1-\delta C^{2}_{1}C^{2}_{2}C^{2}_{3}C^{2}_{4} \bigr) \Vert p_{H}-\Pi _{H}p_{H} \Vert ^{2}_{0}, \end{aligned}$$
provided that \(0<\delta <\min \{\frac{\nu }{C_{3}^{2}(\nu^{2}+C_{D} ^{2}|\mathbf{a}|_{\infty }^{2})},\frac{1}{C^{2}_{1}C^{2}_{2}C^{2}_{3}C ^{2}_{4}}\}\). Denote
$$\begin{aligned} C(\nu)\equiv \min \biggl\{ \nu -\delta C_{3}^{2}\bigl( \nu^{2}+C_{D}^{2} \vert \mathbf{a} \vert _{ \infty }^{2}\bigr),\frac{1}{4}\delta \biggr\} ,\qquad C( \delta)\equiv \max \bigl\{ 2,1+2\delta^{2}\bigr\} . \end{aligned}$$
Then we have
$$\begin{aligned} \Vert \boldsymbol{\nabla} \mathbf{v}_{H} \Vert ^{2}_{0}+ \Vert q_{H} \Vert ^{2}_{0} =& \Vert \boldsymbol{\nabla}\mathbf{u} _{H}-\delta \boldsymbol{\nabla}\mathbf{w}_{H} \Vert ^{2}_{0}+ \Vert p_{H} \Vert ^{2}_{0} \\ \leq & 2 \Vert \boldsymbol{\nabla}\mathbf{u}_{H} \Vert ^{2}_{0}+\bigl(1+2\delta^{2}\bigr) \Vert p_{H} \Vert ^{2} _{0} \leq C(\delta) \bigl( \Vert \boldsymbol{\nabla}\mathbf{u}_{H} \Vert ^{2}_{0}+ \Vert p_{H} \Vert ^{2}_{0}\bigr). \end{aligned}$$
Taking \(\beta^{*}=C(\nu)/C(\delta)\), we obtain the desired result (2.7).
From Theorem 2.2 we see that problem (2.2) admits a unique solution. □
2.2 Stabilized finite volume method
Let \(\mathcal{N}_{H}\) be the set of all interior vertices of the triangulation and \(\mathcal{E}_{H}\) be the set of all interior edges. To define the finite volume method, we introduce a dual partition \(\mathcal{T}_{H}^{*}\) based on \(\mathcal{T}_{H}\), the elements in \(\mathcal{T}_{H}^{*}\) are called control volumes. The dual mesh is constructed by the following rule: For each element \(K\in \mathcal{T} _{H}\) with vertices \(P_{j}\), \(j=1,2,3\), select its barycenter O and the midpoint \(M_{j}\) on each of the edges of K. We can construct the control volumes \(\widetilde{K}_{i}\in \mathcal{T}_{H}^{*}\) by connecting O to \(M_{j}\) as shown in Fig. 1.
The dual finite element space is defined by
$$\begin{aligned} \widetilde{X}_{H}=\bigl\{ \mathbf{v}\in L^{2}( \Omega)^{2}:\mathbf{v}|_{ \widetilde{K}_{i}}\in P_{0}( \widetilde{K}_{i})^{2}, \mathbf{v}|_{\partial \widetilde{K}_{i}\cap \partial \Omega }=0, \forall \widetilde{K} _{i}\in \mathcal{T}_{H}^{*}\bigr\} . \end{aligned}$$
Note that \(P_{0}(\widetilde{K}_{i})\) denotes a piecewise constant on each control volume \(\widetilde{K}_{i}\). The two finite dimensional spaces \(X_{H}\) and \(\widetilde{X}_{H}\) have the same dimension. Furthermore, there exists an invertible linear mapping \(\Gamma_{H}:X _{H}\rightarrow \widetilde{X}_{H}\) such that
$$\begin{aligned} \Gamma_{H}\mathbf{v}_{H}(x)=\sum _{i=1}^{\mathcal{N}_{H}}\mathbf{v}_{H}(x _{i}) \phi_{i}(x), x_{i}\quad \text{is the node of } \mathcal{T}_{H}, x\in \Omega, v _{H}\in X_{H}, \end{aligned}$$
where \(\phi_{i}(x)\) is the characteristic function associated with the dual partition \(\mathcal{T}_{H}^{*}\):
$$\begin{aligned} \phi_{i}(x)=\textstyle\begin{cases} 1& x\in \widetilde{K}_{i}, \\ 0& \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$
The idea of connecting the different spaces through the mapping \(\Gamma_{H}\) was introduced by Li and Zhu in [27] for the elliptic problem. The following properties hold.
Lemma 2.3
(See [7, 26, 34])
For any
\(\mathbf{v}_{H} \in X_{H}\)
and
\(\mathbf{v}^{*}_{H}=\Gamma_{H}\mathbf{v}_{H}\in \widetilde{X}_{H}\), for each interior element
\(K\in \mathcal{T}_{H}\)
with its boundary
\(\partial K\in \mathcal{E}_{H}\), there hold
$$\begin{aligned}& \int_{K}\bigl(\mathbf{v}_{H}-\mathbf{v}^{*}_{H} \bigr)\,dx=0,\qquad \int_{\partial K}\bigl(\mathbf{v}_{H}-\mathbf{v}^{*}_{H} \bigr)\,ds=0,\qquad \bigl\Vert \mathbf{v}^{*}_{H} \bigr\Vert _{0}\leq C \Vert v_{H} \Vert _{0}, \\& \bigl\Vert \mathbf{v}_{H}-\mathbf{v}^{*}_{H} \bigr\Vert _{0,r,K}\leq Ch_{K} \Vert v_{H} \Vert _{1,r,K},\qquad \bigl\Vert \mathbf{v}_{H}- \mathbf{v}^{*}_{H} \bigr\Vert _{0,r,\partial K}\leq CH_{K}^{1-1/r} \Vert \mathbf{v}_{h} \Vert _{1,r,K},\quad r\in [1,\infty). \end{aligned}$$
Analogous to (2.2), the stabilized finite volume method for problem (1.1) is to find \((\mathbf{u}^{v}_{H},p^{v} _{H})\in (X_{H},M_{H})\) satisfying
$$\begin{aligned} \textstyle\begin{cases} A(\mathbf{u}^{v}_{H},\Gamma_{H}\mathbf{v}_{H})+D(\Gamma_{H}\mathbf{v}_{H},p ^{v}_{H}) +(\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u}^{v}_{H},\Gamma_{H}\mathbf{v}_{H})=( \mathbf{f},\Gamma_{h}\mathbf{v}_{H}), &\forall \mathbf{v}_{H}\in X_{H}, \\ (\nabla \cdot \mathbf{u}^{v}_{H},q_{H})+G_{H}(p ^{v}_{H},q_{H})=0, &\forall q_{H}\in M_{H}, \end{cases}\displaystyle \end{aligned}$$
(2.10)
where
$$\begin{aligned}& A\bigl(\mathbf{u}^{v}_{H},\Gamma_{h} \mathbf{v}_{H}\bigr) =-\nu \sum_{j=1}^{ \mathcal{N}_{H}} \mathbf{v}_{H}(P_{j})\cdot \int_{\partial \widetilde{K} _{j}}\frac{\partial \mathbf{u}^{v}_{H}}{\partial \mathbf{n}}\,ds, \\& D\bigl(\Gamma_{H}\mathbf{v}_{H},p^{v}_{H} \bigr) =\sum_{j=1}^{\mathcal{N}_{H}}\mathbf{v} _{H}(P_{j})\cdot \int_{\partial \widetilde{K}_{j}}p^{v}_{H}\cdot \mathbf{n}\,ds, \\& (\mathbf{f},\Gamma_{H}\mathbf{v}_{H}) =\sum _{j=1}^{\mathcal{N}_{H}}\mathbf{v} _{H}(P_{j}) \cdot \int_{\widetilde{K}_{j}}\mathbf{f}\,dx. \end{aligned}$$
Lemma 2.4
It holds that, for all
\(\mathbf{u}^{v}_{H},\mathbf{v} _{H}\in X_{H}\), \(p_{H}^{v}\in M_{H}\),
$$\begin{aligned}& \begin{aligned} &A\bigl(\mathbf{u}_{H}^{v}, \Gamma_{H}\mathbf{v}_{H}\bigr)=\nu \bigl(\nabla \mathbf{u}^{v}_{H}, \nabla \mathbf{v}_{H}\bigr), \\ &D\bigl(\Gamma_{H}\mathbf{v}_{H},p_{H}^{v} \bigr)=\bigl(\nabla \cdot \mathbf{v}_{H},p_{H} ^{v}\bigr). \end{aligned} \end{aligned}$$
(2.11)
Moreover, we have
$$\begin{aligned} \bigl(\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u}^{v}_{H}, \Gamma \mathbf{u}^{v}_{H}\bigr)=0. \end{aligned}$$
(2.12)
Proof
The equations in (2.11) have been shown in [23, 34, 35]. It suffices to prove (2.12). For \(\mathbf{u}_{H}^{v}\in X_{H}\), it follows from the definition of \(\Gamma_{h}\) and (2.1) that
$$\begin{aligned} \bigl(\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u}^{v}_{H}, \Gamma \mathbf{u}^{v}_{H}\bigr) =&\sum _{j=1} ^{N_{H}} \int_{\widetilde{K}_{j}}\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u}_{H}^{v}\cdot \Gamma_{H} \mathbf{u}_{H}^{v}\,dx \\ =&-\sum_{j=1}^{N_{H}} \Gamma_{H}\mathbf{u}_{H} ^{v}(P_{j}) \cdot \int_{\widetilde{K}_{j}}\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u}_{H} ^{v}\,dx \\ =& \sum_{j=1}^{N_{H}}\Gamma_{H} \mathbf{u}_{H}^{v}(P_{j})\cdot \biggl[ \int_{\widetilde{K}_{j}}{(\nabla \cdot \mathbf{u}) u}_{H}^{v} \,dx - \int_{\partial \widetilde{K}_{j}} \mathbf{a} \mathbf{u}_{H}^{v} \cdot \textbf{n}\,ds \biggr]. \end{aligned}$$
With the weakly divergence free of a and the continuity of a, \(\textbf{u}_{H}^{v}\), we complete the proof. □
We denote the generalized bilinear form \(\mathscr{C}((\cdot,\cdot),( \cdot,\cdot))\) on \((X_{H},M_{H})\times (X_{H},M_{H})\) as
$$\begin{aligned}& \mathscr{C}\bigl(\bigl(\mathbf{u}_{H}^{v},p_{H}^{v} \bigr),(\mathbf{v}_{H},q_{H})\bigr) \\& \quad =A\bigl(\mathbf{u} _{H}^{v},\Gamma_{H} \mathbf{v}_{H}\bigr)+D\bigl(\Gamma_{H}\mathbf{v}_{H},p_{H}^{v} \bigr)+d\bigl( \mathbf{u}^{v}_{H},q_{H} \bigr)+G_{H}\bigl(p^{v}_{H},q_{H} \bigr)+\bigl(\mathbf{a}\cdot\boldsymbol{\nabla} \mathbf{u} ^{v}_{H}, \Gamma_{H}\mathbf{v}_{H}\bigr). \end{aligned}$$
By applying the relationships between finite element and finite volume methods presented in Lemma 2.4, and following the proof of Theorem 2.2, we can establish the continuity and weak coercivity for the generalized bilinear form \(\mathscr{C}((\cdot,\cdot),(\cdot,\cdot))\). Here we omit the proof for simplification and present its continuity and weak coercivity.
Theorem 2.5
For all
\((\mathbf{u}_{H}^{v},p_{H}^{v}),(\mathbf{v} _{H},q_{H})\in X_{H}\times M_{H}\), it holds
$$\begin{aligned} \bigl\vert \mathscr{C}\bigl(\bigl(\mathbf{u}_{H}^{v},p_{H}^{v} \bigr),(\mathbf{v}_{H},q_{H})\bigr) \bigr\vert \leq C \bigl( \bigl\Vert \mathbf{u}_{H}^{v} \bigr\Vert _{1}+ \bigl\Vert p_{H}^{v} \bigr\Vert _{0}\bigr) \bigl( \Vert \mathbf{v}_{H} \Vert _{1}+ \Vert q _{H} \Vert _{0}\bigr). \end{aligned}$$
Moreover, there exists a constant
\(\widetilde{\beta }^{*}>0\), independent of
H, such that
$$\begin{aligned} \widetilde{\beta }^{*}\bigl( \bigl\Vert \mathbf{u}_{H}^{v} \bigr\Vert _{1}+ \bigl\Vert p_{H}^{v} \bigr\Vert _{0}\bigr) \leq &\sup _{0\neq (\mathbf{v}_{H},q_{H})\in (X_{H},M_{H})}\frac{ \mathscr{C}((\mathbf{u}_{H}^{v},p_{H}^{v}),(\mathbf{v}_{H},q_{H}))}{ \Vert \mathbf{v} _{H} \Vert _{1}+ \Vert q_{H} \Vert _{0}}. \end{aligned}$$
(2.13)
From Theorem 2.5, we know that problem (2.10) has a unique solution \((\mathbf{u}_{H}^{v},p_{H}^{v})\).
For the stability and convergence results of numerical schemes (2.2) and (2.10), following the proofs provided in [3, 6, 21], by taking different test functions and using the energy method, we can obtain that the numerical schemes (2.2) and (2.10) are unconditionally stable, error estimates for the numerical solutions are also optimal. Here we omit these proofs for simplification.