In the section, we firsthand formulate the fully discretized SMFE model via the time semi-discrete form so that we could avoid the semi-discrete MFE method about spatial variables.
Let \(\Im_{h} = \{K\}\) be the quasi-regular triangulation of Ω (see [12, 18]). The MFE subspaces are chosen as
$$\begin{aligned}& W_{h}= \bigl\{ \boldsymbol{\tau}_{h}\in W\cap\bigl[C(\overline{ \Omega})\bigr]^{2\times2}; \boldsymbol{\tau}_{h}|_{K}\in\bigl[ \mathscr {P}_{1}(K)\bigr]^{2\times2}, \forall K\in\Im _{h} \bigr\} , \\& H_{h}= \bigl\{ \boldsymbol{v}_{h}\in H\cap\bigl[C(\overline{\Omega}) \bigr]^{2}; \boldsymbol{v}_{h}|_{K}\in \bigl[ \mathscr {P}_{1}(K)\bigr]^{2}, \forall K\in\Im_{h} \bigr\} , \\& M_{h}= \bigl\{ q_{h}\in M\cap C(\overline{\Omega}); q_{h}|_{K}\in \mathscr {P}_{1}(K), \forall K\in \Im_{h} \bigr\} , \end{aligned}$$
where \(\mathscr {P}_{1}(K)\) represents the bivariate linear polynomial set on K.
The next lemma is classical and very serviceable (see [12]).
Lemma 3
Suppose that
\(P_{h}: H\to H_{h}\)
represents an elliptic operator, i.e., \(\forall\boldsymbol{u}\in H\), there is one and only one
\(P_{h}\boldsymbol{u}\in X_{h}\)
that satisfies
$$\bigl(\nabla(P_{h}\boldsymbol{u}-\boldsymbol{u}),\nabla\boldsymbol{v}_{h}\bigr)=0, \quad \forall\boldsymbol{v}_{h}\in X_{h}. $$
Then the following error estimates hold:
$$\Vert P_{h}\boldsymbol{u}-\boldsymbol{u} \Vert _{s}\le Ch^{2-s} \Vert \boldsymbol{u} \Vert _{2},\quad s=0,1, \forall\boldsymbol{u}\in \bigl[H^{2}(\Omega)\bigr]^{2}, $$
where
C
used hereinafter represents a general positive constant which is possibly different at different occurrence and does not rely on
h
and
k.
Suppose that
\(Q_{h}: M\to M_{h}\)
represents an
\(L^{2}\)-operator, i.e., \(\forall\omega\in M\), there is one and only one
\(Q_{h}\omega\in M_{h}\)
that satisfies
$$(Q_{h}\omega-\omega,\omega_{h})=0,\quad \forall \omega_{h}\in M_{h}. $$
When
\(\omega\in H^{l}(\Omega)\), the following error estimates hold:
$$\|Q_{h}\omega-\omega\|_{s}\le Ch^{l-s}\|\omega \|_{l},\quad s=0,1, l=1,2. $$
Suppose that
\(R_{h}: W\to W_{h}\)
represents also an
\(L^{2}\)-operator, i.e., \(\forall\boldsymbol{\tau}\in W\), there is one and only one
\(R_{h}\boldsymbol{\tau} \in W_{h}\)
that satisfies
$$(R_{h}\boldsymbol{\tau}-\boldsymbol{\tau},\boldsymbol{\tau}_{h})=0,\quad \forall\boldsymbol{\tau}_{h}\in W_{h}. $$
Then the following error estimates hold:
$$\|R_{h}\boldsymbol{\tau}-\boldsymbol{\tau}\|_{s}\le Ch^{2-s}\|\boldsymbol{\tau} \|_{2},\quad s=0,1, \forall\boldsymbol{\tau}\in\bigl[H^{2}(\Omega) \bigr]^{2\times2}. $$
Then the fully discrete SMFE model based on parameter-free and two local Gauss integrals is described in the following.
Problem IV
Find \((\boldsymbol{u}_{h}^{n}, p_{h}^{n}, T_{h}^{n})\in X_{h}\times M_{h}\times W_{h}\) (\(1\le n\le N\)) such that
$$ \textstyle\begin{cases} (\bar{\partial}_{t}\boldsymbol{u}_{h}^{n},\boldsymbol{v}_{h}) + A({\boldsymbol{u}}_{h}^{n},\boldsymbol{v}_{h}) +A_{1}({\boldsymbol{u}}_{h}^{n},{\boldsymbol{u}}_{h}^{n},\boldsymbol{v}_{h})-B(p_{h}^{n},\boldsymbol{v}_{h}) \\ \quad = (\operatorname{div}({\boldsymbol{F}}_{h}^{n}\boldsymbol{F}_{h}^{nT}),\boldsymbol{v}_{h}),\quad \forall\boldsymbol{v}_{h}\in X_{h}, \\ B(q_{h},\boldsymbol{u}_{h}^{n})+D_{h}(p_{h}^{n},q_{h})=0, \quad \forall q_{h}\in M_{h}, \\ (\bar{\partial}_{t}\boldsymbol{F}_{h}^{n},\boldsymbol{\tau}_{h})+A_{2}({\boldsymbol{u}}_{h}^{n-1},{\boldsymbol{F}}_{h}^{n},\boldsymbol{\tau}_{h})=(\nabla\boldsymbol{u}_{h}^{n-1}\boldsymbol{F}_{h}^{n},\boldsymbol{\tau}_{h}),\quad \forall\boldsymbol{\tau}_{h}\in W_{h}, \\ \boldsymbol{u}_{h}^{0}=P_{h}\boldsymbol{\varphi}, \qquad \boldsymbol{F}_{h}^{0}=R_{h}\boldsymbol{\psi}(x,y),\quad (x,y)\in \Omega, \end{cases} $$
(17)
where
$$ D_{h}\bigl(p_{h}^{n},q_{h} \bigr)=\varepsilon\sum_{K\in \Im_{h}} \biggl\{ \int_{K,2}p_{h}^{n}q_{h}\, \mathrm{d}x\,\mathrm{d}y- \int _{K,1}p_{h}^{n}q_{h}\, \mathrm{d}x\,\mathrm{d}y \biggr\} , \quad p_{h},q_{h}\in M_{h}, $$
(18)
here \(\varepsilon>0\), called parameter-free, represents a constant, \(\int_{K,i} \lambda(x,y)\,\mathrm{d}x\,\mathrm{d}y\) (\(i = 1, 2\)) are two proper Gauss integrals on K and accurate for ith order multinomials (\(i = 1, 2\)), and \(\lambda(x,y) =q_{h}p_{h}\) is the ith order multinomial (\(i = 1, 2\)).
Hence, if \(q_{h}\in M_{h}\), then \(p_{h} \in M_{h}\) is just a piecewise constant as \(i = 1\). Suppose that the operator \(\varrho_{h}: L^{2}(\Omega )\rightarrow\hat{M}_{h}:= \{q_{h}\in L^{2}(\Omega): q_{h}|_{K}\in \mathscr {P}_{0}(K) \forall K\in\Im_{h} \}\) that satisfies, \(\forall p\in L^{2}(\Omega)\),
$$ (p, q_{h}) = (\varrho_{h}p, q_{h}), \quad \forall q_{h}\in\hat{M}_{h}, $$
(19)
where \(\mathscr {P}_{0}(K)\) is the zero degree polynomial set on K. Therefore, the operator \(\varrho_{h}\) has the following properties (see [12, 19]):
$$\begin{aligned}& \|\varrho_{h}p\|_{0}\le C\|p \|_{0},\quad \forall p\in L^{2}(\Omega), \end{aligned}$$
(20)
$$\begin{aligned}& \|p-\varrho_{h}p\|_{0}\le Ch\|p \|_{1} , \quad \forall p\in H^{1}(\Omega). \end{aligned}$$
(21)
Thus, by using \(\varrho_{h}\), the bilinear function \(D_{h}(\cdot,\cdot)\) may be indicated into:
$$ D_{h}(q_{h}, p_{h}) = \varepsilon(q_{h}-\varrho_{h}q_{h}, p_{h}) = \varepsilon(q_{h} -\varrho_{h}q_{h},p_{h}- \varrho_{h}p_{h}). $$
(22)
To discuss the existence, uniqueness, and convergence of the SMFE solutions, it is necessary to use the following discrete Gronwall lemma (see [12, 18]).
Lemma 4
Discrete Gronwall lemma
Suppose that the positive sequences
\(\{\alpha_{n}\}\)
and
\(\{\beta_{n}\}\)
and the monotone positive sequence
\(\{\epsilon_{n}\}\)
satisfy
\(\alpha_{n}+\beta_{n}\le\epsilon_{n}+\bar{\lambda}\sum_{i=0}^{n-1}\alpha_{i}\) (\(\bar{\lambda}>0\)) and
\(\alpha_{0}+\beta_{0}\le\epsilon_{0}\), then
\(\alpha_{n}+\beta_{n}\le\epsilon_{n}\exp(n\bar{\lambda})\) (\(n\ge0\)).
There is the next main conclusion for Problem IV.
Theorem 5
Under the assumptions of Theorem
2, if there exists a constant
\(\alpha>0\)
such that
\(\|\nabla u_{h}^{n-1}\|_{0,\infty }\le\alpha\)
and when
k
is sufficiently small such that
\((1-2k\alpha )>0\), Problem
IV
has one and only one sequence of solutions
\((\boldsymbol{u}_{h}^{n}, \boldsymbol{F}_{h}^{n}, p_{h}^{n})\in H_{h}\times W_{h}\times M_{h}\) (\(n=1,2, \ldots, N\)) that satisfies
$$\begin{aligned}& \bigl\Vert \boldsymbol{F}_{h}^{n} \bigr\Vert _{0}\le\frac{1}{(\sqrt{1-2k\alpha})^{n}} \Vert \boldsymbol{\psi} \Vert _{0}, \end{aligned}$$
(23)
$$\begin{aligned}& \bigl\Vert \boldsymbol{u}_{h}^{n} \bigr\Vert _{0}^{2}+ \bigl\Vert p_{h}^{n} \bigr\Vert _{0}^{2}+\mu k\sum_{i=1}^{n} \bigl\Vert \nabla\boldsymbol{u}_{h}^{i} \bigr\Vert _{0}^{2}\le C\bigl( \bigl\Vert \boldsymbol{F}_{h}^{n} \boldsymbol{F}_{h}^{nT} \bigr\Vert _{0}^{2}+ \Vert \boldsymbol{\varphi} \Vert _{0}^{2}\bigr), \end{aligned}$$
(24)
where
C
used hereinafter represents a constant that does not rely on
k
and
h, but relies on
ψ, φ, and
Re. And if the exact solution to Problem
I
satisfies
\(({\boldsymbol{u}},\boldsymbol{F},p)\in [H_{0}^{1}(\Omega)\cap H^{2}(\Omega)]^{2}\times[H_{0}^{1}(\Omega)\cap H^{2}(\Omega)]^{2\times2}\times[H^{1 }(\Omega)\cap M]\)
and
\(N_{0}\mu ^{-1}\|\nabla\boldsymbol{u}^{n}\|_{0}\le1/4\)
and
\(N_{0}\mu^{-1}\|\nabla{\boldsymbol{u}}_{h}^{n}\|_{0}\le1/4\)
as well as
\(\|{\boldsymbol{u}}_{h}^{n-1}\|_{0,\infty}\)
and
\(\| \operatorname{div}\boldsymbol{F}^{n}\|_{0,\infty}\)
are bounded (\(n=1, 2, \ldots, N\)), we have the following error formulas:
$$\begin{aligned}& \bigl\Vert \boldsymbol{F}(t_{n})-\boldsymbol{F}_{h}^{n} \bigr\Vert _{0}^{2}+ \bigl\Vert \boldsymbol{u}(t_{n})-\boldsymbol{u}_{h}^{n} \bigr\Vert _{0}^{2}+k\sum _{i=1}^{n} \bigl[ \bigl\Vert \nabla\bigl( \boldsymbol{u}(t_{i})-\boldsymbol{u}_{h}^{i}\bigr) \bigr\Vert _{0}^{2}+ \bigl\Vert p(t_{i})-p_{h}^{i} \bigr\Vert _{0}^{2} \bigr] \\& \quad \le C \bigl(k^{2}+h^{2}\bigr),\quad 1\le n\le N. \end{aligned}$$
(25)
Proof
Because the third equation in Problem IV is linear, in order to deduce that the third equation in Problem IV has one and only one sequence of solutions \(\{\boldsymbol{F}_{h}^{n}\}_{n=1}^{N}\subset W_{h}\), we only need to prove that when \(\boldsymbol{\psi}(x,y)=\mathbf{0}\), there holds \(\boldsymbol{F}_{h}^{n}=\mathbf{0}\) (\(n=1,2, \ldots, N\)). Thus, we only prove that (23) is correct. To this end, by choosing \(\boldsymbol{\tau}_{h}=\boldsymbol{F}_{h}^{n}\) in the third equation in Problem IV, if \(\|\nabla u_{h}^{n-1}\| _{0}\le\alpha\), using (6) and Hölder and Cauchy inequalities, we have
$$ \bigl\Vert \boldsymbol{F}_{h}^{n} \bigr\Vert _{0}^{2}=k\bigl(\nabla\boldsymbol{u}^{n-1}\boldsymbol{F}_{h}^{n},\boldsymbol{F}_{h}^{n}\bigr)+\bigl(\boldsymbol{F}_{h}^{n-1},\boldsymbol{F}_{h}^{n}\bigr)\le k \alpha \bigl\Vert \boldsymbol{F}_{h}^{n} \bigr\Vert _{0}^{2}+\frac{1}{2}\bigl( \bigl\Vert \boldsymbol{F}_{h}^{n} \bigr\Vert _{0}^{2}+ \bigl\Vert \boldsymbol{F}_{h}^{n-1} \bigr\Vert _{0}^{2} \bigr). $$
(26)
When k is sufficiently small such that \((1-2k\alpha)>0\), we have
$$ \bigl\Vert \boldsymbol{F}_{h}^{n} \bigr\Vert _{0}\le\frac{ \Vert \boldsymbol{F}_{h}^{n-1} \Vert _{0}}{\sqrt{1-2k\alpha}}\le \cdots\le\frac{ \Vert \boldsymbol{\psi} \Vert _{0}}{(\sqrt{1-2k\alpha})^{n}}\le \frac{ \Vert \boldsymbol{\psi} \Vert _{0}}{(\sqrt{1-2k\alpha})^{N}}. $$
(27)
Thus, if \(\boldsymbol{\psi}=\mathbf{0}\), then the third equation in Problem IV has only a sequence of zero solutions, therefore, it has one and only one sequence of solutions \(\{\boldsymbol{F}_{h}^{n}\}_{n=1}^{N}\).
After we have obtained \(\{\boldsymbol{F}_{h}^{n}\}_{n=1}^{N}\) from the third equation in Problem IV, the first and second equations in Problem IV constitute the fully discrete SMFE model for the nonstationary Navier-Stokes equations. Thus, with the SMFE methods for the nonstationary Navier-Stokes equations (see, e.g., [16, 18–20]), it is obtained that the first and second equations in Problem IV have one and only one sequence of solutions \(\{ (\boldsymbol{u}_{h}^{n}, p_{n}^{n})\}_{n=1}^{N}\subset H_{h}\times M_{h}\) that satisfies (24).
If we let Problem III to subtract Problem IV, and then choose \(\boldsymbol{v}=\boldsymbol{v}_{h}\), \(q=q_{h}\), and \(\boldsymbol{\tau}=\boldsymbol{\tau}_{h}\), we gain three error equations:
$$\begin{aligned}& \bigl(\boldsymbol{u}^{n}-\boldsymbol{u}_{h}^{n}, \boldsymbol{v}_{h}\bigr)+ kA\bigl( {\boldsymbol{u}}^{n}- {\boldsymbol{u}}_{h}^{n}, \boldsymbol{v}_{h}\bigr)+kA_{1}\bigl( {\boldsymbol{u}}^{n}, {\boldsymbol{u}}^{n}, \boldsymbol{v}_{h}\bigr) -kA_{1}\bigl( {\boldsymbol{u}}_{h}^{n}, {\boldsymbol{u}}_{h}^{n}, \boldsymbol{v}_{h}\bigr) \\& \quad =k\bigl(\operatorname{div}\bigl({\boldsymbol{F}}^{n}\boldsymbol{F}^{nT}-{\boldsymbol{F}}_{h}^{n}\boldsymbol{F}_{h}^{nT} \bigr), \boldsymbol{v}_{h}\bigr)+kB\bigl(p^{n}-p_{h}^{n}, \boldsymbol{v}_{h}\bigr) \\& \qquad {}+\bigl(\boldsymbol{u}^{n-1}-\boldsymbol{u}_{h}^{n-1}, \boldsymbol{v}_{h}\bigr), \quad \forall\boldsymbol{v}_{h}\in H_{h}, \end{aligned}$$
(28)
$$\begin{aligned}& B\bigl(q_{h},\boldsymbol{u}^{n}-\boldsymbol{u}_{h}^{n}\bigr)-\varepsilon\bigl(p_{h}^{n}- \varrho_{h}p_{h}^{n}, q_{h}- \varrho_{h}q_{h}\bigr)=0,\quad \forall q_{h}\in M_{h}, \end{aligned}$$
(29)
$$\begin{aligned}& \bigl(\boldsymbol{F}^{n}-\boldsymbol{F}_{h}^{n}, \boldsymbol{\tau}_{h}\bigr)+kA_{2}\bigl( {\boldsymbol{u}}^{n-1}, {\boldsymbol{F}}^{n}, \boldsymbol{\tau}_{h}\bigr)-kA_{2}\bigl( {\boldsymbol{u}}_{h}^{n-1}, {\boldsymbol{F}}_{h}^{n}, \boldsymbol{\tau}_{h}\bigr) \\& \quad =k\bigl(\nabla\boldsymbol{u}^{n-1}\boldsymbol{F}^{n}-\nabla\boldsymbol{u}_{h}^{n-1}\boldsymbol{F}_{h}^{n},\boldsymbol{\tau} _{h}\bigr)+\bigl(\boldsymbol{F}^{n-1}-\boldsymbol{F}_{h}^{n-1}, \boldsymbol{\tau}_{h}\bigr), \quad \forall\boldsymbol{\tau}_{h}\in W_{h}, \end{aligned}$$
(30)
where \(n=1,2,\ldots,N\).
Let \(\boldsymbol{\varrho}_{n}=\boldsymbol{F}^{n}-R_{h}\boldsymbol{F}^{n}\), \(\boldsymbol{E}_{n}=R_{h}\boldsymbol{F}^{n}-\boldsymbol{F}_{h}^{n}\), \(\boldsymbol{e}^{n}=P_{h}\boldsymbol{u}^{n}-\boldsymbol{u}_{h}^{n}\), \(\boldsymbol{\rho}^{n}=\boldsymbol{u}^{n}-P_{h}\boldsymbol{u}^{n}\), \(\eta^{n}=Q_{h}p^{n}-p_{h}^{n}\), and \(\xi^{n}=p^{n}-Q_{h}p^{n}\).
First, by noting that \(\operatorname{div}(\boldsymbol{F}_{h}^{n}\boldsymbol{E}_{n})\) is the piecewise bivariate linear polynomial and \(\|{\boldsymbol{u}}_{h}^{n-1}\|_{0,\infty}\) and \(\|\operatorname{div}\boldsymbol{F}^{n}\|_{0,\infty}\) are bounded and by using the error equation (30), Lemma 3, the properties of \(A_{2}(\cdot,\cdot,\cdot)\), Green’s formula, and Hölder and Cauchy inequalities, we have
$$\begin{aligned} \Vert \boldsymbol{E}_{n} \Vert _{0}^{2} =&( \boldsymbol{E}_{n}, {\boldsymbol{E}}_{n})=-(\boldsymbol{\varrho}_{n}, {\boldsymbol{E}}_{n})+\bigl(\boldsymbol{F}^{n}-\boldsymbol{F}_{h}^{n},{ \boldsymbol{E}}_{n}\bigr) \\ =&k\bigl(\nabla\boldsymbol{u}^{n-1}\boldsymbol{F}^{n}-\nabla\boldsymbol{u}_{h}^{n-1}\boldsymbol{F}_{h}^{n},\boldsymbol{E}_{n}\bigr)+\bigl(\boldsymbol{F}^{n-1}-\boldsymbol{F}_{h}^{n-1}, \boldsymbol{E}_{n}\bigr) \\ & {}+kA_{2}\bigl( {\boldsymbol{u}}_{h}^{n-1}, {\boldsymbol{F}}_{h}^{n},{\boldsymbol{E}}_{n}\bigr)-kA_{2} \bigl( {\boldsymbol{u}}^{n-1}, {\boldsymbol{F}}^{n}, {\boldsymbol{E}}_{n}\bigr) \\ =&k\bigl(\nabla\boldsymbol{u}^{n-1}(\boldsymbol{\varrho}_{n}+\boldsymbol{E}_{n}),\boldsymbol{E}_{n}\bigr)-k\bigl(\boldsymbol{e}^{n-1}, \operatorname{div}\bigl(\boldsymbol{F}_{h}^{n}\boldsymbol{E}_{n} \bigr)\bigr)+(\boldsymbol{E}_{n-1}, \boldsymbol{E}_{n}) \\ &{}+kA_{2}\bigl( {\boldsymbol{u}}_{h}^{n-1}, {\boldsymbol{F}}_{h}^{n},{\boldsymbol{E}}_{n}\bigr)-kA_{2} \bigl( {\boldsymbol{u}}^{n-1}, {\boldsymbol{F}}^{n}, {\boldsymbol{E}}_{n}\bigr) \\ =&k\bigl(\nabla\boldsymbol{u}^{n-1}(\boldsymbol{\varrho}_{n}+\boldsymbol{E}_{n}),\boldsymbol{E}_{n}\bigr)-k\bigl(\boldsymbol{e}^{n-1} \operatorname{div}\boldsymbol{F}^{n},\boldsymbol{\varrho}^{n}\bigr)+( \boldsymbol{E}_{n-1}, \boldsymbol{E}_{n}) \\ & {}+kA_{2}\bigl( {\boldsymbol{u}}_{h}^{n-1}, {\boldsymbol{F}}_{h}^{n},{\boldsymbol{E}}_{n}\bigr)-kA_{2} \bigl( {\boldsymbol{u}}^{n-1}, {\boldsymbol{F}}^{n}, {\boldsymbol{E}}_{n}\bigr) \\ \le& Ckh^{2}+\alpha k \Vert \boldsymbol{E}_{n} \Vert _{0}^{2}+\frac{k}{4} \bigl\Vert {\boldsymbol{e}}^{n-1} \bigr\Vert _{0}^{2}+ \frac{1}{2} \Vert \boldsymbol{E}_{n-1} \Vert _{0}^{2}+ \frac{1}{2} \Vert \boldsymbol{E}_{n} \Vert _{0}^{2}. \end{aligned}$$
(31)
By simplifying (31), we obtain
$$ \Vert \boldsymbol{E}_{n} \Vert _{0}^{2} \le Ckh^{2}+\alpha k \Vert \boldsymbol{E}_{n} \Vert _{0}^{2}+\frac{k}{2} \bigl\Vert {\boldsymbol{e}}^{n-1} \bigr\Vert _{0}^{2}+ \Vert \boldsymbol{E}_{n-1} \Vert _{0}^{2}. $$
(32)
By summing (32) from 1 to n, using Lemma 3, and noting that \(\|\boldsymbol{E}_{0}\|_{0}=\|\boldsymbol{\tau}^{n}-R_{h}\boldsymbol{\tau}\|_{0}\le Ch^{2}\), we obtain
$$ \Vert E_{n} \Vert _{0}^{2} \le Cnkh^{2}+\alpha k\sum_{i=1}^{n} \Vert {\boldsymbol{E}}_{i} \Vert _{0}^{2}+ \frac{k}{2}\sum_{i=1}^{n-1} \bigl\Vert {\boldsymbol{e}}^{i} \bigr\Vert _{0}^{2}. $$
(33)
When k is sufficiently small so that \(\alpha k\le1/2\), from (33), we obtain
$$ \Vert E_{n} \Vert _{0}^{2} \le Cnkh^{2}+\alpha k\sum_{i=0}^{n-1} \Vert {\boldsymbol{E}}_{i} \Vert _{0}^{2}+{k}\sum _{i=1}^{n-1} \bigl\Vert {\boldsymbol{e}}^{i} \bigr\Vert _{0}^{2}. $$
(34)
Thus, applying Lemma 4 (Gronwall lemma) to (34) yields
$$ \Vert E_{n} \Vert _{0}^{2} \le \Biggl[Ch^{2}+{k}\sum_{i=1}^{n-1} \bigl\Vert {\boldsymbol{e}}^{i} \bigr\Vert _{0}^{2} \Biggr]\exp (\alpha nk)\le Ch^{2}+C{k}\sum_{i=1}^{n-1} \bigl\Vert {\boldsymbol{e}}^{i} \bigr\Vert _{0}^{2}. $$
(35)
Next, by using the error equations (28) and (29), (5), and Lemma 3, Hölder and Cauchy inequalities, we have
$$\begin{aligned}& \bigl\Vert \boldsymbol{e}^{n} \bigr\Vert _{0}^{2}+k\mu \bigl\Vert \nabla\boldsymbol{e}^{n} \bigr\Vert _{0}^{2} \\& \quad = \bigl(P_{h}\boldsymbol{u}^{n}- \boldsymbol{u}_{h}^{n}, \boldsymbol{e}^{n}\bigr)+ka \bigl(P_{h}\boldsymbol{u}^{n}-\boldsymbol{u}_{h}^{n},\boldsymbol{e}^{n}\bigr) \\& \quad = -\bigl(\boldsymbol{\rho}^{n}, \boldsymbol{e}^{n}\bigr)+\bigl(\boldsymbol{u}^{n}-\boldsymbol{u}_{h}^{n}, \boldsymbol{e}^{n}\bigr)+kA \bigl(\boldsymbol{u}^{n}-\boldsymbol{u}_{h}^{n},\boldsymbol{e}^{n} \bigr)+kA\bigl(P_{h}\boldsymbol{u}^{n}-\boldsymbol{u}^{n},\boldsymbol{e}^{n}\bigr) \\& \quad = \bigl(\boldsymbol{\rho}^{n-1}-\boldsymbol{\rho}^{n},\boldsymbol{e}^{n} \bigr)+kB\bigl(p^{n}-p_{h}^{n},\boldsymbol{e}^{n}\bigr)+k\bigl(\operatorname{div}\bigl({\boldsymbol{F}}^{n}\boldsymbol{F}^{nT}-{\boldsymbol{F}}_{h}^{n}\boldsymbol{F}_{h}^{nT} \bigr),\boldsymbol{e}^{n}\bigr) \\& \qquad {}+\bigl(\boldsymbol{e}^{n-1}, \boldsymbol{e}^{n}\bigr)-kA_{1} \bigl( {\boldsymbol{u}}^{n}, {\boldsymbol{u}}^{n}, \boldsymbol{e}^{n} \bigr)+kA_{1}\bigl( {\boldsymbol{u}}_{h}^{n},{\boldsymbol{u}}_{h}^{n}, \boldsymbol{e}^{n}\bigr) \\& \quad = \bigl(\boldsymbol{\rho}^{n-1}-\boldsymbol{\rho}^{n},\boldsymbol{e}^{n} \bigr)-kA_{1}\bigl( {\boldsymbol{u}}^{n}, {\boldsymbol{u}}^{n}, \boldsymbol{e}^{n}\bigr)+kA_{1}\bigl({\boldsymbol{u}}_{h}^{n},{ \boldsymbol{u}}_{h}^{n}, \boldsymbol{e}^{n}\bigr) +\bigl(\boldsymbol{e}^{n-1}, \boldsymbol{e}^{n}\bigr) \\& \qquad {}+k\bigl(\operatorname{div}\bigl({\boldsymbol{F}}^{n}\boldsymbol{F}^{nT}-{ \boldsymbol{F}}_{h}^{n}\boldsymbol{F}_{h}^{nT}\bigr), \boldsymbol{e}^{n}\bigr)-k\varepsilon\bigl(\eta^{n}- \varrho_{h}\eta^{n}, \eta^{n}-\varrho _{h}\eta^{n}\bigr)+kB\bigl(\xi^{n},\boldsymbol{e}^{n}\bigr) \\& \qquad {}-kB\bigl(\eta^{n},\boldsymbol{\rho}^{n}\bigr)+k\varepsilon \bigl(p_{h}^{n}-\varrho_{h}p^{n}, \eta ^{n}-\varrho_{h}\eta^{n}\bigr)+k\varepsilon \bigl(Q_{h}p^{n}-p^{n}, \eta^{n}- \varrho _{h}\eta^{n}\bigr) \\& \quad \le C\bigl(k^{-1} \bigl\Vert \boldsymbol{\rho}^{n-1}-\boldsymbol{\rho}^{n} \bigr\Vert _{-1}^{2}\bigr)+ \frac{k}{8} \bigl\Vert \eta^{n} \bigr\Vert _{0}^{2}+Ck \bigl\Vert \xi^{n} \bigr\Vert _{0}^{2}+Ckh\varepsilon \bigl\Vert \eta^{n}- \varrho_{h}\eta ^{n} \bigr\Vert _{0}^{2} \\& \qquad {}+Ck \bigl\Vert \nabla\rho^{n} \bigr\Vert _{0}^{2}+ \frac{k\mu}{8} \bigl\Vert \nabla\boldsymbol{e}^{n} \bigr\Vert _{0}^{2}+\frac{1}{2} \bigl\Vert \boldsymbol{e}^{n-1} \bigr\Vert _{0}^{2} +\frac{1}{2} \bigl\Vert \boldsymbol{e}^{n} \bigr\Vert _{0}^{2}-k\varepsilon \bigl\Vert \eta^{n}-\varrho_{h}\eta ^{n} \bigr\Vert _{0}^{2} \\& \qquad {}-kA_{1}\bigl( {\boldsymbol{u}}^{n}, {\boldsymbol{u}}^{n}, \boldsymbol{e}^{n}\bigr) +kA_{1}\bigl( {\boldsymbol{u}}_{h}^{n}, {\boldsymbol{u}}_{h}^{n}, \boldsymbol{e}^{n}\bigr) +k\bigl( \operatorname {div}\bigl({\boldsymbol{F}}^{n}\boldsymbol{F}^{nT}-{\boldsymbol{F}}_{h}^{n}\boldsymbol{F}_{h}^{nT}\bigr), \boldsymbol{e}^{n}\bigr) \\& \quad \le C\bigl(k^{-1}h^{4}+kh^{2}\bigr)+ \frac{k}{8} \bigl\Vert \eta^{n} \bigr\Vert _{0}^{2}+Ck \bigl\Vert \xi^{n} \bigr\Vert _{0}^{2} +\frac{k\mu}{8} \bigl\Vert \nabla\boldsymbol{e}^{n} \bigr\Vert _{0}^{2} \\& \qquad {}+\frac{1}{2} \bigl\Vert \boldsymbol{e}^{n-1} \bigr\Vert _{0}^{2}+\frac{1}{2} \bigl\Vert \boldsymbol{e}^{n} \bigr\Vert _{0}^{2}-\frac {k\varepsilon}{4}\bigl( \bigl\Vert \eta^{n} \bigr\Vert _{0}^{2}- \bigl\Vert \varrho_{h}\eta^{n} \bigr\Vert _{0}^{2} \bigr) \\& \qquad {}-kA_{1}\bigl( {\boldsymbol{u}}^{n}, {\boldsymbol{u}}^{n}, \boldsymbol{e}^{n}\bigr) +kA_{1}\bigl( {\boldsymbol{u}}_{h}^{n}, {\boldsymbol{u}}_{h}^{n}, \boldsymbol{e}^{n}\bigr) +k\bigl( \operatorname {div}\bigl({\boldsymbol{F}}^{n}\boldsymbol{F}^{nT}-{\boldsymbol{F}}_{h}^{n}\boldsymbol{F}_{h}^{nT}\bigr), \boldsymbol{e}^{n}\bigr). \end{aligned}$$
(36)
If \(N_{0}\mu^{-1}\|\nabla\boldsymbol{u}^{i}\|_{0}\le1/4\) and \(N_{0}\mu^{-1}\| \nabla{\boldsymbol{u}}_{h}^{i}\|_{0}\le1/4\) (\(i=1, 2, \ldots, N\)), by using the properties of \(A_{1}(\cdot,\cdot,\cdot)\), Hölder and Cauchy inequalities, and Lemma 3, we have
$$ kA_{1}\bigl( {\boldsymbol{u}}_{h}^{n}, { \boldsymbol{u}}_{h}^{n}, \boldsymbol{e}^{n}\bigr)-kA_{1} \bigl( {\boldsymbol{u}}^{n}, {\boldsymbol{u}}^{n}, \boldsymbol{e}^{n}\bigr) \le Ck \bigl\Vert \nabla\boldsymbol{\rho}^{n} \bigr\Vert _{0}^{2}+ \frac{k\mu}{4} \bigl\Vert \nabla\boldsymbol{e}^{n} \bigr\Vert _{0}^{2}. $$
(37)
By using Green’s formula and Hölder and Cauchy inequalities, we have
$$\begin{aligned}& k\bigl(\operatorname{div}\bigl({\boldsymbol{F}}^{n}\boldsymbol{F}^{nT}-{\boldsymbol{F}}_{h}^{n}\boldsymbol{F}_{h}^{nT} \bigr), \boldsymbol{e}^{n}\bigr) \\& \quad =-k\bigl({\boldsymbol{F}}^{n}\boldsymbol{F}^{nT}-{\boldsymbol{F}_{h}}^{n}\boldsymbol{F}^{nT}+{\boldsymbol{F}_{h}}^{n} \boldsymbol{F}^{nT}-{\boldsymbol{F}}_{h}^{n}\boldsymbol{F}_{h}^{nT}, \nabla\boldsymbol{e}^{n}\bigr) \\& \quad \le Ck \bigl\Vert {\boldsymbol{F}}^{n}- {\boldsymbol{F}}_{h}^{n} \bigr\Vert _{0} \bigl\Vert \nabla\boldsymbol{e}^{n} \bigr\Vert _{0}\le Ck \bigl\Vert {\boldsymbol{F}}^{n}- {\boldsymbol{F}}_{h}^{n} \bigr\Vert _{0}^{2}+ \frac{k\mu}{8} \bigl\Vert \nabla\boldsymbol{e}^{n} \bigr\Vert _{0}^{2}. \end{aligned}$$
(38)
If \(\eta^{n}\neq0\), it is easily deduced that \(\|\eta^{n}\|_{0}^{2}>\| \varrho_{h}\eta\|_{0}^{2}\) from (22). Therefore, there exists a constant \(\delta\in(0,1)\) such that \(\delta\|\eta^{n}\|_{0}^{2}\ge\| \varrho_{h}\eta\|_{0}^{2}\). By choosing \(\varepsilon=(1-\delta)^{-1}\), combining (37) with (38) and (36), and using Lemma 3, we obtain
$$ \bigl\Vert \boldsymbol{e}^{n} \bigr\Vert _{0}^{2}+k \mu \bigl\Vert \nabla\boldsymbol{e}^{n} \bigr\Vert _{0}^{2}+2k \bigl\Vert \eta^{n} \bigr\Vert _{0}^{2} \le Ckh^{2} + Ck \bigl\Vert {\boldsymbol{F}}^{n}- {\boldsymbol{F}}_{h}^{n} \bigr\Vert _{0}^{2}+ \bigl\Vert \boldsymbol{e}^{n-1} \bigr\Vert _{0}^{2}. $$
(39)
Because \(k\|\eta^{n}\|_{0}^{2}\le2k\|\eta^{n}\|_{0}^{2}\), by summing (39) from 1 to n, we have
$$\begin{aligned} \bigl\Vert \boldsymbol{e}^{n} \bigr\Vert _{0}^{2}+k\sum_{i=1}^{n} \bigl( \bigl\Vert \nabla\boldsymbol{e}^{i} \bigr\Vert _{0}^{2} + \bigl\Vert \eta^{i} \bigr\Vert _{0}^{2}\bigr) \le& Cnkh^{2}+Ck \sum_{i=1}^{n} \bigl\Vert {\boldsymbol{F}}^{n}- {\boldsymbol{F}}_{h}^{n} \bigr\Vert _{0}^{2} \\ \le& Ch^{2}+Ck \sum_{i=1}^{n} \Vert {\boldsymbol{E}}_{i} \Vert _{0}^{2}. \end{aligned}$$
(40)
When k is adequately small that satisfies \(Ck\le1/2\), from (35) and (40), we obtain
$$ \Vert \boldsymbol{E}_{n} \Vert _{0}^{2}+ \bigl\Vert \boldsymbol{e}^{n} \bigr\Vert _{0}^{2}+k\sum _{i=1}^{n}\bigl( \bigl\Vert \nabla\boldsymbol{e}^{i} \bigr\Vert _{0}^{2} + \bigl\Vert \eta^{i} \bigr\Vert _{0}^{2}\bigr) \le Ch^{2}+Ck \sum_{i=0}^{n-1}\bigl( \Vert {\boldsymbol{E}}_{i} \Vert _{0}^{2}+ \bigl\Vert {\boldsymbol{e}}^{i} \bigr\Vert _{0}^{2}\bigr). $$
(41)
Applying Lemma 4 to (41) gains
$$ \Vert \boldsymbol{E}_{n} \Vert _{0}^{2}+ \bigl\Vert \boldsymbol{e}^{n} \bigr\Vert _{0}^{2}+k\sum _{i=1}^{n}\bigl( \bigl\Vert \nabla\boldsymbol{e}^{i} \bigr\Vert _{0}^{2} + \bigl\Vert \eta^{i} \bigr\Vert _{0}^{2}\bigr) \le Ch^{2}\exp(Ck)\le Ch^{2}. $$
(42)
Combining (42) with Theorem 2 yields (25). If \(\eta ^{n}=0\), (25) is obviously correct, which accomplishes the proof of Theorem 5. □
Remark 2
Theorem 5 implies that the sequence of solutions for Problem IV is stabilized and convergent. This signifies that it is theoretically valid that the SMFE model is used to solve the 2D nonlinear incompressible viscoelastic fluid system. Moreover, it is known from Theorems 2 and 5 and their proofs that when \(\|\boldsymbol{\psi}\|_{1}\) and \(\|\boldsymbol{\varphi}\|_{1}\) are sufficiently small, the assumptions that \(N_{0}\mu^{-1}\|\nabla{\boldsymbol{u}}^{n}\|_{0}\le1/4\) and \(N_{0}\mu^{-1}\|\nabla{\boldsymbol{u}}_{h}^{n}\|_{0}\le1/4\) (\(n=1, 2, \ldots, N\)) in Theorem 5 are reasonable.