First, by the symmetry and positive definiteness of the matrices
\(\hat{\boldsymbol {A}}\) and
A we conclude that Problem
9 has a unique series of the coefficient vector solutions
\((\boldsymbol{\omega}^{n},\boldsymbol{\psi}^{n})\in \mathbb{R}^{4N}\times \mathbb{R}^{4N}\) (
\(n=1, 2, \ldots, M\)). Thus, by
\(\omega_{N}^{n}=\tilde{\boldsymbol {N}}\cdot\boldsymbol{\omega}^{n}\),
\(\psi_{N}^{n}=\tilde{\boldsymbol {N}}\cdot\boldsymbol{\psi}^{n}\) we can immediately conclude that Problem
8 has a unique series of the CNFSE solutions
\((\omega_{N}^{n}, \psi_{N}^{n})\) (
\(n=1, 2, \ldots, M\)).
Next, we analyze the stability of the CNFSE solutions. From (
31) and (
32) we can attain the following:
$$ \textstyle\begin{cases} \boldsymbol{\psi}^{n-1}={\hat{\boldsymbol {A}}^{-1}}\boldsymbol {C}\boldsymbol{\omega}^{n-1},& 1\leq n\leq M+1; \\ \boldsymbol{\omega}^{n}=\boldsymbol{\omega}^{n-1}+\Delta t{\boldsymbol {A}}^{-1}\boldsymbol {B}\boldsymbol{\omega}^{n-1}+\Delta t{\boldsymbol {A}}^{-1}\boldsymbol {F}^{n},& 1\leq n \leq M. \end{cases} $$
(37)
Moreover, from the FE method (see, e.g., [
11, Lemmas 1.18 and 1.22]) and FSE method (see, e.g., [
17, Chapters II and III]) we can attain the following inequalities:
$$ \begin{aligned} & \bigl\Vert \hat{\boldsymbol {A}}^{-1} \bigr\Vert _{\infty}\le\sigma N^{-1};\qquad \bigl\Vert { \boldsymbol {A}}^{-1} \bigr\Vert _{\infty}\le\sigma N^{-1}; \qquad \Vert {\boldsymbol {B}} \Vert _{\infty}\le\sigma N; \\ & \Vert {\boldsymbol {C}} \Vert _{\infty}\le\sigma,\qquad \bigl\Vert { \boldsymbol {C}}^{-1} \bigr\Vert _{\infty}\le\sigma. \end{aligned} $$
(38)
Thus, by (
37) and (
38) we obtain
$$\begin{aligned}& \bigl\Vert \boldsymbol{\psi}^{n} \bigr\Vert _{\infty}\le\sigma N^{-1} \bigl\Vert \boldsymbol {\omega}^{n} \bigr\Vert _{\infty},\quad n=0, 1, 2, \ldots, M; \end{aligned}$$
(39)
$$\begin{aligned}& \bigl\Vert \boldsymbol{\omega}^{n} \bigr\Vert _{\infty}\le \bigl\Vert \boldsymbol{\omega}^{n-1} \bigr\Vert _{\infty }+ \sigma\Delta t \bigl\Vert \boldsymbol{\omega}^{n-1} \bigr\Vert _{\infty}+\sigma\Delta t N^{-1} \bigl\Vert \boldsymbol {F}^{n} \bigr\Vert _{\infty},\quad n=1, 2, \ldots, M. \end{aligned}$$
(40)
Summing (
40) from 1 to
n, we attain
$$ \bigl\Vert \boldsymbol{\omega}^{n} \bigr\Vert _{\infty}\le \bigl\Vert \boldsymbol{\omega}^{0} \bigr\Vert _{\infty}+ \sigma \Delta t\sum_{i=0}^{n-1} \bigl\Vert \boldsymbol{\omega}^{i} \bigr\Vert _{\infty}+\sigma \Delta t N^{-1}\sum_{i=1}^{n} \bigl\Vert \boldsymbol {F}^{i} \bigr\Vert _{\infty},\quad n=1, 2, \ldots, M. $$
(41)
By the discrete Gronwall inequality (Lemma
4) and from (
41) we obtain
$$ \bigl\Vert \boldsymbol{\omega}^{n} \bigr\Vert _{\infty}\le \Biggl( \bigl\Vert \boldsymbol{\omega}^{0} \bigr\Vert _{\infty }+\sigma\Delta t N^{-1}\sum _{i=1}^{n} \bigl\Vert {\boldsymbol {F}^{i}} \bigr\Vert _{\infty} \Biggr)\exp[\sigma n\Delta t], \quad n=1, 2, \ldots, M. $$
(42)
Combining (
42) with (
39), we get
$$ \bigl\Vert \boldsymbol{\psi}^{n} \bigr\Vert _{\infty}\le\sigma N^{-1} \Biggl( \bigl\Vert \boldsymbol {\omega}^{0} \bigr\Vert _{\infty}+\Delta t N^{-1}\sum _{i=1}^{n} \bigl\Vert \boldsymbol {F}^{i} \bigr\Vert _{\infty} \Biggr), \quad n=1, 2, \ldots, M. $$
(43)
Because
\(\omega_{N}^{n}=\tilde{\boldsymbol {N}}\cdot\boldsymbol{\omega}^{n}\),
\(\psi _{N}^{n}=\tilde{\boldsymbol {N}}\cdot\boldsymbol{\psi}^{n}\), and
\(\|\tilde{\boldsymbol {N}}\| _{\infty}\le1\), from (
42) and (
43) we immediately attain (
33) and (
34), respectively.
Finally, we discuss the convergence of the CNFSE solutions. Subtracting (
29) and (
30) from (
9) and (
10) taking
\(w=w_{N}\), respectively, we attain the following equations for determining the error:
$$\begin{aligned}& \int_{\varTheta}\nabla\bigl(\psi^{n-1}-\psi_{N}^{n-1} \bigr)\nabla w_{N}\,\mathrm{d}x\,\mathrm{d}y \\& \quad = \int_{\varTheta}\bigl(\omega^{n-1}-\omega_{N}^{n-1} \bigr)w_{N}\,\mathrm{d}x\,\mathrm{d}y,\quad \forall w_{N}\in V_{N}, n=1, 2, \ldots, M+1; \end{aligned}$$
(44)
$$\begin{aligned}& \int_{\varTheta}\biggl[\bigl(\omega^{n}- \omega_{N}^{n}\bigr)w_{N}+\frac{\mu \Delta t}{2}\nabla \bigl(\omega^{n}-\omega_{N}^{n}\bigr)\nabla w_{N} \biggr]\,\mathrm{d}x\,\mathrm{d}y \\& \quad = \int_{\varTheta}\bigl(\omega^{n-1}-\omega_{N}^{n-1} \bigr)w_{N}\,\mathrm{d}x\,\mathrm{d}y \\& \qquad {}-\frac{\mu\Delta t}{2} \int_{\varTheta}\nabla\bigl(\omega ^{n-1}- \omega_{N}^{n-1}\bigr)\nabla w_{N}\,\mathrm{d}x\,\mathrm{d}y,\quad \forall w_{N}\in V_{N}, n=1, 2, \ldots, M, \end{aligned}$$
(45)
where
\(\omega_{N}^{0}=R_{N}\omega^{0}\).
By (
44) and (
28), the Cauchy–Schwarz, Hölder, and Poincaré inequalities, and Theorem
7, we obtain
$$\begin{aligned}& \bigl\Vert \nabla\bigl(\psi^{n-1}-\psi_{N}^{n-1} \bigr) \bigr\Vert _{0}^{2} \\& \quad = \int_{\varTheta}\nabla \bigl(\psi^{n-1}-\psi_{N}^{n-1} \bigr)\nabla\bigl(\psi^{n-1}-\psi_{N}^{n-1}\bigr)\,\mathrm{d}x\,\mathrm{d}y \\& \quad = \int_{\varTheta}\nabla\bigl(\psi^{n-1}-R_{N} \psi^{n-1}\bigr)\nabla\bigl(\psi ^{n-1}-R_{N} \psi^{n-1}\bigr)\,\mathrm{d}x\,\mathrm{d}y \\& \qquad {}+ \int_{\varTheta}\nabla\bigl(\psi^{n-1}-\psi_{N}^{n-1} \bigr)\nabla\bigl(R_{N}\psi ^{n-1}-\psi_{N}^{n-1} \bigr)\,\mathrm{d}x\,\mathrm{d}y \\& \quad = \bigl\Vert \nabla\bigl(\psi^{n-1}-R_{N} \psi^{n-1}\bigr) \bigr\Vert _{0}^{2}+ \int_{\varTheta}\bigl(\omega ^{n-1}-\omega_{N}^{n-1} \bigr) \bigl(R_{N}\psi^{n-1}-\psi_{N}^{n-1} \bigr)\,\mathrm{d}x\,\mathrm{d}y \\& \quad \le \bigl\Vert \nabla\bigl(\psi^{n-1}-R_{N} \psi^{n-1}\bigr) \bigr\Vert _{0}^{2}+ \bigl\Vert \omega^{n-1}-\omega _{N}^{n-1} \bigr\Vert _{0} \bigl\Vert R_{N}\psi^{n-1}- \psi^{n-1} \bigr\Vert _{0} \\& \qquad {}+ \bigl\Vert \omega^{n-1}-\omega_{N}^{n-1} \bigr\Vert _{0} \bigl\Vert \psi^{n-1}- \psi_{N}^{n-1} \bigr\Vert _{0} \\& \quad \le\sigma\bigl(N^{2-2q}+ \bigl\Vert \omega^{n-1}- \omega_{N}^{n-1} \bigr\Vert _{0}^{2} \bigr) \\& \qquad {}+\frac{1}{2} \bigl\Vert \nabla\bigl(\psi^{n-1}- \psi_{N}^{n-1}\bigr) \bigr\Vert _{0}^{2}, \quad n=1, 2, \ldots, M+1, 2\le q\le N+1. \end{aligned}$$
(46)
Further, we get
$$\begin{aligned}& \bigl\Vert \nabla\bigl(\psi^{n-1}-\psi_{N}^{n-1} \bigr) \bigr\Vert _{0} \\& \quad \le\sigma\bigl(N^{1-q}+ \bigl\Vert \omega^{n-1}- \omega_{N}^{n-1} \bigr\Vert _{0}\bigr), \quad n=1, 2, \ldots, M+1, 2\le q\le N+1. \end{aligned}$$
(47)
By using (
45) and (
28), the Hölder, Poincaré, and Cauchy–Schwarz inequalities, and Theorem
7, we obtain
$$\begin{aligned}& \bigl\Vert \omega^{n}-\omega_{N}^{n} \bigr\Vert _{0}^{2}+\frac{\mu\Delta t}{2} \bigl\Vert \nabla\bigl( \omega^{n}-\omega_{N}^{n}\bigr) \bigr\Vert _{0}^{2} \\& \quad = \int_{\varTheta}\bigl(\omega^{n}-\omega_{N}^{n} \bigr) \bigl(\omega^{n}-\omega _{N}^{n}\bigr)\,\mathrm{d}x\,\mathrm{d}y+\frac{\mu\Delta t}{2} \int_{\varTheta}\nabla\bigl(\omega ^{n}- \omega_{N}^{n}\bigr)\nabla\bigl(\omega^{n}- \omega_{N}^{n}\bigr)\,\mathrm{d}x\,\mathrm{d}y \\& \quad = \int_{\varTheta}\bigl(\omega^{n}-\omega_{N}^{n} \bigr) \bigl(\omega^{n}-R_{N}\omega ^{n}\bigr)\,\mathrm{d}x\,\mathrm{d}y+\frac{\mu\Delta t}{2} \bigl\Vert \nabla\bigl(\omega^{n}-R_{N} \omega ^{n}\bigr) \bigr\Vert _{0}^{2} \\& \qquad {}+ \int_{\varTheta}\bigl(\omega^{n}-\omega_{N}^{n} \bigr) \bigl(R_{N}\omega^{n}-\omega _{N}^{n} \bigr)\,\mathrm{d}x\,\mathrm{d}y \\& \qquad {}+\frac{\mu\Delta t}{2} \int_{\varTheta}\nabla\bigl(\omega ^{n}- \omega_{N}^{n}\bigr)\nabla\bigl(R_{N} \omega^{n}-\omega_{N}^{n}\bigr)\,\mathrm{d}x\,\mathrm{d}y \\& \quad = \int_{\varTheta}\bigl(\omega^{n}-\omega_{N}^{n} \bigr) \bigl(\omega^{n}-R_{N}\omega ^{n}\bigr)\,\mathrm{d}x\,\mathrm{d}y+\frac{\mu\Delta t}{2} \bigl\Vert \nabla\bigl(\omega^{n}-R_{N} \omega ^{n}\bigr) \bigr\Vert _{0}^{2} \\& \qquad {}+ \int_{\varTheta}\bigl(\omega^{n-1}-\omega_{N}^{n-1} \bigr) \bigl(R_{N}\omega^{n}-\omega ^{n}\bigr)\,\mathrm{d}x\,\mathrm{d}y+ \int_{\varTheta}\bigl(\omega^{n-1}-\omega_{N}^{n-1} \bigr) \bigl(\omega ^{n}-\omega_{N}^{n}\bigr)\,\mathrm{d}x\,\mathrm{d}y \\& \qquad {}-\frac{\mu\Delta t}{2} \int_{\varTheta}\nabla\bigl(\omega ^{n-1}-R_{N} \omega^{n-1}\bigr)\nabla\bigl(R_{N}\omega^{n}- \omega^{n}\bigr)\,\mathrm{d}x\,\mathrm{d}y \\& \qquad {}-\frac{\mu\Delta t}{2} \int_{\varTheta}\nabla\bigl(\omega^{n-1}-\omega _{N}^{n-1}\bigr)\nabla\bigl(\omega^{n}- \omega_{N}^{n}\bigr)\,\mathrm{d}x\,\mathrm{d}y \\& \quad \le\sigma N^{-q}\bigl( \bigl\Vert \omega^{n}- \omega_{N}^{n} \bigr\Vert _{0}+ \bigl\Vert \omega ^{n-1}-\omega_{N}^{n-1} \bigr\Vert _{0}\bigr)+\sigma\Delta tN^{2-2q} \\& \qquad {}+\frac{\mu\Delta t}{4} \bigl\Vert \nabla\bigl(\omega^{n-1}-R_{N} \omega^{n-1}\bigr) \bigr\Vert _{0}^{2}+ \frac{\mu\Delta t}{4} \bigl\Vert \nabla\bigl(\omega^{n}-R_{N} \omega^{n}\bigr) \bigr\Vert _{0}^{2} \\& \qquad {}+\frac{1}{2} \bigl\Vert \omega^{n-1}- \omega_{N}^{n-1} \bigr\Vert _{0}^{2}+ \frac{1}{2} \bigl\Vert \omega ^{n}-\omega_{N}^{n} \bigr\Vert _{0}^{2},\quad n=1, 2, \ldots, M, 2\le q\le N+1. \end{aligned}$$
(48)
Further, we get
$$\begin{aligned}& \bigl\Vert \omega^{n}-\omega_{N}^{n} \bigr\Vert _{0}^{2}+\frac{\mu\Delta t}{2} \bigl\Vert \nabla\bigl( \omega^{n}-\omega_{N}^{n}\bigr) \bigr\Vert _{0}^{2} \\& \quad \le \bigl\Vert \omega^{n-1}-\omega _{N}^{n-1} \bigr\Vert _{0}^{2}+\frac{\mu\Delta t}{2} \bigl\Vert \nabla \bigl(\omega^{n-1}-\omega _{N}^{n-1}\bigr) \bigr\Vert _{0}^{2} \\& \qquad {} +\sigma N^{-q}\bigl( \bigl\Vert \omega^{n}- \omega_{N}^{n} \bigr\Vert _{0}+ \bigl\Vert \omega ^{n-1}-\omega_{N}^{n-1} \bigr\Vert _{0}\bigr)+\sigma\Delta tN^{2-2q},\quad n=1, 2, \ldots, M. \end{aligned}$$
(49)
Summing (
49) from 1 to
n and using Theorem
7, we attain
$$\begin{aligned}& \bigl\Vert \omega^{n}-\omega_{N}^{n} \bigr\Vert _{0}^{2}+\frac{\mu\Delta t}{2} \bigl\Vert \nabla\bigl( \omega^{n}-\omega_{N}^{n}\bigr) \bigr\Vert _{0}^{2} \\& \quad \le \bigl\Vert \omega^{0}-R_{N}\omega ^{0} \bigr\Vert _{0}^{2}+\frac{\mu\Delta t}{2} \bigl\Vert \nabla\bigl(\omega^{0}-R_{N}\omega ^{0}\bigr) \bigr\Vert _{0}^{2} \\& \qquad {}+\sigma N^{-1}\sum_{i=0}^{n} \bigl\Vert \omega^{i}-\omega_{N}^{i} \bigr\Vert _{0}^{2}+\sigma \bigl(N^{2-2q}+n\Delta tN^{2-2q}\bigr) \\& \quad \le\sigma\bigl(N^{2-2q}+n\Delta tN^{2-2q}\bigr)+ \frac{1}{2N}\sum_{i=0}^{n} \bigl\Vert \omega^{i}-\omega_{N}^{i} \bigr\Vert _{0}^{2}, \quad n=1, 2, \ldots, M. \end{aligned}$$
(50)
When
N is sufficiently large such that
\(N^{-1}\le1/2\), from (
50) we attain
$$\begin{aligned}& \bigl\Vert \omega^{n}-\omega_{N}^{n} \bigr\Vert _{0}^{2}+\Delta t \bigl\Vert \nabla\bigl(\omega ^{n}-\omega_{N}^{n}\bigr) \bigr\Vert _{0}^{2} \\& \quad \le\sigma N^{2-2q}+ N^{-1}\sum _{i=0}^{n-1} \bigl\Vert \omega ^{i}- \omega_{N}^{i} \bigr\Vert _{0}^{2}, \quad n=1, 2, \ldots, M. \end{aligned}$$
(51)
By the discrete Gronwall inequality (Lemma
4) and from (
51) we obtain
$$\begin{aligned}& \bigl\Vert \omega^{n}-\omega_{N}^{n} \bigr\Vert _{0}^{2}+\Delta t \bigl\Vert \nabla\bigl(\omega ^{n}-\omega_{N}^{n}\bigr) \bigr\Vert _{0}^{2} \\& \quad \le\sigma N^{2-2q}\exp\bigl(nN^{-1}\bigr) \\& \quad \le\sigma N^{1-2q},\quad n=1, 2, \ldots, M, 2\le q\le N+1. \end{aligned}$$
(52)
By (
52) and Theorem
5 we obtain (
35). Combining (
52) with (
47) and Theorem
5, we attain (
36). This finishes the proof of Theorem
10. □