${{\varrho}}_i=1$ for $i=n+1,\ldots,2n$.

We define an approximate solution as a pair of functions $(v_{n,m}; u_n)$: $$\begin{gathered}
v_{n,m}(t)=\sum_{i=1}^m \alpha_i(t)\psi_i +\sum_{j=1}^{2n} \dot{\beta}_j(t)\phi_j, \notag \\  u_n(t)=\sum_{j=1}^{2n}\beta_j(t)\zeta_j+ (0;0;(I-\widehat{P})w_0), \label{approx_sol}\end{gathered}$$ which satisfy the relations $$\dot{\alpha}_k(t) +\sum_{j=1}^{2n} \ddot{\beta}_j(t)(\phi_j,\psi_k)_{{\cal O}}+\nu\mu_k \alpha_k(t)+\nu\sum_{j=1}^{2n} \dot{\beta}_j(t)E(\phi_j,\psi_k)_{{\cal O}}=(G_f, \psi_k)_{{\cal O}}\label{psi_eq}$$ for $k=1...m$, and $$\begin{gathered}
\sum_{i=1}^m \dot{\alpha}_i(t)(\psi_i, \phi_k)_{{\cal O}}+\sum_{j=1}^{2n} \ddot{\beta}_j(t)(\phi_j,\phi_k)_{{\cal O}}+{{\varrho}}_k\ddot{\beta}_k(t)    \\ +
\nu\sum_{i=1}^m  \alpha_i(t)E(\psi_i, \phi_k)_{{\cal O}}+
\nu\sum_{j=1}^{2n} \dot{\beta}_j(t) E(\phi_j,\phi_k)_{{\cal O}}+\kappa_k \beta_k(t) +  \\
+{{\gamma}}_k\dot{\beta}_k(t)   +q(u_n(t), \zeta_k) =
(G_f(t), \phi_k)_{{\cal O}}+(G_{sh}(t), \zeta_k)_{{\Omega}}\label{phi_eq}\end{gathered}$$ for $k=1,\dots,2n$. This system of ordinary differential equations is endowed with the initial data $$v_{v,m}(0)=\Pi_m(v_0-N_0(0;0;w_1))+N_0(0;0;P_n w_1),$$ $$u_n(0)=(R_n(u^1_0; u^2_0); P_n\widehat{P}w_0 + (I-\widehat{P})w_0), \; \dot{u}_n(0)=(R_n(u^1_1; u^2_1); P_n w_1),$$ where $\Pi_m$ is the orthoprojector on $Lin\{\psi_j : j=1,\ldots,m,\}$ in $\widetilde{X}$, $P_n$ is orthoprojector on $Lin\{\xi_i : i=1,\ldots,n\}$ in ${\widehat L_2({{\Omega}})}$ and $R_n$ is orthoprojector on $Lin\{\eta_i : i=1,\ldots,n\}$ in $L_2({{\Omega}})\times L_2({{\Omega}})$. Since $\Pi_m$, $R_n$ and $P_n$ are spectral projectors we have that $$\label{id-conv}
(v_{v,m}(0);u_n(0);  \dot{u}_n(0))\to  (v_0;u_0;u_1)~~
\mbox{strongly in ${{\cal H}}$ as $m,n\to\infty$.}$$

We can rewrite system and as $$M\frac{d}{dt}\left(\begin{matrix}\alpha(t) \\ \dot{\beta}(t)\end{matrix}\right) +g(\alpha(t), \beta(t), \dot{\beta}(t))=G(t)$$ for some locally Lipschitz function $g\,: {{\mathbb R}}^{m+4n}\mapsto {{\mathbb R}}^{m+2n}$, where the function $G$ lies in $L_2(0,T; {{\mathbb R}}^{m+2n})$ and $$\label{matr-M}
M=
\left[\begin{matrix}0&0\\0&  {\mathscr{R}}\end{matrix}\right]+
\left[\begin{matrix} \{(\psi_i, \psi_j)_{{\cal O}}\}_{j,k=1}^m & \{(\psi_l, \phi_k)_{{\cal O}}\}_{l,k=1}^{m,2n} \\ \{(\phi_k, \psi_l)_{{\cal O}}\}_{l,k=1}^{2n,m} &
\{ (\phi_i, \phi_j))_{{\cal 