may be, that belong to $C^{r}$. The inner product for real or complex-valued functions in $L^{2}$ is denoted by $(\cdot,\cdot)$ and the associated norm by $||\cdot ||$.

The hybrid spectral-finite element scheme {#sec2}
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We consider the periodic initial-value problem for the Benjamin equation, i. e. for $t\geq 0$ we seek a $2L-$periodic real function $u=u(x,t)$ such that $$\begin{aligned}
& u_t+\alpha u_x+\beta u u_x-\gamma \mathcal{G}u_{xx}-\delta u_{xxx}=0,\quad x\in[-L,L], \quad t>0,  \label{E21}\\
& u(x,0)=u_0(x),\quad x\in [-L,L]\nonumber\end{aligned}$$ where $u_{0}$ is a given smooth $2L-$periodic function and $\alpha,\beta, \gamma, \delta$ positive constants. The operator $\mathcal{G}$ is the Hilbert transform acting on $2L-$periodic functions; for the purposes of this section it will be represented by its principal-value integral form [@ABFS] $$\begin{aligned}
\mathcal{G}f(x):=\frac{1}{2L}p.v.\int_{-L}^{L}{\it cot}\left(\frac{\pi (x-y)}{2L}\right)f(y)dy,\label{E22}\end{aligned}$$ where $f$ is $2L-$periodic. In the sequel we will assume that the solution of (\[E21\]) is sufficiently smooth. For simplicity, we assume that the problem (\[E21\]) has been transformed onto the spatial interval $[0,2\pi]$.

The semidiscrete hybrid scheme {#sec21}
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For integer $r\geq 3$ and an even integer $N$, let $h=2\pi/N$, $x_{j}=jh, j=0,\ldots,N$, and consider the finite dimensional spaces $$S_N=\mbox{span}\left\{e^{{\rm i}kx}:\, k\in \mathbb{Z},\,-N/2\leq k\leq N/2-1 \right\},$$ and $$S_h=\left\{\phi\in C^{r-2}_p: \phi_{|_{[x_j,x_{j+1}]}} \in\mathbb{P}_{r-1},\, 0\leq j\leq N-1 \right\}.$$ The hybrid spectral-finite element approximation $u_{h}$ 