  We consider the ill-posed operator equation $Ax=y$ with an injective and
bounded linear operator $A$ mapping between $\ell^2$ and a Hilbert space $Y$,
possessing the unique solution \linebreak $x^\dag=\{x^\dag_k\}_{k=1}^\infty$.
For the cases that sparsity $x^\dag \in \ell^0$ is expected but often slightly
violated in practice, we investigate in comparison with the
$\ell^1$-regularization the elastic-net regularization, where the penalty is a
weighted superposition of the $\ell^1$-norm and the $\ell^2$-norm square, under
the assumption that $x^\dag \in \ell^1$. There occur two positive parameters in
this approach, the weight parameter $\eta$ and the regularization parameter as
the multiplier of the whole penalty in the Tikhonov functional, whereas only
one regularization parameter arises in $\ell^1$-regularization. Based on the
variational inequality approach for the description of the solution smoothness
with respect to the forward operator $A$ and exploiting the method of
approximate source conditions, we present some results to estimate the rate of
convergence for the elastic-net regularization. The occurring rate function
contains the rate of the decay $x^\dag_k \to 0$ for $k \to \infty$ and the
classical smoothness properties of $x^\dag$ as an element in $\ell^2$.
