  In this paper, we study a newly developed shearlet system on bounded domains
which yields frames for $H^s(\Omega)$ for some $s\in \mathbb{N}$, $\Omega
\subset \mathbb{R}^2$. We will derive approximation rates with respect to
$H^s(\Omega)$ norms for functions whose derivatives admit smooth jumps along
curves and demonstrate superior rates to those provided by pure wavelet
systems. These improved approximation rates demonstrate the potential of the
novel shearlet system for the discretization of partial differential equations.
Therefore, we implement an adaptive shearlet-based algorithm for the solution
of an elliptic PDE and analyze its computational complexity and convergence
properties.
