  Spectral element methods (SEM), which are extensions of finite element
methods (FEM), are important emerging techniques for solving partial
differential equations in physics and engineering. SEM can potentially deliver
better accuracy due to the potential superconvergence for well-shaped
tensor-product elements. However, for complex geometries, the accuracy of SEM
often degrades due to a combination of geometric inaccuracies near curved
boundaries and the loss of superconvergence with simplicial or
non-tensor-product elements. We propose to overcome the first issue by using
$h$- and $p$-geometric refinement, to refine the mesh near high-curvature
regions and increase the degree of geometric basis functions, respectively. We
show that when using mixed-meshes with tensor-product elements in the interior
of the domain, curvature-based geometric refinement near boundaries can improve
the accuracy of the interior elements by reducing pollution errors and
preserving the superconvergence. To overcome the second issue, we apply a
post-processing technique to recover the accuracy near the curved boundaries by
using the adaptive extended stencil finite element method (AES-FEM). The
combination of curvature-based geometric refinement and accurate
post-processing delivers an effective and easier-to-implement alternative to
other methods based on exact geometries. We demonstrate our techniques by
solving the convection-diffusion equation in 2D and show one to two orders of
magnitude of improvement in the solution accuracy, even when the elements are
poorly shaped near boundaries.
