Preserving Superconvergence of Spectral Elements for Curved Domains via $h$ and $p$-Geometric Refinement

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.