Preserving superconvergence of spectral elements in three-dimensional curved domains via ApSEM

Spectral element methods (SEM), extensions of finite element methods (FEM), have emerged as significant techniques for solving partial differential equations in physics and engineering. SEM can deliver superior accuracy due to the potential superconvergence in nodal solutions for well-shaped tensor-product elements. However, the accuracy of SEM often degrades in complex geometries due to geometric inaccuracies near curved boundaries and the loss of superconvergence with non-tensor-product or poorly shaped tensor-product elements. To overcome this, we introduce ApSEM (AES-FEM post-processed Spectral Element Method) to recover the accuracy near the curved boundaries when solving elliptic PDEs. The combination of curvature-based geometric refinement (Jones et al. in Eng Comput 40:1877–1892, 2024) and accurate post-processing offers an effective alternative to methods that rely on exact geometries. We demonstrate our techniques by solving the convection-diffusion equation in 3D and show up to two orders of magnitude improvement in solution accuracy, even when the elements are poorly shaped near curved boundaries. We also show the efficiency of ApSEM as it can recover superconvergence in nodal solutions without significantly increasing computational cost.