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.

翻译：谱元法（SEM）作为有限元法（FEM）的推广，是解决物理与工程中偏微分方程的重要新兴技术。对于形状良好的张量积单元，SEM凭借其潜在的超收敛性可实现更高的计算精度。然而在复杂几何中，曲面边界附近的几何不准确性以及单纯形或非张量积单元超收敛性的丧失，常导致SEM精度下降。针对首个问题，我们提出采用$h$几何细化与$p$几何细化策略——前者对高曲率区域进行网格加密，后者提升几何基函数的阶次。研究表明：当域内部采用混合网格（含张量积单元）时，基于曲率的边界几何细化可通过降低污染误差并保留超收敛性，提升内部单元的计算精度。针对第二个问题，我们引入自适应扩展模板有限元法（AES-FEM）的后处理技术，恢复曲面边界附近的计算精度。曲率导向的几何细化与精确后处理的结合，为实现精确几何的替代方法提供了高效且易操作的方案。通过求解二维对流-扩散方程验证该技术，结果表明：即使边界单元形状不佳，求解精度仍可提升1-2个数量级。