An $h$-adaptive Tetrahedral Spectral Element Method with Applications to Kohn-Sham Density Functional Theory

Abstract

High-order $h$-adaptive spectral element methods on tetrahedral meshes provide an effective framework for resolving localized singularities and multiscale structures in complex three-dimensional geometries. However, their development is often hindered by difficulties in maintaining $C^0$ continuity across refinement interfaces and efficiently transferring solutions between adaptive meshes. Such limitations are particularly relevant in demanding applications such as all-electron Kohn-Sham density functional theory, which place stringent requirements on the accurate resolution of both nuclear singularities and multiple physical scales. In this paper, we present an efficient $h$-adaptive tetrahedral spectral element framework. To address the continuity challenge, we develop an adaptive strategy that combines element orientation alignment with geometric red-green refinement, thereby eliminating the need for algebraic hanging-node constraints while preserving inter-element continuity. Furthermore, an efficient topology-based point-location algorithm is introduced to accelerate interpolation between adaptive meshes. Numerical experiments on Poisson and Laplacian eigenvalue problems confirm the spectral convergence of the proposed method. Applications to all-electron Kohn-Sham equations further demonstrate its capability to accurately resolve nuclear singularities. Moreover, parallel performance studies exhibit excellent scalability, with matrix assembly and adaptivity modules generally achieving speedups above 15 times and the proposed interpolation algorithm attaining speedups ranging from 25 to 35 on 64-core configurations compared to the single-core performance. These results indicate that the proposed framework provides an accurate, robust, and efficient solution for large-scale, high-resolution simulations.