用于Kohn-Sham密度泛函理论的h自适应四面体谱元方法
An $h$-adaptive Tetrahedral Spectral Element Method with Applications to Kohn-Sham Density Functional Theory
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中文总结 AI 辅助
本文提出结合单元方向对齐与几何红绿加密的h自适应四面体谱元框架，引入拓扑点定位插值算法，在Kohn-Sham密度泛函理论等问题中实现高效高精度求解，具备优异并行可扩展性。
中文摘要 AI 辅助
四面体网格上的高阶h自适应谱元方法为求解复杂三维几何中的局域奇异性和多尺度结构提供了有效框架。然而，其发展常受限于难以在加密界面保持C⁰连续性以及在自适应网格间高效传递解的问题，这些限制在全电子Kohn-Sham密度泛函理论等严苛应用中尤为突出，该应用对核奇异性和多物理尺度的精确求解有严格要求。本文提出一种高效的h自适应四面体谱元框架：为解决连续性问题，开发了结合单元方向对齐与几何红绿加密的自适应策略，无需代数悬挂节点约束即可保持单元间连续性；此外，引入高效的基于拓扑的点定位算法以加速自适应网格间的插值。对泊松和拉普拉斯特征值问题的数值实验证实了该方法的谱收敛性；对全电子Kohn-Sham方程的应用进一步证明其精确求解核奇异性的能力；并行性能研究显示其具备优异的可扩展性，在64核配置下，与单核性能相比，矩阵组装和自适应模块通常实现15倍以上的加速，所提插值算法实现25至35倍的加速。这些结果表明，该框架为大规模高分辨率模拟提供了精确、鲁棒且高效的解决方案。
英文摘要
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.
