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首页> 外文期刊>Journal of Computational Physics >Higher-order adaptive finite-element methods for orbital-free density functional theory
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Higher-order adaptive finite-element methods for orbital-free density functional theory

机译：无轨道密度泛函理论的高阶自适应有限元方法

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In the present work, we study various numerical aspects of higher-order finite-element discretizations of the non-linear saddle-point formulation of orbital-free density-functional theory. We first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem. We find that a staggered solution procedure where the potential fields are computed consistently for every trial electron-density is a robust solution procedure for higher-order finite-element discretizations. We next study the convergence properties of higher-order finite-element discretizations of orbital-free density functional theory by considering benchmark problems that include calculations involving both pseudopotential as well as Coulomb singular potential fields. Our numerical studies suggest close to optimal rates of convergence on all benchmark problems for various orders of finite-element approximations considered in the present study. We finally investigate the computational efficiency afforded by various higher-order finite-element discretizations, which constitutes the main aspect of the present work, by measuring the CPU time for the solution of discrete equations on benchmark problems that include large Aluminum clusters. In these studies, we use mesh coarse-graining rates that are derived from error estimates and an a priori knowledge of the asymptotic solution of the far-field electronic fields. Our studies reveal a significant 100-1000 fold computational savings afforded by the use of higher-order finite-element discretization, alongside providing the desired chemical accuracy. We consider this study as a step towards developing a robust and computationally efficient discretization of electronic structure calculations using the finite-element basis.
机译：在目前的工作中，我们研究了无轨道密度泛函理论的非线性鞍点公式的高阶有限元离散化的各个数值方面。我们首先通过分析离散问题的可解性条件来研究可行解决方案的鲁棒性。我们发现，对于每个试验电子密度一致地计算势场的交错求解程序，对于高阶有限元离散化来说是一个鲁棒的求解程序。接下来，我们通过考虑基准问题来研究无轨密度泛函理论的高阶有限元离散化的收敛性质，其中包括涉及伪势和库仑奇异势场的计算。我们的数值研究表明，对于本研究中考虑的各种阶次的有限元近似值，所有基准问题的收敛速度都接近最佳值。最后，我们通过测量用于解决包含大型铝团簇的基准问题的离散方程的CPU时间，来研究构成本工作主要方面的各种高阶有限元离散化所提供的计算效率。在这些研究中，我们使用了网格粗粒度速率，该速率是从误差估计和远场电子场渐近解的先验知识中得出的。我们的研究表明，通过使用高阶有限元离散化技术，以及提供所需的化学精度，可节省100-1000倍的计算量。我们认为这项研究是朝着使用有限元基础发展鲁棒且计算效率高的电子结构计算离散化迈出的一步。

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