Multilevel correction type of adaptive finite element method for Hartree-Fock equation
Fei Xu
TL;DR
This work addresses the computational bottleneck of solving the 3D Hartree–Fock equation by introducing a multilevel correction adaptive finite element method. It replaces large nonlinear eigenvalue solves with a sequence of linearized boundary-value problems and a low-dimensional correction space per orbital, with a fixed coarse space to enable preprocessing. The method achieves substantial speedups and memory savings and exhibits strong parallel scalability by decoupling orbitals and enabling tensor-based reuse. This approach lowers the resource barrier for accurate 3D HF and provides a solid foundation for extending to hybrid DFT with exact exchange.
Abstract
This paper proposes an efficient algorithm for solving the Hartree--Fock equation combining a multilevel correction scheme with an adaptive refinement technique to improve computational efficiency. The algorithm integrates a multilevel correction framework with an optimized implementation strategy. Within this framework, a series of linearized boundary value problems are solved, and their approximate solutions are corrected by solving small-scale Hartree--Fock equations in low-dimensional correction spaces. The correction space comprises a coarse space and the solution to the linearized boundary value problem, enabling high accuracy while preserving low-dimensional characteristics. The proposed algorithm efficiently addresses the inherent computational complexity of the Hartree--Fock equation. Innovative correction strategies eliminate the need for direct computation of large-scale nonlinear eigenvalue systems and dense matrix operations. Furthermore, optimization techniques based on precomputations within the correction space render the total computational workload nearly independent of the number of self-consistent field iterations. This approach significantly accelerates the solution process of the Hartree--Fock equation, effectively mitigating the traditional exponential scaling demands on computational resources while maintaining precision.