Multilevel correction type of adaptive finite element method for Hartree-Fock equation
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917007144779776 |
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| author | Xu, Fei |
| author_facet | Xu, Fei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10879 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multilevel correction type of adaptive finite element method for Hartree-Fock equation Xu, Fei Numerical Analysis 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. |
| title | Multilevel correction type of adaptive finite element method for Hartree-Fock equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2510.10879 |