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学者姓名：黄秋梅
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Abstract ：
In this article, an improved self-consistent field iteration scheme is introduced. The proposed method has essential applications in Kohn--Sham density functional theory and relies on an extrapolation scheme and the least squares method. Moreover, the proposed solution is easy to implement and can accelerate the convergence of self-consistent field iteration. The main idea is to fit out a polynomial based on the errors of the derived approximate solutions and then extrapolate the errors into zero to obtain a new approximation. The developed scheme can be applied not only to the Kohn--Sham density functional theory but also to accelerate the self-consistent field iterations of other nonlinear equations. Some numerical results for the Kohn--Sham equation and general nonlinear equations are presented to validate the efficiency of the new method.
Keyword ：
self -consistent field iteration self -consistent field iteration acceleration acceleration Kohn --Sham density functional theory Kohn --Sham density functional theory
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| GB/T 7714 | Xu, Fei , Huang, Qiumei . IMPROVED SELF-CONSISTENT FIELD ITERATION FOR KOHN--SHAM DENSITY FUNCTIONAL THEORY\ast [J]. | MULTISCALE MODELING & SIMULATION , 2024 , 22 (1) : 142-154 . |
| MLA | Xu, Fei 等. "IMPROVED SELF-CONSISTENT FIELD ITERATION FOR KOHN--SHAM DENSITY FUNCTIONAL THEORY\ast" . | MULTISCALE MODELING & SIMULATION 22 . 1 (2024) : 142-154 . |
| APA | Xu, Fei , Huang, Qiumei . IMPROVED SELF-CONSISTENT FIELD ITERATION FOR KOHN--SHAM DENSITY FUNCTIONAL THEORY\ast . | MULTISCALE MODELING & SIMULATION , 2024 , 22 (1) , 142-154 . |
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Abstract ：
In this paper a nonpolynomial Jacobi spectral-collocation (NJSC) method for the second kind Fredholm integral equations (FIEs) with weakly singular kernel |s-t|(-gamma) is proposed. By dividing the integral interval symmetrically into two parts and applying the NJSC method symmetrically to the two weakly singular FIEs respectively, the mild singularities of the interval endpoints can be captured and the exponential convergence can be obtained. A detailed L-infinity convergence analysis of the numerical solution is derived. The NJSC method is then extended to two dimensional case and similar exponential convergence results are obtained for low regular solutions. Numerical examples are presented to demonstrate the efficiency of the proposed method.
Keyword ：
exponential convergence exponential convergence Fredholm integral equations Fredholm integral equations weakly singular weakly singular Nonpolynomial Jacobi spectral-collocation method Nonpolynomial Jacobi spectral-collocation method
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| GB/T 7714 | Huang, Qiumei , Wang, Min . Nonpolynomial Jacobi Spectral-Collocation Method for Weakly Singular Fredholm Integral Equations of the Second Kind [J]. | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS , 2024 . |
| MLA | Huang, Qiumei 等. "Nonpolynomial Jacobi Spectral-Collocation Method for Weakly Singular Fredholm Integral Equations of the Second Kind" . | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS (2024) . |
| APA | Huang, Qiumei , Wang, Min . Nonpolynomial Jacobi Spectral-Collocation Method for Weakly Singular Fredholm Integral Equations of the Second Kind . | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS , 2024 . |
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Abstract ：
数值计算方法是高等学校数学专业和理工科专业必修的公共基础课,通过数值方法可以解决许多实际问题.介绍了国产通用型科学计算软件——北太天元(baltamatica)在数值计算方法教学中的应用.从介绍求解Helmholtz方程的有限差分法出发,探讨了在数值计算方法课程中使用baltamatica进行算法实现、可视化和实际问题求解的教学方法.通过编写代码、绘制图形和求解实际问题,学生能够深入理解数值计算方法的原理和应用,有助于培养他们解决实际问题的能力.最后,总结了baltamatica在数值计算方法教学中的优点.
Keyword ：
Helmholtz方程 Helmholtz方程 北太天元 北太天元 可视化 可视化 有限差分法 有限差分法 算法实现 算法实现
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| GB/T 7714 | 江雪 , 黄秋梅 . 北太天元在数值计算方法教学中的应用 [J]. | 数学的实践与认识 , 2024 , 54 (2) : 226-231 . |
| MLA | 江雪 等. "北太天元在数值计算方法教学中的应用" . | 数学的实践与认识 54 . 2 (2024) : 226-231 . |
| APA | 江雪 , 黄秋梅 . 北太天元在数值计算方法教学中的应用 . | 数学的实践与认识 , 2024 , 54 (2) , 226-231 . |
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Abstract ：
In this paper, a discontinuous Galerkin (DG) time stepping method combined with the standard finite element method in space is proposed to solve a class of semilinear parabolic differential equations with time constant delay. The time semi-discretization and the relevant global convergence of the DG solution under suitable uniform meshes are derived. The standard Galerkin method in space is used to obtain the fully discrete scheme and the optimal global convergence of the full discretization is presented. Numerical experiments for one-dimensional and two-dimensional equations are provided to demonstrate the theoretical results.
Keyword ：
Finite element method Finite element method Time constant delay Time constant delay Discontinuous Galerkin time stepping Discontinuous Galerkin time stepping Fully discrete scheme Fully discrete scheme Parabolic equations Parabolic equations
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| GB/T 7714 | Xu, Xiuxiu , Huang, Qiumei . Discontinuous Galerkin Time Stepping for Semilinear Parabolic Problems with Time Constant Delay [J]. | JOURNAL OF SCIENTIFIC COMPUTING , 2023 , 96 (2) . |
| MLA | Xu, Xiuxiu 等. "Discontinuous Galerkin Time Stepping for Semilinear Parabolic Problems with Time Constant Delay" . | JOURNAL OF SCIENTIFIC COMPUTING 96 . 2 (2023) . |
| APA | Xu, Xiuxiu , Huang, Qiumei . Discontinuous Galerkin Time Stepping for Semilinear Parabolic Problems with Time Constant Delay . | JOURNAL OF SCIENTIFIC COMPUTING , 2023 , 96 (2) . |
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Abstract ：
This study used a multigrid-based convolutional neural network architecture known as MgNet in operator learning to solve numerical partial differential equations (PDEs). Given the property of smoothing iterations in multigrid methods where low-frequency errors decay slowly, we introduced a low-frequency correction structure for residuals to enhance the standard V-cycle MgNet. The enhanced MgNet model can capture the low-frequency features of solutions considerably better than the standard V-cycle MgNet. The numerical results obtained using some standard operator learning tasks are better than those obtained using many state-of-the-art methods, demonstrating the efficiency of our model. Moreover, numerically, our new model is more robust in case of low- and high-resolution data during training and testing, respectively.
Keyword ：
Low-frequency correction Low-frequency correction Numerical partial differential equations Numerical partial differential equations Operator learning Operator learning MgNet MgNet
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| GB/T 7714 | Zhu, Jianqing , He, Juncai , Huang, Qiumei . An enhanced V-cycle MgNet model for operator learning in numerical partial differential equations [J]. | COMPUTATIONAL GEOSCIENCES , 2023 , 28 (5) : 809-820 . |
| MLA | Zhu, Jianqing 等. "An enhanced V-cycle MgNet model for operator learning in numerical partial differential equations" . | COMPUTATIONAL GEOSCIENCES 28 . 5 (2023) : 809-820 . |
| APA | Zhu, Jianqing , He, Juncai , Huang, Qiumei . An enhanced V-cycle MgNet model for operator learning in numerical partial differential equations . | COMPUTATIONAL GEOSCIENCES , 2023 , 28 (5) , 809-820 . |
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Abstract ：
In this paper, we propose a type of adaptive multigrid method for eigenvalue problem based on the multilevel correction method and adaptive multigrid method. Different from the standard adaptive finite element method applied to eigenvalue problem, with our method we only need to solve a linear boundary value problem on each adaptive space and then correct the approximate solution by solving a low dimensional eigenvalue problem. Further, the involved boundary value problems are solved by some adaptive multigrid iteration steps. The proposed adaptive algorithm can reach the same accuracy as the standard adaptive finite element method for eigenvalue problem but evidently reduces the computational work. In addition, the corresponding convergence and optimal complexity analysis are derived theoretically and numerically, respectively.
Keyword ：
convergence convergence optimal complexity optimal complexity Eigenvalue problem Eigenvalue problem adaptive multigrid method adaptive multigrid method multilevel correction multilevel correction
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| GB/T 7714 | Xu, Fei , Huang, Qiumei , Chen, Shuangshuang et al. ADAPTIVE MULTIGRID METHOD FOR EIGENVALUE PROBLEM [J]. | INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING , 2022 , 19 (1) : 1-18 . |
| MLA | Xu, Fei et al. "ADAPTIVE MULTIGRID METHOD FOR EIGENVALUE PROBLEM" . | INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING 19 . 1 (2022) : 1-18 . |
| APA | Xu, Fei , Huang, Qiumei , Chen, Shuangshuang , Ma, Hongkun . ADAPTIVE MULTIGRID METHOD FOR EIGENVALUE PROBLEM . | INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING , 2022 , 19 (1) , 1-18 . |
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Abstract ：
This paper aims to introduce a novel adaptive multigrid method for the elasticity eigenvalue problem. Different from the developing adaptive algorithms for the elasticity eigenvalue problem, the proposed approach transforms the elasticity eigenvalue problem into a series of boundary value problems in the adaptive spaces and some small-scale elasticity eigenvalue problems in a low-dimensional space. As our algorithm avoids solving large-scale elasticity eigenvalue problems, which is time-consuming, and the boundary value problem can be solved efficiently by the adaptive multigrid method, our algorithm can evidently improve the overall solving efficiency for the elasticity eigenvalue problem. Meanwhile, we present a rigorous theoretical analysis of the convergence and optimal complexity. Finally, some numerical experiments are presented to validate the theoretical conclusions and verify the numerical efficiency of our approach.
Keyword ：
Elasticity eigenvalue problem Elasticity eigenvalue problem Adaptive multigrid method Adaptive multigrid method Convergence and optimal complexity Convergence and optimal complexity Finite element method Finite element method
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| GB/T 7714 | Xu, Fei , Huang, Qiumei , Xie, Manting . An efficient adaptive multigrid method for the elasticity eigenvalue problem [J]. | BIT NUMERICAL MATHEMATICS , 2022 , 62 (4) : 2005-2033 . |
| MLA | Xu, Fei et al. "An efficient adaptive multigrid method for the elasticity eigenvalue problem" . | BIT NUMERICAL MATHEMATICS 62 . 4 (2022) : 2005-2033 . |
| APA | Xu, Fei , Huang, Qiumei , Xie, Manting . An efficient adaptive multigrid method for the elasticity eigenvalue problem . | BIT NUMERICAL MATHEMATICS , 2022 , 62 (4) , 2005-2033 . |
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Abstract ：
In this study, a multilevel local defect-correction method is designed for solving nonsymmetric eigenvalue problems. The main feature of our approach is the transformation of the nonsymmetric eigenvalue problems into several symmetric boundary value problems defined in a multilevel finite element space sequence and some low-dimensional nonsymmetric eigenvalue problems defined in a specially designed correction space. Moreover, the symmetric boundary value problems involved in our algorithm are solved by the local defect-correction strategy that divides the computing domain into small-scale subdomains. Since solving the high-dimensional nonsymmetric eigenvalue problems is avoided which is quite time-consuming compared with that of solving boundary value problems, the presented algorithm greatly improves the solving efficiency for nonsymmetric eigenvalue problems. Rigorous theoretical analysis and several numerical experiments are given to demonstrate the efficiency of our algorithm.
Keyword ：
Multilevel correction method Multilevel correction method Nonsymmetric eigenvalue problems Nonsymmetric eigenvalue problems Local defect-correction method Local defect-correction method
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| GB/T 7714 | Xu, Fei , Huang, Qiumei , Dai, Haishen et al. Multilevel Local Defect-Correction Method for Nonsymmetric Eigenvalue Problems [J]. | JOURNAL OF SCIENTIFIC COMPUTING , 2022 , 92 (3) . |
| MLA | Xu, Fei et al. "Multilevel Local Defect-Correction Method for Nonsymmetric Eigenvalue Problems" . | JOURNAL OF SCIENTIFIC COMPUTING 92 . 3 (2022) . |
| APA | Xu, Fei , Huang, Qiumei , Dai, Haishen , Ma, Hongkun . Multilevel Local Defect-Correction Method for Nonsymmetric Eigenvalue Problems . | JOURNAL OF SCIENTIFIC COMPUTING , 2022 , 92 (3) . |
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Abstract ：
In this study, a multilevel correction-type goal-oriented adaptive finite element method is designed for semilinear elliptic equations. Concurrently, the corresponding convergence property is theoretically proved. In the novel goal-oriented adaptive finite element method, only a linearized primal equation and a linearized dual equation are required to be solved in each adaptive finite element space. To ensure convergence, the approximate solution of the primal equation was corrected by solving a small-scale semilinear elliptic equation after the central solving process in each adaptive finite element space. Since solving of the large-scale semilinear elliptic equations is avoided and the goal-oriented technique is absorbed, there has been a significant improvement in the solving efficiency for the goal functional of semilinear elliptic equations. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.
Keyword ：
Adaptive finite element method Adaptive finite element method Multilevel correction method Multilevel correction method Convergence Convergence Goal-oriented Goal-oriented
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| GB/T 7714 | Xu, Fei , Huang, Qiumei , Yang, Huiting et al. Multilevel correction goal-oriented adaptive finite element method for semilinear elliptic equations [J]. | APPLIED NUMERICAL MATHEMATICS , 2022 , 172 : 224-241 . |
| MLA | Xu, Fei et al. "Multilevel correction goal-oriented adaptive finite element method for semilinear elliptic equations" . | APPLIED NUMERICAL MATHEMATICS 172 (2022) : 224-241 . |
| APA | Xu, Fei , Huang, Qiumei , Yang, Huiting , Ma, Hongkun . Multilevel correction goal-oriented adaptive finite element method for semilinear elliptic equations . | APPLIED NUMERICAL MATHEMATICS , 2022 , 172 , 224-241 . |
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Abstract ：
In this paper, we study an efficient multigrid method to solve the semilinear interface problems. We first give an optimal finite element error estimate for the semilinear interface problems under a weak condition for the nonlinear term compared with the existing conclusions. Then next based on the finite element error estimate, we design a novel multigrid method for semilinear elliptic problems. The proposed multigrid method only requires to solve a linear interface problem in each level of the multilevel space sequence and a small-scale semilinear interface problem in a correction space. The involved linear interface problem can be solved efficiently by the multigrid iteration. The dimension of the correction space is small and fixed, which is independent from the fine spaces. Thus the computational time of the correction step is negligible compared with that of the linear interface problems in the fine spaces. On the whole, the efficiency of the presented multigrid method is nearly the same as that of the multigrid method for linear interface problems. Additionally, unlike the existing finite element error estimates and the multigrid methods for semilinear interface problems, which always require the bounded second order derivatives of the nonlinear terms, all the analysis in our paper only requires a Lipschitz continuous condition. (C) 2022 IMACS. Published by Elsevier B.V. All rights reserved.
Keyword ：
Multigrid method Multigrid method Finite element method Finite element method Semilinear interface problem Semilinear interface problem
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| GB/T 7714 | Xu, Fei , Guo, Yasai , Huang, Qiumei et al. An efficient multigrid method for semilinear interface problems [J]. | APPLIED NUMERICAL MATHEMATICS , 2022 , 179 : 238-254 . |
| MLA | Xu, Fei et al. "An efficient multigrid method for semilinear interface problems" . | APPLIED NUMERICAL MATHEMATICS 179 (2022) : 238-254 . |
| APA | Xu, Fei , Guo, Yasai , Huang, Qiumei , Ma, Hongkun . An efficient multigrid method for semilinear interface problems . | APPLIED NUMERICAL MATHEMATICS , 2022 , 179 , 238-254 . |
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