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学者姓名：黄秋梅
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摘要 ：
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.
关键词 ：
Adaptive finite element method Adaptive finite element method Convergence Convergence Goal-oriented Goal-oriented Multilevel correction method Multilevel correction method
引用：
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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 . |
| 导入链接 | NoteExpress RIS BibTex |
摘要 ：
In this paper we propose and analyze a second order accurate (in time) numerical scheme for the square phase field crystal equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. Its primary difference with the standard phase field crystal model is an introduction of the 4-Laplacian term in the free energy potential, which in turn leads to a much higher degree of nonlinearity. To make the numerical scheme linear while preserving the nonlinear energy stability, we make use of the scalar auxiliary variable (SAV) approach, in which a second order backward differentiation formula is applied in the temporal stencil. Meanwhile, a direct application of the SAV method faces certain difficulties, due to the involvement of the 4-Laplacian term, combined with a derivation of the lower bound of the nonlinear energy functional. In the proposed numerical method, an appropriate decomposition for the physical energy functional is formulated, so that the nonlinear energy part has a well-established global lower bound, and the rest terms lead to constant-coefficient diffusion terms with positive eigenvalues. In turn, the numerical scheme could be very efficiently implemented by constant-coefficient Poisson-like type solvers (via FFT), and energy stability is established by introducing an auxiliary variable, and an optimal rate convergence analysis is provided for the proposed SAV method. A few numerical experiments are also presented, which confirm the efficiency and accuracy of the proposed scheme.
关键词 ：
Energy stability Energy stability Fourier pseudo-spectral approximation Fourier pseudo-spectral approximation Optimal rate convergence analysis Optimal rate convergence analysis Second order BDF stencil Second order BDF stencil Square phase field crystal equation Square phase field crystal equation The Scalar auxiliary variable (SAV)method The Scalar auxiliary variable (SAV)method
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| GB/T 7714 | Wang, Min , Huang, Qiumei , Wang, Cheng . A Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Square Phase Field Crystal Equation [J]. | JOURNAL OF SCIENTIFIC COMPUTING , 2021 , 88 (2) . |
| MLA | Wang, Min et al. "A Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Square Phase Field Crystal Equation" . | JOURNAL OF SCIENTIFIC COMPUTING 88 . 2 (2021) . |
| APA | Wang, Min , Huang, Qiumei , Wang, Cheng . A Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Square Phase Field Crystal Equation . | JOURNAL OF SCIENTIFIC COMPUTING , 2021 , 88 (2) . |
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摘要 ：
This paper presents a new type of local and parallel multigrid method to solve semilinear elliptic equations. The proposed method does not directly solve the semilinear elliptic equations on each layer of the multigrid mesh sequence, but transforms the semilinear elliptic equations into several linear elliptic equations on the multigrid mesh sequence and some low-dimensional semilinear elliptic equations on the coarsest mesh. Furthermore, the local and parallel strategy is used to solve the involved linear elliptic equations. Since solving large-scale semilinear elliptic equations in fine space, which can be fairly time-consuming, is avoided, the proposed local and parallel multigrid scheme will significantly improve the solving efficiency for the semilinear elliptic equations. Besides, compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only requires the Lipschitz continuation property of the nonlinear term. We make a rigorous theoretical analysis of the presented local and parallel multigrid scheme, and propose some numerical experiments to support the theory. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.
关键词 ：
Local and parallel Local and parallel Multigrid method Multigrid method Multilevel correction method Multilevel correction method Semilinear elliptic equations Semilinear elliptic equations
引用：
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| GB/T 7714 | Xu, Fei , Huang, Qiumei , Jiang, Kun et al. Local and parallel multigrid method for semilinear elliptic equations [J]. | APPLIED NUMERICAL MATHEMATICS , 2021 , 162 : 20-34 . |
| MLA | Xu, Fei et al. "Local and parallel multigrid method for semilinear elliptic equations" . | APPLIED NUMERICAL MATHEMATICS 162 (2021) : 20-34 . |
| APA | Xu, Fei , Huang, Qiumei , Jiang, Kun , Ma, Hongkun . Local and parallel multigrid method for semilinear elliptic equations . | APPLIED NUMERICAL MATHEMATICS , 2021 , 162 , 20-34 . |
| 导入链接 | NoteExpress RIS BibTex |
摘要 ：
In this paper, we discuss the superconvergence of the "interpolated" collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution u(h), two different interpolation postprocessing approximations of higher accuracy: I-2h(2m-1) u(h) based on the collocation points and I(2h)(m)u(h) based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods.
关键词 ：
Collocation Collocation Hybrid collocation Hybrid collocation Interpolation postprocessing Interpolation postprocessing Supercloseness Supercloseness Superconvergence Superconvergence Volterra integral equations Volterra integral equations Weakly singular kernels Weakly singular kernels
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| GB/T 7714 | Huang, Qiumei , Wang, Min . Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind [J]. | COMPUTATIONAL & APPLIED MATHEMATICS , 2021 , 40 (3) . |
| MLA | Huang, Qiumei et al. "Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind" . | COMPUTATIONAL & APPLIED MATHEMATICS 40 . 3 (2021) . |
| APA | Huang, Qiumei , Wang, Min . Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind . | COMPUTATIONAL & APPLIED MATHEMATICS , 2021 , 40 (3) . |
| 导入链接 | NoteExpress RIS BibTex |
摘要 ：
In this paper we propose and analyze a (temporally) third order accurate backward differentiation formula (BDF) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. The surface diffusion term is treated implicitly, while the nonlinear chemical potential is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont regularization term, in the form of -A Delta t(2)Delta N-2(un+1-un), is added in the numerical scheme. A careful energy stability estimate, combined with Fourier eigenvalue analysis, results in the energy stability in a modified version, and a theoretical justification of the coefficient A becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the l(infinity)(0, T;l(2)) boolean AND l(2)(0,T;Hh(2)) norm, with the help of a linearized estimate for the nonlinear error terms. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for epsilon = 0.02 (up to T = 3x10(5)) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.
关键词 ：
energy stability energy stability Epitaxial thin film growth Epitaxial thin film growth no-slope-selection no-slope-selection optimal rate convergence analysis optimal rate convergence analysis third order backward differentiation formula third order backward differentiation formula
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| GB/T 7714 | Hao, Yonghong , Huang, Qiumei , Wang, Cheng . A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model [J]. | COMMUNICATIONS IN COMPUTATIONAL PHYSICS , 2021 , 29 (3) : 905-929 . |
| MLA | Hao, Yonghong et al. "A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model" . | COMMUNICATIONS IN COMPUTATIONAL PHYSICS 29 . 3 (2021) : 905-929 . |
| APA | Hao, Yonghong , Huang, Qiumei , Wang, Cheng . A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model . | COMMUNICATIONS IN COMPUTATIONAL PHYSICS , 2021 , 29 (3) , 905-929 . |
| 导入链接 | NoteExpress RIS BibTex |
摘要 ：
In this paper, we develop the superconvergence analysis of the implicit second-order two-grid discrete scheme with the lowest Nedelec element for wave propagation with Debye Polarization in nonlinear Dielectric materials. Our main contribution will have two parts. On one hand, in order to overcome the difficulty of misconvergence of classical two-grid algorithm by the lowest Nedelec elements, we employ the Newton-type Taylor expansion at the superconvergent solutions for the nonlinear terms on coarse mesh, which is different from the classical numerical solution on the coarse mesh. On the other hand, we push the two-grid solution to high accuracy by the interpolation post-processing technique. Such a design can both improve the computational accuracy in spatial and decrease time consumption simultaneously. Based on this design, we can obtain the convergent rate O (tau(2) + h(2) + H-3), and the spatial convergence can be obtained by choosing the mesh size h = O (H-3/2). At last, one numerical experiment is illustrated to verify our theoretical results. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.
关键词 ：
Nedelec element Nedelec element Nonlinear Nonlinear Post-processing Post-processing Two-grid algorithm Two-grid algorithm Wave propagation Wave propagation
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| GB/T 7714 | Yao, Changhui , Wei, Yifan , Huang, Qiumei . Post-processing technique of two-grid algorithm for wave propagation with Debye polarization in nonlinear dielectric materials [J]. | APPLIED NUMERICAL MATHEMATICS , 2020 , 157 : 405-418 . |
| MLA | Yao, Changhui et al. "Post-processing technique of two-grid algorithm for wave propagation with Debye polarization in nonlinear dielectric materials" . | APPLIED NUMERICAL MATHEMATICS 157 (2020) : 405-418 . |
| APA | Yao, Changhui , Wei, Yifan , Huang, Qiumei . Post-processing technique of two-grid algorithm for wave propagation with Debye polarization in nonlinear dielectric materials . | APPLIED NUMERICAL MATHEMATICS , 2020 , 157 , 405-418 . |
| 导入链接 | NoteExpress RIS BibTex |
摘要 ：
In this study, we present a new type of domain decomposition algorithm to obtain the ground state solution for Bose-Einstein condensates. Using our proposed algorithm, instead of directly solving the nonlinear eigenvalue problem, one only needs to solve a series of linear boundary value problems on a finite element space sequence using the domain decomposition method, and subsequently solve a small-scale nonlinear eigenvalue problem on an enriched space simultaneously. Because solving large-scale nonlinear eigenvalue problem directly is time intensive, our algorithm can obviously improve the efficiency of producing simulation for Bose-Einstein condensates. In addition, any domain decomposition algorithm for linear boundary value problems can be applied to our algorithm framework, which makes the algorithm considerably flexible. Two numerical experiments are presented in the paper to demonstrate the efficiency and scalability of our proposed algorithm. (C) 2020 Elsevier Ltd. All rights reserved.
关键词 ：
Bose-Einstein condensates Bose-Einstein condensates Domain decomposition method Domain decomposition method Multilevel correction Multilevel correction Parallel computing Parallel computing
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| GB/T 7714 | Xu, Fei , Huang, Qiumei , Ma, Hongkun . A novel domain decomposition framework for the ground state solution of Bose-Einstein condensates [J]. | COMPUTERS & MATHEMATICS WITH APPLICATIONS , 2020 , 80 (5) : 1287-1300 . |
| MLA | Xu, Fei et al. "A novel domain decomposition framework for the ground state solution of Bose-Einstein condensates" . | COMPUTERS & MATHEMATICS WITH APPLICATIONS 80 . 5 (2020) : 1287-1300 . |
| APA | Xu, Fei , Huang, Qiumei , Ma, Hongkun . A novel domain decomposition framework for the ground state solution of Bose-Einstein condensates . | COMPUTERS & MATHEMATICS WITH APPLICATIONS , 2020 , 80 (5) , 1287-1300 . |
| 导入链接 | NoteExpress RIS BibTex |
摘要 ：
In this paper, a type of cascadic adaptive finite element method is proposed for eigenvalue problem based on the complementary approach. In this new scheme, instead of solving the eigenvalue problem in each adaptive finite element space directly, we only need to do some smoothing steps for a boundary value problems on each adaptive space and solve some eigenvalue problems on a low dimensional space. Hence the efficiency can be improved since we do not need to solve the eigenvalue problems on each adaptive space which is time-consuming. Further, the complementary error estimate for eigenvalue problem will be introduced. This estimate can not only provide an accurate error estimate for eigenvalue problem but also provide the way to refine mesh and control the number of smoothing steps for the cascadic adaptive algorithm. Some numerical examples are presented to validate the efficiency of the proposed algorithm in this paper.
关键词 ：
Adaptive finite element method Adaptive finite element method cascadic multigrid method cascadic multigrid method complementary method complementary method eigenvalue problem eigenvalue problem
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| GB/T 7714 | Xu, Fei , Huang, Qiumei , Chen, Shuangshuang et al. A Type of Cascadic Adaptive Finite Element Method for Eigenvalue Problem [J]. | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS , 2020 , 12 (3) : 774-796 . |
| MLA | Xu, Fei et al. "A Type of Cascadic Adaptive Finite Element Method for Eigenvalue Problem" . | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS 12 . 3 (2020) : 774-796 . |
| APA | Xu, Fei , Huang, Qiumei , Chen, Shuangshuang , Ma, Hongkun . A Type of Cascadic Adaptive Finite Element Method for Eigenvalue Problem . | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS , 2020 , 12 (3) , 774-796 . |
| 导入链接 | NoteExpress RIS BibTex |
摘要 ：
In this paper, a new type of local and parallel algorithm is proposed to solve nonlinear eigenvalue problem based on multigrid discretization. Instead of solving the nonlinear eigenvalue problem directly in each level mesh, our method converts the nonlinear eigenvalue problem in the finest mesh to a linear boundary value problem on each level mesh and some nonlinear eigenvalue problems on the coarsest mesh. Further, the involved linear boundary value problems are solved using the local and parallel strategy. As no nonlinear eigenvalue problem is being solved directly on the fine spaces, which is time-consuming, this new type of local and parallel multigrid method evidently improves the efficiency of nonlinear eigenvalue problem solving. We provide a rigorous theoretical analysis for our algorithm and present details on numerical simulations to support our theory.
关键词 ：
Finite element method Finite element method Local and parallel Local and parallel Multilevel correction method Multilevel correction method Nonlinear eigenvalue problem Nonlinear eigenvalue problem
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| GB/T 7714 | Xu, Fei , Huang, Qiumei . Local and Parallel Multigrid Method for Nonlinear Eigenvalue Problems [J]. | JOURNAL OF SCIENTIFIC COMPUTING , 2020 , 82 (1) . |
| MLA | Xu, Fei et al. "Local and Parallel Multigrid Method for Nonlinear Eigenvalue Problems" . | JOURNAL OF SCIENTIFIC COMPUTING 82 . 1 (2020) . |
| APA | Xu, Fei , Huang, Qiumei . Local and Parallel Multigrid Method for Nonlinear Eigenvalue Problems . | JOURNAL OF SCIENTIFIC COMPUTING , 2020 , 82 (1) . |
| 导入链接 | NoteExpress RIS BibTex |
摘要 ：
This paper is to introduce a type of cascadic multigrid method for coupled semilinear elliptic equations. Instead of solving the coupled semilinear elliptic equation on a very fine finite element space directly, the new scheme needs to solve a decoupled linear system by some smoothing steps on each of multilevel finite element spaces and solve a coupled semilinear elliptic equation on a coarse space. By choosing the appropriate number of smoothing steps on different finite element spaces, the corresponding optimal convergence rate and optimal computation work can be derived. Besides, the adaptive cascadic multigrid method for coupled semilinear elliptic equations and its analysis are also presented theoretically and numerically. Moreover, the requirement of bounded second order derivatives of the nonlinear term in the existing multigrid methods is reduced to the Lipschitz continuation property in the presented new scheme. Some numerical experiments are presented to validate our theoretical analysis.
关键词 ：
Adaptive multigrid method Adaptive multigrid method Cascadic multigrid method Cascadic multigrid method Coupled semilinear elliptic equation Coupled semilinear elliptic equation Finite element method Finite element method
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| GB/T 7714 | Xu, Fei , Huang, Qiumei . A type of cascadic multigrid method for coupled semilinear elliptic equations [J]. | NUMERICAL ALGORITHMS , 2020 , 83 (2) : 485-510 . |
| MLA | Xu, Fei et al. "A type of cascadic multigrid method for coupled semilinear elliptic equations" . | NUMERICAL ALGORITHMS 83 . 2 (2020) : 485-510 . |
| APA | Xu, Fei , Huang, Qiumei . A type of cascadic multigrid method for coupled semilinear elliptic equations . | NUMERICAL ALGORITHMS , 2020 , 83 (2) , 485-510 . |
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