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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 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 . |
| 导入链接 | NoteExpress RIS BibTex |
摘要 ：
In this paper, we study numerical methods for solving a multi-dimensional fracture model, which couples n-dimensional Darcy flow in matrix with (n - 1)-dimensional Brinkman flow on fracture. A two-grid decoupled algorithm is proposed, in which the mixed model is decoupled by using the coarse grid approximation to the interface conditions, and then efficient singlemodel solvers are applied for decoupled Darcy and Brinkman problems on the fine mesh. Error estimates show that the two-grid decoupled algorithm retains the same order of approximation accuracy as the coupled one. Numerical experiments in two-dimensional (2D) and three-dimensional (3D) geometries are conducted, and their results confirm our theoretical analysis to illustrate the efficiency and effectiveness of the proposed method for solving multi-domain problems.
关键词 ：
Two-grid decoupled algorithm Two-grid decoupled algorithm Error estimates Error estimates Darcy-Brinkman model Darcy-Brinkman model Numerical experiments Numerical experiments
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| GB/T 7714 | Chen, Shuangshuang , Huang, Qiumei , Xu, Fei . A Two-Grid Decoupled Algorithm for a Multi-Dimensional Darcy-Brinkman Fracture Model [J]. | JOURNAL OF SCIENTIFIC COMPUTING , 2022 , 90 (3) . |
| MLA | Chen, Shuangshuang et al. "A Two-Grid Decoupled Algorithm for a Multi-Dimensional Darcy-Brinkman Fracture Model" . | JOURNAL OF SCIENTIFIC COMPUTING 90 . 3 (2022) . |
| APA | Chen, Shuangshuang , Huang, Qiumei , Xu, Fei . A Two-Grid Decoupled Algorithm for a Multi-Dimensional Darcy-Brinkman Fracture Model . | JOURNAL OF SCIENTIFIC COMPUTING , 2022 , 90 (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 ：
third order backward differentiation formula third order backward differentiation formula energy stability energy stability optimal rate convergence analysis optimal rate convergence analysis no-slope-selection no-slope-selection Epitaxial thin film growth Epitaxial thin film growth
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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 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) . |
| 导入链接 | NoteExpress RIS BibTex |
摘要 ：
The Peng-Robison equation of state, one of the most extensively applied equations of state in the petroleum industry and chemical engineering, has an excellent appearance in predicting the thermodynamic properties of a wide variety of materials. It has been a great challenge on how to design numerical schemes with preservation of mass conservation and energy dissipation law. Based on the exponential time difference combined with the stabilizing technique and added Lagrange multiplier enforcing the mass conservation, we develop the efficient first-and second-order numerical schemes with preservation of maximum bound principle (MBP) to solve the single-component two-phase diffuse interface model with Peng-Robison equation of state. Convergence analyses as well as energy stability are also proven. Several twodimensional and three-dimensional experiments are performed to verify these theoretical results.
关键词 ：
maximum bound principle maximum bound principle exponential time differencing exponential time differencing Peng-Robinson equation of state Peng-Robinson equation of state diffuse interface model diffuse interface model Lagrange multiplier Lagrange multiplier
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| GB/T 7714 | Huang, Qiumei , Jiang, Kun , Li, Jingwei . Exponential Time Differencing Schemes for the Peng-Robinson Equation of State with Preservation of Maximum Bound Principle [J]. | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS , 2021 , 14 (2) : 494-527 . |
| MLA | Huang, Qiumei et al. "Exponential Time Differencing Schemes for the Peng-Robinson Equation of State with Preservation of Maximum Bound Principle" . | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS 14 . 2 (2021) : 494-527 . |
| APA | Huang, Qiumei , Jiang, Kun , Li, Jingwei . Exponential Time Differencing Schemes for the Peng-Robinson Equation of State with Preservation of Maximum Bound Principle . | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS , 2021 , 14 (2) , 494-527 . |
| 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 ：
Interpolation postprocessing Interpolation postprocessing Weakly singular kernels Weakly singular kernels Volterra integral equations Volterra integral equations Supercloseness Supercloseness Superconvergence Superconvergence Collocation Collocation Hybrid collocation Hybrid collocation
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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 ：
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 Semilinear elliptic equations Semilinear elliptic equations Multilevel correction method Multilevel correction method
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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 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 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 type of cascadic adaptive finite element method for nonlinear eigenvalue problem is proposed based on the complementary approach. In this new scheme, we only need to do some smoothing steps for the linearized boundary value problems on a series of adaptive finite element spaces and solve some nonlinear eigenvalue problems on a low dimensional space. Hence the efficiency can be improved since we do not need to solve the nonlinear eigenvalue problems on each adaptive space directly. Besides, the complementary error estimate for nonlinear eigenvalue problem will be introduced in our paper. This estimate can not only provide an accurate error estimate for the nonlinear 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. (C) 2020 Elsevier B.V. All rights reserved.
关键词 ：
Adaptive method Adaptive method Cascadic multigrid method Cascadic multigrid method Complementary method Complementary method Finite element method Finite element method Nonlinear eigenvalue problem Nonlinear eigenvalue problem
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| GB/T 7714 | Xu, Fei , Huang, Qiumei . Cascadic adaptive finite element method for nonlinear eigenvalue problem based on complementary approach [J]. | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS , 2020 , 372 . |
| MLA | Xu, Fei et al. "Cascadic adaptive finite element method for nonlinear eigenvalue problem based on complementary approach" . | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 372 (2020) . |
| APA | Xu, Fei , Huang, Qiumei . Cascadic adaptive finite element method for nonlinear eigenvalue problem based on complementary approach . | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS , 2020 , 372 . |
| 导入链接 | NoteExpress RIS BibTex |
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