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学者姓名：黄秋梅
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Abstract ：
Prostate cancer (PCa) is a significant global health concern that affects the male population. In this study, we present a numerical approach to simulate the growth of PCa tumors and their response to drug therapy. The approach is based on a previously developed model, which consists of a coupled system comprising one phase field equation and two reaction-diffusion equations. To solve this system, we employ the fast second -order exponential time differencing Runge-Kutta (ETDRK2) method with stabilizing terms. This method is a decoupled linear numerical algorithm that preserves three crucial physical properties of the model: a maximum bound principle (MBP) on the order parameter and non -negativity of the two concentration variables. Our simulations allow us to predict tumor growth patterns and outcomes of drug therapy over extended periods, offering valuable insights for both basic research and clinical treatments.
Keyword ：
Maximum bound principle Maximum bound principle Prostate cancer tumor growth Prostate cancer tumor growth Exponential time differencing Runge-Kutta Exponential time differencing Runge-Kutta Phase field equation Phase field equation Non-negativity Non-negativity Drug therapy Drug therapy
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| GB/T 7714 | Huang, Qiumei , Qiao, Zhonghua , Yang, Huiting . Maximum bound principle and non-negativity preserving ETD schemes for a phase field model of prostate cancer growth with treatment [J]. | COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING , 2024 , 426 . |
| MLA | Huang, Qiumei 等. "Maximum bound principle and non-negativity preserving ETD schemes for a phase field model of prostate cancer growth with treatment" . | COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 426 (2024) . |
| APA | Huang, Qiumei , Qiao, Zhonghua , Yang, Huiting . Maximum bound principle and non-negativity preserving ETD schemes for a phase field model of prostate cancer growth with treatment . | COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING , 2024 , 426 . |
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Abstract ：
Weakly singular Volterra integral equations of the second kind typically have nonsmooth solutions near the initial point of the interval of integration, which seriously affects the accuracy of spectral methods. We present Jacobi spectral -collocation method to solve two-dimensional weakly singular Volterra -Hammerstein integral equations based on smoothing transformation and implicitly linear method. The solution of the smoothed equation is much smoother than the original one after smoothing transformation and the spectral method can be used. For the nonlinear Hammerstein term, the implicitly linear method is applied to simplify the calculation and improve the accuracy. The weakly singular integral term is discretized by Jacobi Gauss quadrature formula which can absorb the weakly singular kernel function into the quadrature weight function and eliminate the influence of the weakly singular kernel on the method. Convergence analysis in the L-infinity-norm is carried out and the exponential convergence rate is obtained. Finally, we demonstrate the efficiency of the proposed method by numerical examples.
Keyword ：
Weakly singular Weakly singular Smoothing transformation Smoothing transformation Jacobi spectral-collocation method Jacobi spectral-collocation method Two-dimensional Volterra-Hammerstein integral equations Two-dimensional Volterra-Hammerstein integral equations Exponential convergence Exponential convergence Implicitly linear method Implicitly linear method
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| GB/T 7714 | Huang, Qiumei , Yang, Huiting . Implicitly linear Jacobi spectral-collocation methods for two-dimensional weakly singular Volterra-Hammerstein integral equations [J]. | APPLIED NUMERICAL MATHEMATICS , 2024 , 201 : 159-174 . |
| MLA | Huang, Qiumei 等. "Implicitly linear Jacobi spectral-collocation methods for two-dimensional weakly singular Volterra-Hammerstein integral equations" . | APPLIED NUMERICAL MATHEMATICS 201 (2024) : 159-174 . |
| APA | Huang, Qiumei , Yang, Huiting . Implicitly linear Jacobi spectral-collocation methods for two-dimensional weakly singular Volterra-Hammerstein integral equations . | APPLIED NUMERICAL MATHEMATICS , 2024 , 201 , 159-174 . |
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Abstract ：
We develop an initial-boundary value problem derived from the Maxwell's system with a nonlinear feedback-type boundary mechanism in metamaterials, which both involves polarization, magnetization ef-fect and time-localized delay effect in a bounded domain. Based on the nonlinear semigroup theory and the properties of viscoelasticity theory, we show the well-posedness of solution in an appropriate Hilbert space. Under some suitable assumptions and geometric conditions, we prove the exponential stability of the Maxwell's system.(c) 2023 Elsevier Inc. All rights reserved.
Keyword ：
Exponential stability Exponential stability Maxwell's equations Maxwell's equations Nonlinear boundary condition Nonlinear boundary condition Well-posedness Well-posedness Delay feedback Delay feedback
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| GB/T 7714 | Yao, Changhui , Sun, Rong , Huang, Qiumei . Global well-posedness and exponential stability for Maxwell's equations under delayed boundary condition in metamaterials [J]. | JOURNAL OF DIFFERENTIAL EQUATIONS , 2023 , 365 : 168-198 . |
| MLA | Yao, Changhui 等. "Global well-posedness and exponential stability for Maxwell's equations under delayed boundary condition in metamaterials" . | JOURNAL OF DIFFERENTIAL EQUATIONS 365 (2023) : 168-198 . |
| APA | Yao, Changhui , Sun, Rong , Huang, Qiumei . Global well-posedness and exponential stability for Maxwell's equations under delayed boundary condition in metamaterials . | JOURNAL OF DIFFERENTIAL EQUATIONS , 2023 , 365 , 168-198 . |
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Abstract ：
In this paper, ETD3-Pade and ETD4-Pade Galerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions. An ETD-based RK is used for time integration of the corresponding equation. To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator, the Pade approach is used for such an exponential operator approximation, which in turn leads to the corresponding ETD-Pade schemes. An unconditional L-2 numerical stability is proved for the proposed numerical schemes, under a global Lipshitz continuity assumption. In addition, optimal rate error estimates are provided, which gives the convergence order of O(k(3) + h(r)) (ETD3-Pade) or O(k(4) + h(r)) (ETD4-Pade) in the L-2 norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.
Keyword ：
Nonlinear delayed convection diffusion reaction equations Nonlinear delayed convection diffusion reaction equations Lipshitz continuity Lipshitz continuity ETD-Pade scheme ETD-Pade scheme Convergence analysis and error estimate Convergence analysis and error estimate L-2 stability analysis L-2 stability analysis
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| GB/T 7714 | Dai, Haishen , Huang, Qiumei , Wang, Cheng . EXPONENTIAL TIME DIFFERENCING-PADE FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY [J]. | JOURNAL OF COMPUTATIONAL MATHEMATICS , 2023 , 41 (3) : 350-351 . |
| MLA | Dai, Haishen 等. "EXPONENTIAL TIME DIFFERENCING-PADE FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY" . | JOURNAL OF COMPUTATIONAL MATHEMATICS 41 . 3 (2023) : 350-351 . |
| APA | Dai, Haishen , Huang, Qiumei , Wang, Cheng . EXPONENTIAL TIME DIFFERENCING-PADE FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY . | JOURNAL OF COMPUTATIONAL MATHEMATICS , 2023 , 41 (3) , 350-351 . |
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Abstract ：
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.
Keyword ：
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 等. "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) . |
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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 ：
A new type of adaptive multigrid method is presented for multiple eigenvalue problems based on multilevel correction scheme and adaptive multigrid method. Different from the classical adaptive finite element method which requires to solve eigenvalue problems on the adaptively refined triangulations, with our approach we just need to solve several linear boundary value problems in the current refined space and an eigenvalue problem in a very low dimensional space. Further, the involved boundary value problems are solved by an adaptive multigrid iteration. Since there is no eigenvalue problem to be solved on the refined triangulations, which is quite time-consuming, the proposed method can achieve the same efficiency as that of the adaptive multigrid method for the associated linear boundary value problems. Besides, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically. (C) 2022 Elsevier B.V. All rights reserved.
Keyword ：
Convergence and optimality complexity Convergence and optimality complexity Multiple eigenvalue problems Multiple eigenvalue problems Adaptive finite element method Adaptive finite element method Multigrid method Multigrid method
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| GB/T 7714 | Xu, Fei , Xie, Manting , Huang, Qiumei et al. Convergence and optimality of adaptive multigrid method for multiple eigenvalue problems [J]. | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS , 2022 , 415 . |
| MLA | Xu, Fei et al. "Convergence and optimality of adaptive multigrid method for multiple eigenvalue problems" . | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 415 (2022) . |
| APA | Xu, Fei , Xie, Manting , Huang, Qiumei , Yue, Meiling , Ma, Hongkun . Convergence and optimality of adaptive multigrid method for multiple eigenvalue problems . | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS , 2022 , 415 . |
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Abstract ：
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.
Keyword ：
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 . |
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Abstract ：
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.
Keyword ：
The Scalar auxiliary variable (SAV)method The Scalar auxiliary variable (SAV)method Energy stability Energy stability Fourier pseudo-spectral approximation Fourier pseudo-spectral approximation Second order BDF stencil Second order BDF stencil Square phase field crystal equation Square phase field crystal equation Optimal rate convergence analysis Optimal rate convergence analysis
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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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Abstract ：
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.
Keyword ：
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 . |
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