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Abstract ：
We propose a symmetric interior penalty discontinuous Galerkin (DG) method for nonlinear fully coupled quasi-static thermo-poroelasticity problems. Firstly, a fully implicit nonlinear discrete scheme is constructed by adopting the DG method for the spatial approximation and the backward Euler method for the temporal discretization. Subsequently, the existence and uniqueness of the solution of the numerical scheme is proved, and then we derive the a priori error estimate for the three variables, i.e., the displacement, the pressure and the temperature. Lastly, we carry out numerical experiments to confirm the theoretical findings of our suggested approach.
Keyword ：
symmetric interior penalty discontinuous Galerkin method symmetric interior penalty discontinuous Galerkin method fully implicit nonlinear discrete scheme fully implicit nonlinear discrete scheme a priori error estimate a priori error estimate thermo-poroelasticity thermo-poroelasticity
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| GB/T 7714 | Chen, Fan , Cui, Ming , Zhou, Chenguang . Symmetric interior penalty discontinuous Galerkin method for nonlinear fully coupled quasi-static thermo-poroelasticity problems [J]. | APPLICATIONS OF MATHEMATICS , 2025 , 70 (1) : 97-123 . |
| MLA | Chen, Fan 等. "Symmetric interior penalty discontinuous Galerkin method for nonlinear fully coupled quasi-static thermo-poroelasticity problems" . | APPLICATIONS OF MATHEMATICS 70 . 1 (2025) : 97-123 . |
| APA | Chen, Fan , Cui, Ming , Zhou, Chenguang . Symmetric interior penalty discontinuous Galerkin method for nonlinear fully coupled quasi-static thermo-poroelasticity problems . | APPLICATIONS OF MATHEMATICS , 2025 , 70 (1) , 97-123 . |
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Abstract ：
In this paper, we propose an energy-conservative finite difference time domain (FDTD) method for solving the coupled nonlinear Klein-Gordon equations (CNKGEs) in the nonrelativistic limit regime, involving a small parameter 0 < epsilon << 1 which is inversely proportional to the speed of light. Employing cut-off technique, we analyze rigourously error estimates for the numerical method. Numerical results are reported to confirm the energy-conservative property and the error results in l(2) norm and H-1 norm under different values of epsilon.
Keyword ：
energy-conservative energy-conservative finite difference time domain method finite difference time domain method cut-off technique cut-off technique nonrelativistic regime nonrelativistic regime Coupled nonlinear Klein-Gordon equations Coupled nonlinear Klein-Gordon equations
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| GB/T 7714 | Cui, Ming , Li, Yanfei . ENERGY-CONSERVATIVE FINITE DIFFERENCE METHOD FOR THE COUPLED NONLINEAR KLEIN-GORDON EQUATION IN THE NONRELATIVISTIC LIMIT REGIME [J]. | INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING , 2025 , 21 (2) : 246-267 . |
| MLA | Cui, Ming 等. "ENERGY-CONSERVATIVE FINITE DIFFERENCE METHOD FOR THE COUPLED NONLINEAR KLEIN-GORDON EQUATION IN THE NONRELATIVISTIC LIMIT REGIME" . | INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING 21 . 2 (2025) : 246-267 . |
| APA | Cui, Ming , Li, Yanfei . ENERGY-CONSERVATIVE FINITE DIFFERENCE METHOD FOR THE COUPLED NONLINEAR KLEIN-GORDON EQUATION IN THE NONRELATIVISTIC LIMIT REGIME . | INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING , 2025 , 21 (2) , 246-267 . |
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Abstract ：
This paper proposes and analyzes a type of efficient multigrid method, which is called multilevel correction method, for solving semilinear parabolic interface problems. The core idea of this method is that, at each time step, the semilinear elliptic interface problem's solution is transformed into the same-scale linear elliptic interface problem's solution in each level of multilevel space sequence and the semilinear elliptic interface problem's solution on a newly defined low dimensional augmented subspace. Through analyzing the algebraic error estimate of the method, we design the method to iterate only one step in the intermediate grid layer, which makes our method more efficient than the work of Xu et al. (2022a) without losing accuracy. In addition, in the aspect of theoretical analysis, we present a new technique of analysis to derive the convergence order estimates. Numerical experiments are conducted to validate the precision and effectiveness of our proposed method.
Keyword ：
Finite element method Finite element method Nested multilevel correction method Nested multilevel correction method Semilinear parabolic interface problem Semilinear parabolic interface problem Nonnested multilevel correction method Nonnested multilevel correction method
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| GB/T 7714 | Chen, Fan , Cui, Ming , Zhou, Chenguang . A type of efficient multigrid method for semilinear parabolic interface problems [J]. | COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION , 2025 , 143 . |
| MLA | Chen, Fan 等. "A type of efficient multigrid method for semilinear parabolic interface problems" . | COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 143 (2025) . |
| APA | Chen, Fan , Cui, Ming , Zhou, Chenguang . A type of efficient multigrid method for semilinear parabolic interface problems . | COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION , 2025 , 143 . |
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Abstract ：
This paper is devoted to a discontinuous Galerkin (DG) method for nonlinear quasistatic poroelasticity problems. The fully implicit nonlinear numerical scheme is constructed by utilizing DG method for the spatial approximation and the backward Euler method for the temporal discretization. The existence and uniqueness of the numerical solution is proved. Then we derive the optimal convergence order estimates in a discrete H1 norm for the displacement and in H1 and L2 norms for the pressure. Finally, numerical experiments are supplied to validate the theoretical error estimates of our proposed method.
Keyword ：
Nonlinear quasi-static poroelasticity problem Nonlinear quasi-static poroelasticity problem optimal convergence order estimate optimal convergence order estimate fully implicit nonlinear numerical scheme fully implicit nonlinear numerical scheme discontinuous Galerkin method discontinuous Galerkin method
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| GB/T 7714 | Chen, Fan , Cui, Ming , Zhou, Chenguang . DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR QUASI-STATIC POROELASTICITY PROBLEMS [J]. | INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING , 2024 , 21 (2) : 201-220 . |
| MLA | Chen, Fan 等. "DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR QUASI-STATIC POROELASTICITY PROBLEMS" . | INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING 21 . 2 (2024) : 201-220 . |
| APA | Chen, Fan , Cui, Ming , Zhou, Chenguang . DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR QUASI-STATIC POROELASTICITY PROBLEMS . | INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING , 2024 , 21 (2) , 201-220 . |
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Abstract ：
In this paper, we consider the energy conserving numerical scheme for cou-pled nonlinear Klein-Gordon equations. We propose energy conserving finite element method and get the unconditional superconvergence result O(h2+Delta t2) by using the er-ror splitting technique and postprocessing interpolation. Numerical experiments are carried out to support our theoretical results.
Keyword ：
the nonlinear coupled Klein-Gordon equations the nonlinear coupled Klein-Gordon equations finite element method finite element method unconditional superconvergence result unconditional superconvergence result Energy conserving Energy conserving postprocessing interpolation postprocessing interpolation
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| GB/T 7714 | Cui, Ming , Li, Yanfei , Yao, Changhui . Unconditional Superconvergence Analysis of Energy Conserving Finite Element Methods for the Nonlinear Coupled Klein-Gordon Equations [J]. | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS , 2023 , 15 (3) : 602-622 . |
| MLA | Cui, Ming 等. "Unconditional Superconvergence Analysis of Energy Conserving Finite Element Methods for the Nonlinear Coupled Klein-Gordon Equations" . | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS 15 . 3 (2023) : 602-622 . |
| APA | Cui, Ming , Li, Yanfei , Yao, Changhui . Unconditional Superconvergence Analysis of Energy Conserving Finite Element Methods for the Nonlinear Coupled Klein-Gordon Equations . | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS , 2023 , 15 (3) , 602-622 . |
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Abstract ：
A modified weak Galerkin (MWG) finite element method is developed for solving the biharmonic equation.This method uses the same finite element space as that of the discontinuous Galerkin method,the space of discontinuous polynomials on polytopal meshes.But its formulation is simple,symmetric,positive definite,and parameter independent,without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method.Optimal order error estimates in a discrete H2 norm are established for the corresponding finite element solutions.Error estimates in the L2 norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements.The numerical results are presented to confirm the theory of convergence.
Keyword ：
Weak Laplacian Weak Laplacian Biharmonic equations Biharmonic equations Polytopal meshes Polytopal meshes Finite element methods Finite element methods
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| GB/T 7714 | Ming Cui , Xiu Ye , Shangyou Zhang . A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes [J]. | 应用数学与计算数学学报 , 2021 , 3 (1) : 91-105 . |
| MLA | Ming Cui 等. "A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes" . | 应用数学与计算数学学报 3 . 1 (2021) : 91-105 . |
| APA | Ming Cui , Xiu Ye , Shangyou Zhang . A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes . | 应用数学与计算数学学报 , 2021 , 3 (1) , 91-105 . |
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Abstract ：
A modified weak Galerkin (MWG) finite element method is developed for solving the biharmonic equation. This method uses the same finite element space as that of the discontinuous Galerkin method, the space of discontinuous polynomials on polytopal meshes. But its formulation is simple, symmetric, positive definite, and parameter independent, without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method. Optimal order error estimates in a discrete H2 norm are established for the corresponding finite element solutions. Error estimates in the L2 norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements. The numerical results are presented to confirm the theory of convergence.
Keyword ：
Biharmonic equations Biharmonic equations 65N15 65N15 76D07 76D07 Weak Laplacian Weak Laplacian Finite element methods Finite element methods 65N30 65N30 Polytopal meshes Polytopal meshes
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| GB/T 7714 | Cui, Ming , Ye, Xiu , Zhang, Shangyou . A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes [J]. | COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION , 2021 , 3 (1) : 91-105 . |
| MLA | Cui, Ming 等. "A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes" . | COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION 3 . 1 (2021) : 91-105 . |
| APA | Cui, Ming , Ye, Xiu , Zhang, Shangyou . A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes . | COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION , 2021 , 3 (1) , 91-105 . |
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Abstract ：
Aerosol particles have an important effect on changing of climate and human health, where aerosols scatter and absorb the incoming solar radiation, and thus decrease the precipitation efficiency of warm clouds and can cause an indirect radiative forcing associated with changes in cloud properties. Meanwhile, it has also been recognized that the particles of aerosols in the sub-micrometer size range can be inhaled and thus pose certain health hazards. In this paper we analyze the finite volume method based on linear interpolation and Hermite interpolation combined with the method of characteristics for the nonlinear aerosol dynamic equations on time and particle size, which involve the advection process, condensation process and the nonlinear coagulation process. Numerical experiments for the multiple log-normal aerosol distributions are further given to confirm the theoretical results. (C) 2019 Elsevier B.V. All rights reserved.
Keyword ：
Finite volume method Finite volume method The method of characteristics The method of characteristics Aerosol dynamic equation Aerosol dynamic equation Error estimate Error estimate
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| GB/T 7714 | Cui, Ming , Li, Fangxia , Liang, Dong . High-order characteristic-finite volume methods for aerosol dynamic equations [J]. | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS , 2020 , 370 . |
| MLA | Cui, Ming 等. "High-order characteristic-finite volume methods for aerosol dynamic equations" . | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 370 (2020) . |
| APA | Cui, Ming , Li, Fangxia , Liang, Dong . High-order characteristic-finite volume methods for aerosol dynamic equations . | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS , 2020 , 370 . |
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Abstract ：
For the biharmonic equation or this singularly-perturbed biharmonic equation, lower order nonconforming finite elements are usually used. It is difficult to construct high order C1 conforming, or nonconforming elements, especially in 3D. A family of any quadratic or higher order weak Galerkin finite elements is constructed on 2D polygonal grids and 3D polyhedral grids for solving the singularly-perturbed biharmonic equation. The optimal order of convergence, up to any order the smooth solution can have, is proved for this method, in a discrete H-2 norm. Under a full elliptic regularity H-4 assumption, the L-2 convergence achieves the optimal order as well, in 2D and 3D. Numerical tests are presented verifying the theory.
Keyword ：
Polyhedral grid Polyhedral grid Biharmonic equation Biharmonic equation Weak Galerkin Weak Galerkin Singular perturbation Singular perturbation Polygonal grid Polygonal grid Finite element Finite element
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| GB/T 7714 | Cui, Ming , Zhang, Shangyou . On the Uniform Convergence of the Weak Galerkin Finite Element Method for a Singularly-Perturbed Biharmonic Equation [J]. | JOURNAL OF SCIENTIFIC COMPUTING , 2020 , 82 (1) . |
| MLA | Cui, Ming 等. "On the Uniform Convergence of the Weak Galerkin Finite Element Method for a Singularly-Perturbed Biharmonic Equation" . | JOURNAL OF SCIENTIFIC COMPUTING 82 . 1 (2020) . |
| APA | Cui, Ming , Zhang, Shangyou . On the Uniform Convergence of the Weak Galerkin Finite Element Method for a Singularly-Perturbed Biharmonic Equation . | JOURNAL OF SCIENTIFIC COMPUTING , 2020 , 82 (1) . |
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Abstract ：
Aerosol modeling is very important to study the behavior of aerosol dynamics in atmospheric environment. In this paper we consider numerical methods for the nonlinear aerosol dynamic equations on time and particle size. The finite volume element methods based on the linear interpolation and Hermite interpolation are provided to approximate the aerosol dynamic equation where the condensation and removal processes are considered. Numerical examples are provided to show the efficiency of these numerical methods.
Keyword ：
Aerosol dynamic equation Aerosol dynamic equation removal removal finite volume method finite volume method condensation condensation
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| GB/T 7714 | Cui, Ming , Su, Yanxin , Liang, Dong . High-Order Finite Volume Methods for Aerosol Dynamic Equations [J]. | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS , 2016 , 8 (2) : 213-235 . |
| MLA | Cui, Ming 等. "High-Order Finite Volume Methods for Aerosol Dynamic Equations" . | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS 8 . 2 (2016) : 213-235 . |
| APA | Cui, Ming , Su, Yanxin , Liang, Dong . High-Order Finite Volume Methods for Aerosol Dynamic Equations . | ADVANCES IN APPLIED MATHEMATICS AND MECHANICS , 2016 , 8 (2) , 213-235 . |
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