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Numerical Mathematics(Theory,methods and Applications) 2014年第1期7卷 41-57页

作者： Limei Li Da Xu College of Mathematics and Computer Science Hunan Normal UniversityChangsha 410081China Department of Mathematics Hunan Institute of Science and TechnologyYueyang 414000China

New numerical techniques are presented for the solution of the twodimensional time fractional evolution equation in the unit square.In these methods,galerkin finite element is used for the spatial discretization,and,for the time stepping,new alternating direction implicit(ADI)method based on the backward Euler method combined with the first order convolution quadrature approximating the integral term are considered.The ADI Galerkin Finite Element Method is proved to be convergent in time and in the L2 norm in space.The convergence order is O(k|ln k|+h^(r)),where k is the temporal grid size and h is spatial grid size in the x and y directions,respectively.Numerical results are presented to support our theoretical analysis.

关键词： Fractional evolution equation alternating direction implicit method Galerkin Finite Element Method backward Euler

A symplectic finite element method based on galerkin discretization for solving linear systems

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Applied Mathematics and Mechanics(English Edition) 2023年第8期44卷 1305-1316页

作者： Zhiping QIU Zhao WANG Bo ZHU Institute of Solid Mechanics School of Aeronautic Science and EngineeringBeihang UniversityBeijing 100083China

We propose a novel symplectic finite element method to solve the structural dynamic responses of linear elastic systems.For the dynamic responses of continuous medium structures,the traditional numerical algorithm is the dissipative algorithm and cannot maintain long-term energy conservation.Thus,a symplectic finite element method with energy conservation is constructed in this paper.A linear elastic system can be discretized into multiple elements,and a Hamiltonian system of each element can be constructed.The single element is discretized by the galerkin method,and then the Hamiltonian system is constructed into the Birkhoffian system.Finally,all the elements are combined to obtain the vibration equation of the continuous system and solved by the symplectic difference scheme.Through the numerical experiments of the vibration response of the Bernoulli-Euler beam and composite plate,it is found that the vibration response solution and energy obtained with the algorithm are superior to those of the Runge-Kutta algorithm.The results show that the symplectic finite element method can keep energy conservation for a long time and has higher stability in solving the dynamic responses of linear elastic systems.

关键词： Galerkin Finite Element Method linear system structural dynamic response symplectic difference scheme

galerkin finite element Approximation for Semilinear Stochastic Time-Tempered Fractional Wave Equations with Multiplicative Gaussian Noise and Additive Fractional Gaussian Noise

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Numerical Mathematics(Theory,methods and Applications) 2022年第4期15卷 1063-1098页

作者： Yajing Li Yejuan Wang Weihua Deng Daxin Nie School of Mathematics and Statistics Gansu Key Laboratory of Applied Mathematics and Complex SystemsLanzhou UniversityLanzhou 730000P.R.China College of Science Northwest A&F UniversityYangling 712100ShaanxiP.R.China

To model wave propagation in inhomogeneous media with frequency dependent power-law attenuation,it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time.The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative Gaussian noise and additive fractional Gaussian noise,because of the potential fluctuations of the external sources.The purpose of this work is to discuss the galerkin finite element approximation for the semilinear stochastic fractional wave equation.First,the space-time multiplicative Gaussian noise and additive fractional Gaussian noise are discretized,which results in a regularized stochastic fractional wave equation while introducing a modeling error in the mean-square sense.We further present a complete regularity theory for the regularized equation.A standard finite element approximation is used for the spatial operator,and a mean-square priori estimates for the modeling error and the approximation error to the solution of the regularized problem are established.Finally,numerical experiments are performed to confirm the theoretical analysis.

关键词： Galerkin Finite Element Method semilinear stochastic time-tempered fractional wave equation fractional Laplacian multiplicative Gaussian noise additive fractional Gaussian noise

galerkin finite element analysis of magneto-hydrodynamic natural convection of Cu-water nanoliquid in a baffled U-shaped enclosure

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Propulsion and Power Research 2020年第4期9卷 383-393页

作者： A.Zaim A.Aissa F.Mebarek-Oudina B.Mahanthesh G.Lorenzini M.Sahnoun M.El Ganaoui BLSTE Laboratory Faculty of Science and TechnologyUniversity of MascaraMascaraAlgeria Labor atoire de Physique Quantique de La Matiere et Moderation Mathematique(LPQ3M) Universite Mustapha Stambouli de MascaraMascaraAlgeria Department of Physics Faculty of SciencesUniversity of 20 Aoiit 1955-SkikdaSkikda2100Algeria Labor atoire des Materiaux et Genie Energetique(LMGE) University of 20 Aout 1955-SkikdaSkikda2100Algeria Department of Mathematics CHRIST(Deemed to Be University)Bangalore560029India Department of Engineering and Architecture University of Parma.Parco Area Delle Scienze 1H1/A43124ParmaItaly Labor atoire energetique de Longvy(FJV/LERMAB) Universite de LorraineFrance

In this paper,single-phase homogeneous nanofluid model is proposed to investigate the natural convection of magneto-hydrodynamic(MIID)flow of Newtonian Cu—H20 nanoliquid in a baffled U-shaped enclosure.The Brinkman model and Wasp model are considered to measure the effective dynamic viscosity and effective thermal conductivity of the nanoliquid coreespondingly.Nanoliquid's effective properties such as specific heat,density and thermal expansion coefficient are modeled using mixture theory.The complicated PDS(partial differential system)is treated for numeric solutions via the Galerkin Finite Element Method.The pertinent parameters Hartmann number(1≤Ha≤60),Rayleigh number(10^(3)≤Ra≤10^(6))and nanoparticles volume fraction (0% ≤Ф≤4%) are taken for the parametric analysis, and it is conducted via streamlines and isotherms. Excellent agreement between numerical results and open literature. It is ascertained that heat transfer rate enhances with Rayleigh number Ra and volume fraction 0, however it is diminished for laiger Hartmann number Ha.

关键词： Natural convection Nanoliquid Rayleigh number Baffled U-shaped Nusselt number Galerkin Finite Element Method

Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise

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Numerical Mathematics(Theory,methods and Applications) 2020年第1期13卷 1-26页

作者： Wenju Zhao Max Gunzburger Department of Mathematics Southern University of Science and TechnologyShenzhenGuangdong 518055China Department of Scientific Computing Florida State UniversityTallahasseeFL 32304USA

This paper presents a Martingale regularization method for the stochas-tic Navier–Stokes equations with additive noise.The original system is split into two equivalent parts,the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier–Stokes equations with relatively-higher regular-ities.Meanwhile,a fractional Laplace operator is introduced to regularize the noise term.The stability and convergence of numerical scheme for the pathwise modified Navier–Stokes equations are proved.The comparisons of non-regularized and reg-ularized noises for the Navier–Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.

关键词： Stochastic Navier-Stokes equations Martingale regularization method Galerkin Finite Element Method white noise

ON THE STABILITY OF THE RESIDUAL-FREE BUBBLES FOR THE NAVIER-STOKES EQUATIONS

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Acta Mathematica Scientia 2005年第4期25卷 715-730页

作者： Ali I.Nesliturk Department of Mathematics Izmir Institute of Technology P.O. Box 35430 Izmir Turkey

This paper considers the Calerkin finite element method for the incompressible Navier-Stokes equations in two dimensions, where the finite-dimensional spaces employed consist of piecewise polynomials enriched with residual-free bubble （RFB） functions. The stability features of the residual-free bubble functions for the linearized Navier-Stokes equations are analyzed in this work. It is shown that the enrichment of the velocity space by bubble functions stabilizes the numerical method for any value of the viscosity parameter for triangular elements and for values of the viscosity parameter in the vanishing limit case for quadrilateral elements.

关键词： Galerkin Finite Element Method incompressible Navier-Stokes equations stability

Computational Studies of Bacterial Colony Model

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American Journal of Computational Mathematics 2013年第2期3卷 147-157页

作者： Ondrej Partl Department of Mathematics Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague Prague Czech Republic

Microbiological experiments show that the colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied spatial patterns while the individual cells grow, reproduce and migrate on the dish in clumps. In this paper, we discuss a system of reaction-diffusion equations that can be used with a view to modelling this phenomenon and we solve it numerically by means of the method of lines. For the spatial discretization, we use the finite difference method and Galerkin Finite Element Method. We present how the spatial patterns obtained depend on the spatial discretization employed and we measure the experimental order of convergence of the numerical schemes used. Further, we present the numerical results obtained by solving the model in a cubic domain.

关键词： Bacterial Colony Model Reaction-Diffusion Equations method of Lines Galerkin Finite Element Method