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A VARIATIONAL ANALYSIS FOR THE moving finite element method FOR GRADIENTFLOWS

Journal of Computational Mathematics 2023年第2期41卷 191-210页

作者： Xianmin Xu LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingNCMISAMSSChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

By using the Onsager principle as an approximation tool,we give a novel derivation for the moving finite element method for gradient flow equations.We show that the discretized problem has the same energy dissipation structure as the continuous one.This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials.We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity.The global minimizer,once it is detected by the discrete scheme,approximates the continuous stationary solution in optimal order.Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.

关键词： moving finite element method Convergence analysis Onsager principle

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A Simplified moving finite element method

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Journal of Hydrodynamics 1993年第3期5卷 25-31页

作者： Liu Ru-xun Xuan Ben-jin Department of Mathematics University of Science and Technology of China,Hefei,Anhui 230026,P.R.China

In this paper,we present a simplified moving finite element method (abbr.SM- FEM) and an element analysis approach to SMFEM.Using SMFEM,we successfully carry out a computation for Burgers' equation with two-nodal elem... 详细信息

关键词： moving finite element method quasi-characteristics element analysis Burgers' equation

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moving finite element Simulations for Reaction-Diffusion Systems

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Advances in Applied Mathematics and Mechanics 2012年第3期4卷 365-381页

作者： Guanghui Hu Zhonghua Qiao Tao Tang Department of Mathematics Michigan State UniversityEast LansingMI 48824-1027USA Department of Applied Mathematics The Hong Kong Polytechnic UniversityHung HomKowloonHong Kong Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong

This work is concerned with the numerical simulations for two reactiondiffusion systems,i.e.,the Brusselator model and the Gray-Scott model.The numerical algorithm is based upon a moving finite element method which helps to resolve large solution gradients.High quality meshes are obtained for both the spot replication and the moving wave along boundaries by using proper monitor functions.Unlike[33],this work finds out the importance of the boundary grid redistribution which is particularly important for a class of problems for the Brusselator model.Several ways for verifying the quality of the numerical solutions are also proposed,which may be of important use for comparisons.

关键词： Reaction-diffusion systems Brusselator model Gray-Scott model moving finite element method

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Numerical Investigations of the Dynamical Behaviors and Instabilities for the Gierer-Meinhardt System

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Communications in Computational Physics 2008年第2期3卷 406-426页

作者： Zhonghua Qiao Center for Research in Scientific Computation North Carolina StateUniversityRaleighNorth Carolina27695-8205USA.

This work is concerned with the numerical simulations on the Gierer-Meinhardt activator-inhibitor models. We consider the case when the inhibitor timeconstant τ is non-zero. In this case, oscillations and pulse splitting are observed numerically.Numerical experiments are carried out to investigate the dynamical behaviorsand instabilities of the spike patterns. The numerical schemes used are based upon anefficient moving mesh finite element method which distributes more grid points nearthe localized spike regions.

关键词： Gierer-Meinhardtmodel moving finite element method pulse splitting spikes.

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