With the aim of simulating the blow-up solutions, a moving finite element method, based on nonuni- form meshes both in time and in space, is proposed in this paper to solve time fractional partial differential equatio...With the aim of simulating the blow-up solutions, a moving finite element method, based on nonuni- form meshes both in time and in space, is proposed in this paper to solve time fractional partial differential equations （FPDEs）. The unconditional stability and convergence rates of 2 -a for time and r for space are proved when the method is used for the linear time FPDEs with a-th order time derivatives. Numerical exam-ples are provided to support the theoretical findings, and the blow-up solutions for the nonlinear FPDEs are simulated by the method.展开更多
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 ...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 element,a linear basis function.展开更多
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 ...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.展开更多
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 he...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.展开更多
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 split...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.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos.10901027 and 11171274)Foundation of Hunan Educational Committee (Grant No. 10C0370)
文摘With the aim of simulating the blow-up solutions, a moving finite element method, based on nonuni- form meshes both in time and in space, is proposed in this paper to solve time fractional partial differential equations （FPDEs）. The unconditional stability and convergence rates of 2 -a for time and r for space are proved when the method is used for the linear time FPDEs with a-th order time derivatives. Numerical exam-ples are provided to support the theoretical findings, and the blow-up solutions for the nonlinear FPDEs are simulated by the method.
文摘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 element,a linear basis function.
基金supported in part by NSFC grants DMS-11971469the National Key R&D Program of China under Grant 2018YFB0704304 and Grant 2018YFB0704300.
文摘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.
基金The first and the third authors are partially supported by HKBU FRG grants and the Hong Kong Research Grant CouncilThe second author is partially supported by the Hong Kong RGC grant(No.201710).
文摘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.
文摘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.
