The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusionwave equation with spatial variable coefficient,which...The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusionwave equation with spatial variable coefficient,which contains a time-space coupled derivative.The nonconforming EQ^(rot)_(1)element and Raviart-Thomas element are employed for spatial discretization,and L1 time-stepping method combined with the Crank-Nicolson scheme are applied for temporal discretization.Firstly,based on some significant lemmas,the unconditional stability analysis of the fully-discrete scheme is acquired.With the assistance of the interpolation operator I_(h)and projection operator Rh,superclose and convergence results of the variable u in H^(1)-norm and the flux~p=k_(5)(x)ru(x,t)in L^(2)-norm are obtained,respectively.Furthermore,the global superconvergence results are derived by applying the interpolation postprocessing technique.Finally,the availability and accuracy of the theoretical analysis are corroborated by experimental results of numerical examples on anisotropic meshes.展开更多
By employing EQ^(ROT)_(1) nonconforming finite element,the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes.Comparing with the m...By employing EQ^(ROT)_(1) nonconforming finite element,the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes.Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation,the mixed case contains a special time-space coupled derivative,which leads to many difficulties in numerical analysis.Firstly,a fully discrete scheme is established by using nonconforming finite element method(FEM)in spatial direction and L1 approximation coupled with Crank-Nicolson(L1-CN)scheme in temporal direction.Furthermore,the fully discrete scheme is proved to be unconditional stable.Besides,convergence and superclose results are derived by using the properties of EQ^(ROT)_(1) nonconforming finite element.What's more,the global superconvergence is obtained via the interpolation postprocessing technique.Finally,several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.展开更多
In this paper, high-order numerical analysis of finite element method（FEM） is presented for twodimensional multi-term time-fractional diffusion-wave equation（TFDWE）. First of all, a fully-discrete approximate sche...In this paper, high-order numerical analysis of finite element method（FEM） is presented for twodimensional multi-term time-fractional diffusion-wave equation（TFDWE）. First of all, a fully-discrete approximate scheme for multi-term TFDWE is established, which is based on bilinear FEM in spatial direction and Crank-Nicolson approximation in temporal direction, respectively. Then the proposed scheme is proved to be unconditionally stable and convergent. And then, rigorous proofs are given here for superclose properties in H-1-norm and temporal convergence in L-2-norm with order O（h-2＋ τ-（3-α））, where h and τ are the spatial size and time step, respectively. At the same time, theoretical analysis of global superconvergence in H-1-norm is derived by interpolation postprocessing technique. At last, numerical example is provided to demonstrate the theoretical analysis.展开更多
基金The work is supported by the National Natural Science Foundation of China(Nos.11971416 and 11871441)the Scientific Research Innovation Team of Xuchang University(No.2022CXTD002)the Foundation for University Key Young Teacher of Henan Province(No.2019GGJS214).
文摘The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusionwave equation with spatial variable coefficient,which contains a time-space coupled derivative.The nonconforming EQ^(rot)_(1)element and Raviart-Thomas element are employed for spatial discretization,and L1 time-stepping method combined with the Crank-Nicolson scheme are applied for temporal discretization.Firstly,based on some significant lemmas,the unconditional stability analysis of the fully-discrete scheme is acquired.With the assistance of the interpolation operator I_(h)and projection operator Rh,superclose and convergence results of the variable u in H^(1)-norm and the flux~p=k_(5)(x)ru(x,t)in L^(2)-norm are obtained,respectively.Furthermore,the global superconvergence results are derived by applying the interpolation postprocessing technique.Finally,the availability and accuracy of the theoretical analysis are corroborated by experimental results of numerical examples on anisotropic meshes.
基金National Natural Science Foundation of China(No.11971416)Scientific Research Innovation Team of Xuchang University(No.2022CXTD002)+3 种基金Foundation for University Key Young Teacher of Henan Province(No.2019GGJS214)Key Scientific Research Projects in Universities of Henan Province(Nos.21B110007,22A110022)National Natural Science Foundation of China(International cooperation key project:No.12120101001)Australian Research Council via the Discovery Project(DP190101889).
文摘By employing EQ^(ROT)_(1) nonconforming finite element,the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes.Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation,the mixed case contains a special time-space coupled derivative,which leads to many difficulties in numerical analysis.Firstly,a fully discrete scheme is established by using nonconforming finite element method(FEM)in spatial direction and L1 approximation coupled with Crank-Nicolson(L1-CN)scheme in temporal direction.Furthermore,the fully discrete scheme is proved to be unconditional stable.Besides,convergence and superclose results are derived by using the properties of EQ^(ROT)_(1) nonconforming finite element.What's more,the global superconvergence is obtained via the interpolation postprocessing technique.Finally,several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.
基金Supported by the National Natural Science Foundation of China(Nos.11771438,11471296)the Key Scientific Research Projects in Universities of Henan Province(No.19B110013)the Program for Scientific and Technological Innovation Talents in Universities of Henan Province(No.19HASTIT025)
文摘In this paper, high-order numerical analysis of finite element method（FEM） is presented for twodimensional multi-term time-fractional diffusion-wave equation（TFDWE）. First of all, a fully-discrete approximate scheme for multi-term TFDWE is established, which is based on bilinear FEM in spatial direction and Crank-Nicolson approximation in temporal direction, respectively. Then the proposed scheme is proved to be unconditionally stable and convergent. And then, rigorous proofs are given here for superclose properties in H-1-norm and temporal convergence in L-2-norm with order O（h-2＋ τ-（3-α））, where h and τ are the spatial size and time step, respectively. At the same time, theoretical analysis of global superconvergence in H-1-norm is derived by interpolation postprocessing technique. At last, numerical example is provided to demonstrate the theoretical analysis.
