分布阶波方程全离散有限元方法的高精度分析新途径
|
摘要
本文研究了时间分布阶波方程的全离散有限元数值逼近及其高精度误差分析的新途径.首先,基于L1公式离散Caputo时间分数阶导数,构造了时间分布阶波方程的有限元全离散格式,证明了格式的无条件稳定性.然后,利用双线性元的Ritz投影算子R_h和插值算子I_h之间的高精度误差估计,再借助于插值后处理技术得到了在全离散格式下单独利用插值或投影所无法得到的超逼近和超收敛结果.进一步地,将该方法应用于变系数分布阶波方程,也证明了格式的无条件稳定性和超收敛性.最后,对一些常见的单元作了进一步探讨.
In this paper,a new approach of numerical fully discrete scheme based on the finite element approximation for the distributed order time fractional diffusion equations is developed and high accuracy error analysis is provided.Firstly,based on the L1 formula for the approximation of the time distributed order fractional derivative,the fully discrete finite element scheme is derived and the unconditional stability of the scheme is obtained.Secondly,by use of the supercolse estimate between the Ritz projection operator Rh and interpolation operator I_h of the bilinear element and the interpolated post-processing technique,the superclose and superconvergence results for the fully discrete scheme are obtained,which can't be deduced by the interpolation or Ritz projection alone.Furthermore,the proposed method is applied to the equations with variable cofficienet and the unconditional stability and superconvergent eatimates are also proved.Finally,some popular finite elements are investigated.
In this paper,a new approach of numerical fully discrete scheme based on the finite element approximation for the distributed order time fractional diffusion equations is developed and high accuracy error analysis is provided.Firstly,based on the L1 formula for the approximation of the time distributed order fractional derivative,the fully discrete finite element scheme is derived and the unconditional stability of the scheme is obtained.Secondly,by use of the supercolse estimate between the Ritz projection operator Rh and interpolation operator I_h of the bilinear element and the interpolated post-processing technique,the superclose and superconvergence results for the fully discrete scheme are obtained,which can't be deduced by the interpolation or Ritz projection alone.Furthermore,the proposed method is applied to the equations with variable cofficienet and the unconditional stability and superconvergent eatimates are also proved.Finally,some popular finite elements are investigated.
引文
[1]Diethelm K.The Analysis of Fractional Differential Equations.Lecture Notes in Mathematics.Berlin:Springer,2004
[2]Podlubny I.Fractional Differential Equations.Academic Press,San Diego,CA,1999
[3]Chen W,Sun H G.Numerical algorithms for fractional differential equations:current situation and problems.Comput.Aided Engrg.,2010,19:1-2
[4]Chen C M,Liu F W,Turner I,et al.A Fourier method for the fractional diffusion equation describing sub-diffusion.J.Comput.Phys.,2007,227:886-897
[5]Sun Z Z,Wu X N.A fully discrete difference scheme for a diffusion-wave system.Appl.Numer.Math.,2006,56:193-209
[6]Zhang Y N,Sun Z Z,Zhao X.Compact alternating direction implicit schemes for the two-dimensional fractional diffusion-wave equation.SIAM J.Numer.Anal., 2012,50:1535-1555
[7]Wang Z B,Vong S W.Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation.J.Comput. Phys.,2014,277:1-15
[8]Ren J C,Mao S P,Zhang J W.Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation,Numer.Meth.Partial Differential Equations,2018,34(2):705-730
[9]Ren J C,Long X N,Mao S P,Zhang J W.Superconvergence of finite element approximations for the fractional diffusion-wave equation.J.Sci.Comput.,2017,72(3):917-935
[10]Zhao X,Zhang Z M.Superconvergence points of fractional spectral interpolation.SIAM J.Sci.Comput.,2016,38:A598-A614
[11]Zhao X,Hu X Z,Cai W,Karniadakis G E.Adaptive finite element method for fractional differential equations using Hierarchical matrices.Comput.Methods Appl.Mech.Engrg.,2017,325:56-76
[12]Gao G H,Sun Z Z,Zhang H W.A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and it applications.J.Comput.Phy.,2014,259:33-50
[13]Alikhanov A A.A new difference scheme for the time fractional diffusion equation.J.Comput.Phy.,2015,280:424-438
[14]Sun Z Z,Gao G H.The difference method of fractional differential equations.Beijing:Science press,2015
[15]Liao H L,Zhang Y N,Zhao Y,Shi H S.Stability and convergence of modified Du Fort-Frankel schemes for solving time-fractioal subfiffusion equations.J.Sci.Comput. 2014,61:629-648
[16]Celik C,Duman M.Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative.J.Comput.Ph-ys.,2012,231:1743-1750
[17]Ye H P,Liu F W,Anh V,Turner I.Numerical analysis for the time distributed order and Riesz space fractional diffusions on bounded domains.IMA J.Appl.Math.,2015,80:825-838
[18]Gao G H,Sun H W,Sun Z Z.Some high-order difference schemes for the distributed-order differential equations.J.Comput.Phys., 2015,298:337-359
[19]Gao G H,Sun Z Z.Two alternating direction implicit difference schemes for two-Dimensional distributedorder fractional diffusion equations.J.Sci.Comput.,2016,66:1281-1312
[20]Mashayekhia S,Razzaghi M.Numerical solution of distributed order fractional differential equations by hybrid functions.J.Comput.Phys.,2016,315:169-181
[21]Bu W P,Xiao A G,Zeng W.Finite difference/finite element methods for distributed-order time fractional diffusion equations.J.Sci.Comput. 2017,72:422-441
[22]Yang X H,Zhang H X,Xu D.WSGD-OSC scheme for two-dimensional distributed order fractional reaction-diffusion equation.J.Sci.Comput.,2018,76:1502-1520
[23]Kincaid D,Cheney W.Numerical Analysis:Mathematics of Scientific Computing.Wadsworth,Belmont,CA,1991
[24]Thomée,V.Galerkin finite element methods for parabolic problems.Springer series in computational mathematics,Sweden,2000
[25]Lin Q,Yan N.N.The Construction and Analysis of High Efficient Elements.Hebei:Hebei University Press,1996
[26]Shi D Y,Wang P L,Zhao Y M.Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schr(o|¨)dinger equation.Appl.Math.Lett.,2014,38:129-134
[27]Lin Q.Global error expansion and superconvergence for higher order interpolation of finite elements.J.Comp.Math.,1992,10:286-289
[28]Shi D Y,Liang H.Superconvergence analysis and extrapolation of a new unconventional Hermite-type anisotropic rectangular element.Math.Numer.Sin.,2005,27:369-382
[29]Chen S C,Shi D Y.Accuracy analysis for quasi-Wison element.Acta Math.Sci.,2000,20(1):44-48
[30]Chen S C,Shi D Y.Zhao Y.C.Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes.IMA J.Numer.Anal.,2004,24(1):77-95
[31]Shi D Y,Wang F L,Zhao Y M.Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations.Acta Math.Appl.Sin.,2013,29(2):403-414
[32]Shi D Y,Pei L F.Nonconforming quadrilateral finite element method for a class of nonlinear sineGordon equations.Appl.Math.Comput.,2013,.219(17):9447-9460
[2]Podlubny I.Fractional Differential Equations.Academic Press,San Diego,CA,1999
[3]Chen W,Sun H G.Numerical algorithms for fractional differential equations:current situation and problems.Comput.Aided Engrg.,2010,19:1-2
[4]Chen C M,Liu F W,Turner I,et al.A Fourier method for the fractional diffusion equation describing sub-diffusion.J.Comput.Phys.,2007,227:886-897
[5]Sun Z Z,Wu X N.A fully discrete difference scheme for a diffusion-wave system.Appl.Numer.Math.,2006,56:193-209
[6]Zhang Y N,Sun Z Z,Zhao X.Compact alternating direction implicit schemes for the two-dimensional fractional diffusion-wave equation.SIAM J.Numer.Anal., 2012,50:1535-1555
[7]Wang Z B,Vong S W.Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation.J.Comput. Phys.,2014,277:1-15
[8]Ren J C,Mao S P,Zhang J W.Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation,Numer.Meth.Partial Differential Equations,2018,34(2):705-730
[9]Ren J C,Long X N,Mao S P,Zhang J W.Superconvergence of finite element approximations for the fractional diffusion-wave equation.J.Sci.Comput.,2017,72(3):917-935
[10]Zhao X,Zhang Z M.Superconvergence points of fractional spectral interpolation.SIAM J.Sci.Comput.,2016,38:A598-A614
[11]Zhao X,Hu X Z,Cai W,Karniadakis G E.Adaptive finite element method for fractional differential equations using Hierarchical matrices.Comput.Methods Appl.Mech.Engrg.,2017,325:56-76
[12]Gao G H,Sun Z Z,Zhang H W.A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and it applications.J.Comput.Phy.,2014,259:33-50
[13]Alikhanov A A.A new difference scheme for the time fractional diffusion equation.J.Comput.Phy.,2015,280:424-438
[14]Sun Z Z,Gao G H.The difference method of fractional differential equations.Beijing:Science press,2015
[15]Liao H L,Zhang Y N,Zhao Y,Shi H S.Stability and convergence of modified Du Fort-Frankel schemes for solving time-fractioal subfiffusion equations.J.Sci.Comput. 2014,61:629-648
[16]Celik C,Duman M.Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative.J.Comput.Ph-ys.,2012,231:1743-1750
[17]Ye H P,Liu F W,Anh V,Turner I.Numerical analysis for the time distributed order and Riesz space fractional diffusions on bounded domains.IMA J.Appl.Math.,2015,80:825-838
[18]Gao G H,Sun H W,Sun Z Z.Some high-order difference schemes for the distributed-order differential equations.J.Comput.Phys., 2015,298:337-359
[19]Gao G H,Sun Z Z.Two alternating direction implicit difference schemes for two-Dimensional distributedorder fractional diffusion equations.J.Sci.Comput.,2016,66:1281-1312
[20]Mashayekhia S,Razzaghi M.Numerical solution of distributed order fractional differential equations by hybrid functions.J.Comput.Phys.,2016,315:169-181
[21]Bu W P,Xiao A G,Zeng W.Finite difference/finite element methods for distributed-order time fractional diffusion equations.J.Sci.Comput. 2017,72:422-441
[22]Yang X H,Zhang H X,Xu D.WSGD-OSC scheme for two-dimensional distributed order fractional reaction-diffusion equation.J.Sci.Comput.,2018,76:1502-1520
[23]Kincaid D,Cheney W.Numerical Analysis:Mathematics of Scientific Computing.Wadsworth,Belmont,CA,1991
[24]Thomée,V.Galerkin finite element methods for parabolic problems.Springer series in computational mathematics,Sweden,2000
[25]Lin Q,Yan N.N.The Construction and Analysis of High Efficient Elements.Hebei:Hebei University Press,1996
[26]Shi D Y,Wang P L,Zhao Y M.Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schr(o|¨)dinger equation.Appl.Math.Lett.,2014,38:129-134
[27]Lin Q.Global error expansion and superconvergence for higher order interpolation of finite elements.J.Comp.Math.,1992,10:286-289
[28]Shi D Y,Liang H.Superconvergence analysis and extrapolation of a new unconventional Hermite-type anisotropic rectangular element.Math.Numer.Sin.,2005,27:369-382
[29]Chen S C,Shi D Y.Accuracy analysis for quasi-Wison element.Acta Math.Sci.,2000,20(1):44-48
[30]Chen S C,Shi D Y.Zhao Y.C.Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes.IMA J.Numer.Anal.,2004,24(1):77-95
[31]Shi D Y,Wang F L,Zhao Y M.Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations.Acta Math.Appl.Sin.,2013,29(2):403-414
[32]Shi D Y,Pei L F.Nonconforming quadrilateral finite element method for a class of nonlinear sineGordon equations.Appl.Math.Comput.,2013,.219(17):9447-9460