拟线性积分微分方程的hp-时间间断Galerkin方法
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摘要
本文将hp-时间间断Galerkin有限元方法应用于拟线性常积分微分方程,然后进一步推广用于偏积分微分方程.首先考虑了拟线性带弱奇异核的Volterra积分微分方程,利用线性化的手段对原问题进行处理,证明了原问题和线性化问题的等价性,最后给出了有限元解的L_2模误差估计.对于拟线性抛物型积分微分方程,同样采用线性化的方法,利用原问题和线性化问题的等价性,证明了在hp-时间间断Galerkin有限元方法下,拟线性抛物型积分微分方程的有限元解的存在唯一性,又对此近似解做出了L_2模误差估计.
In this paper the hp-discontinuous Galerkin time-stepping method for quasi-linear ordinary integro-differential equations and then for partial integro-differential equations are considered. Firstly, we consider quasi-linear Volterra integro-differential equations with weakly singular kernels. The primary problems are linearized and the equivalence of the primary problems and the linearized ones are proved. And the error estimates in L_2 are derived. For quasi-linear parabolic integro-differential equations, similarly, using the linearization method and the equivalence of the primary problems and the linearized ones, under the hp-discontinuous Galerkin time-stepping method the existence and uniqueness of the numerical solution of quasi-linear parabolic integro-differential equations are proved. And the error estimates in L_2 are derived.
In this paper the hp-discontinuous Galerkin time-stepping method for quasi-linear ordinary integro-differential equations and then for partial integro-differential equations are considered. Firstly, we consider quasi-linear Volterra integro-differential equations with weakly singular kernels. The primary problems are linearized and the equivalence of the primary problems and the linearized ones are proved. And the error estimates in L_2 are derived. For quasi-linear parabolic integro-differential equations, similarly, using the linearization method and the equivalence of the primary problems and the linearized ones, under the hp-discontinuous Galerkin time-stepping method the existence and uniqueness of the numerical solution of quasi-linear parabolic integro-differential equations are proved. And the error estimates in L_2 are derived.
引文
[1] P.Lesaint and P.A.Raviart, On a finite element method for solving the neutron transport equation[A]. In:Mathematical Aspects of Finite Elements in Partial Differential Equations, (C.de Boor,ed.)Academic Press. New York, 1974:89-145.
[2] W.H.Reed and T.R.Hill,Triangular Mesh Methods for Neutron Transport Equation [A].Research Report LA-UR-73-479,Los Alamos Scientific Laboratory, 1973.
[3] V.Thomée, Galerkin finite element methods for parabolic equations[M]. Springer-Verlag,New York, 1997.
[4] K.Bottcher and R.Rannacher,Adaptive Error Control in Solving Ordinary Differential Equations by the Discontinuous Galerkin Method[M],Preprint96-53,IWR,Universitat Heidel-berg,Heidelberg,Germany,1996.
[5] M.Delfour,W.Hager,and F.rochu.Discontinuous Galerkin methods for ordinary differential equations[J],Math.Comp.,1981,36:455-473.
[6] D.Estep,A posteriori error bounds and global error control for approximation of ordinary differential equations[J],SIAM J.Numer.Anal., 1995,32:1-48.
[7] C.Johnson,Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations[J],SIAM J.Numer Anal.,1988,25:908-926.
[8] P.Jamet,Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain [J], SIAM J.Numer.Anal.,1978,15:912-928.
[9] I.Babuska and M.Suri,The p and hp-versions of the finite element method[J],basic principles and properties [J], SIAM Rev.,1994,36:578-632.
[10] C.Schwab,p- and hp- Finite Element Methods[M],Oxford University Press,New York,1998.
[11] C.E.Baumann,J.T.Oden,A discontinuous hp finite element method for convection diffusion problems,in Special Issue: Spectral,Spectral Element, and hp methods in CFD,Ed.,G.E.Karniadakis.Computer Methods in Applied Mechanics and Engineering. 1999,175:311-341.
[12] W.Gui and I.Babuska, the h-,p- and hp-versions of the finite element method in one dimension,Ⅰ: The error analysis of the p-version[J]. Numer.Math.,1986,49:577-612.
[13] W.Gui and I.Babuska, the h-,p- and hp-versions of the finite element method in one dimension, Ⅱ: The error analysis of the h- and hp-versions [J]. Numer.Math.,1986,49:613-657.
[14] W.Gui and I.Babuska, the h-,p- and hp-versions of the finite element method in one dimension,Ⅲ: The adaptive hp-version [J]. Numer.Math.,1986,49:659-683.
[15]D.Schotzau,C.Schwab,Time discretization of parabolic problems by the hp version of the discontinuous Galerkin finite element method[J].SIAM J.Numer.Anal.,2000,38:837-875.
[16]T.Werder,K.Gerdes,D.Schotzau,C.Schwab,hp-discontinuous Galerkin time-stepping for parabolic problems[J].Comput.Methods Appl.Mech.Engrg.2001,190:6685-6708.
[17]D.Schotzau,C.Schwab,An hp a-priori analysis of the DG time-stepping for initial value problems [J].Calcolo,2000,37:207-232.
[18]李宏,王焕清,拟线性抛物型积分微分方程的间断时空有限元方法[J].内蒙古大学学报(自然科学版),2005,36(6):630-635.
[19]H.Brunner,D.Schotzau,hp-discontinuous Galerkin time-stepping tbr Volterra integrodifferential equations[J].SIAM J.Numer.Anal.2006,44(1):224-245.
[20]S.Larsson,V.Thomee,L.Wahlbin,Numerical solution of parabolic integrodifferential equations by the discontinuous Galerkin method[J].Math.comp.1998,67:45-71.
[21]D.Schotzau,C.Schwab,hp-discontinuous Galerkin time-stepping for parabolic problems [J]C.R.Acad.Sci.Paris Ser.Ⅰ,2001,333:1121-1126.
[2] W.H.Reed and T.R.Hill,Triangular Mesh Methods for Neutron Transport Equation [A].Research Report LA-UR-73-479,Los Alamos Scientific Laboratory, 1973.
[3] V.Thomée, Galerkin finite element methods for parabolic equations[M]. Springer-Verlag,New York, 1997.
[4] K.Bottcher and R.Rannacher,Adaptive Error Control in Solving Ordinary Differential Equations by the Discontinuous Galerkin Method[M],Preprint96-53,IWR,Universitat Heidel-berg,Heidelberg,Germany,1996.
[5] M.Delfour,W.Hager,and F.rochu.Discontinuous Galerkin methods for ordinary differential equations[J],Math.Comp.,1981,36:455-473.
[6] D.Estep,A posteriori error bounds and global error control for approximation of ordinary differential equations[J],SIAM J.Numer.Anal., 1995,32:1-48.
[7] C.Johnson,Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations[J],SIAM J.Numer Anal.,1988,25:908-926.
[8] P.Jamet,Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain [J], SIAM J.Numer.Anal.,1978,15:912-928.
[9] I.Babuska and M.Suri,The p and hp-versions of the finite element method[J],basic principles and properties [J], SIAM Rev.,1994,36:578-632.
[10] C.Schwab,p- and hp- Finite Element Methods[M],Oxford University Press,New York,1998.
[11] C.E.Baumann,J.T.Oden,A discontinuous hp finite element method for convection diffusion problems,in Special Issue: Spectral,Spectral Element, and hp methods in CFD,Ed.,G.E.Karniadakis.Computer Methods in Applied Mechanics and Engineering. 1999,175:311-341.
[12] W.Gui and I.Babuska, the h-,p- and hp-versions of the finite element method in one dimension,Ⅰ: The error analysis of the p-version[J]. Numer.Math.,1986,49:577-612.
[13] W.Gui and I.Babuska, the h-,p- and hp-versions of the finite element method in one dimension, Ⅱ: The error analysis of the h- and hp-versions [J]. Numer.Math.,1986,49:613-657.
[14] W.Gui and I.Babuska, the h-,p- and hp-versions of the finite element method in one dimension,Ⅲ: The adaptive hp-version [J]. Numer.Math.,1986,49:659-683.
[15]D.Schotzau,C.Schwab,Time discretization of parabolic problems by the hp version of the discontinuous Galerkin finite element method[J].SIAM J.Numer.Anal.,2000,38:837-875.
[16]T.Werder,K.Gerdes,D.Schotzau,C.Schwab,hp-discontinuous Galerkin time-stepping for parabolic problems[J].Comput.Methods Appl.Mech.Engrg.2001,190:6685-6708.
[17]D.Schotzau,C.Schwab,An hp a-priori analysis of the DG time-stepping for initial value problems [J].Calcolo,2000,37:207-232.
[18]李宏,王焕清,拟线性抛物型积分微分方程的间断时空有限元方法[J].内蒙古大学学报(自然科学版),2005,36(6):630-635.
[19]H.Brunner,D.Schotzau,hp-discontinuous Galerkin time-stepping tbr Volterra integrodifferential equations[J].SIAM J.Numer.Anal.2006,44(1):224-245.
[20]S.Larsson,V.Thomee,L.Wahlbin,Numerical solution of parabolic integrodifferential equations by the discontinuous Galerkin method[J].Math.comp.1998,67:45-71.
[21]D.Schotzau,C.Schwab,hp-discontinuous Galerkin time-stepping for parabolic problems [J]C.R.Acad.Sci.Paris Ser.Ⅰ,2001,333:1121-1126.