一类半线性抛物方程混合有限元方法的超逼近分析
|
摘要
采用双线性元及零阶Raviart-Thomas元(Q_(11)+Q_(10)×Q_(01))分析了一类半线性抛物方程的H~1-Galerkin格式下的无网格比超逼近性质.首先,引入一个时间离散方程,将误差拆分成时间误差和空间误差两部分.其次,通过时间误差给出时间离散方程解的正则性,再利用空间误差得到了有限元解U_h~n的W~(0,∞)(Ω)模有界,整个过程避免时间步长τ和空间剖分参数h的比值,即网格比的出现.最后,当原始方程右端项f(u)满足局部Lipschitz条件时,有技巧地导出了原始变量u在H~1(Ω)模意义下及流量p=▽u在L~2(Ω)模意义下的O(h~2+τ~2)的无网格比超逼近性质.当f(u)为二阶可导时,给出▽·p在L~2(Ω)模意义下的O(h~2+τ~2)的无网格比超逼近结果.数值算例验证了理论的正确性.
An H~1-Galerkin mixed finite element method(MFEM) is discussed for a kind of semilinear parabolic equations with the bilinear element and zero-order Raviart-Thomas element(Q_(11)+Q_(10)×Q_(01)).Firstly, a linearized scheme is given and the time discrete equations split the error into two parts, which are called the temporal error and spatial error. Secondly, the regularities of the time discrete equations are reduced by the temporal error and the FE solution ‖U_h~n‖_(0,∞) is bounded unconditionally by the spatial error. Lastly, when the right hand part of the original equation f(u) is local Lipschitz, the unconditional superclose results of u in H~1(Ω) and p = ▽u in L~2(Ω) are obtained skillfully. When f(u) is twicely continuously differentiable with respective to u, the unconditional superclose results of ▽ ·p in L~2(Ω) are gained. Numerical experiment is included to illustrate the feasibility of the proposed method.
An H~1-Galerkin mixed finite element method(MFEM) is discussed for a kind of semilinear parabolic equations with the bilinear element and zero-order Raviart-Thomas element(Q_(11)+Q_(10)×Q_(01)).Firstly, a linearized scheme is given and the time discrete equations split the error into two parts, which are called the temporal error and spatial error. Secondly, the regularities of the time discrete equations are reduced by the temporal error and the FE solution ‖U_h~n‖_(0,∞) is bounded unconditionally by the spatial error. Lastly, when the right hand part of the original equation f(u) is local Lipschitz, the unconditional superclose results of u in H~1(Ω) and p = ▽u in L~2(Ω) are obtained skillfully. When f(u) is twicely continuously differentiable with respective to u, the unconditional superclose results of ▽ ·p in L~2(Ω) are gained. Numerical experiment is included to illustrate the feasibility of the proposed method.
引文
[1] HAYES L J. A modified backward time discretization for nonlinear parabolic equations using patch approximations[J]. SIAM J. Numer. Anal., 1981, 18:781-793.
[2] RACHFORD H H, JR. Two-level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations[J]. SIAM J. Numer. Anal., 1973, 10(6):1010-1026.
[3] LUSKIN M. A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions[J]. SIAM J. Numer. Anal., 1979, 16(2):284-299.
[4] THOMEE V. Galerkin Finite Element Methods for Parabolic Problems[M]. Sweden:Springer Series in Computational Mathematics, 2000.
[5]董自明,李宏,赵智慧.一类抛物方程的降基连续时空有限元方法[J].应用数学,2017, 30(3):706-714.
[6]杨晓侠,石东洋,张芳.一类四阶抛物方程一个低阶非协调混合元方法的超收敛分析[J].应用数学,2016,29(2):370-380.
[7]白秀琴,张厚超,杨楠.一类变系数四阶抛物方程一个低阶非协调混合元方法的超收敛分析[J].应用数学,2017, 30(1):209-220.
[8] SHI D, WANG J. Unconditional super convergence analysis of conforming finite element for nonlinear parabolic equation[J]. Appl. Math. Comput., 2017, 294:216-226.
[9] SHI D, WANG J, YAN F. Unconditional superconvergence analysis for nonlinear parabolic equation with EQ_1~(Tot)nonconforming finite element[J]. J. Sci. Comput., 2017, 70(1):85-111.
[10] SHI D, WANG J. Unconditional superconvergence analysis of an H~1-Galerkin mixed finite element method for nonlinear Sobolev equations[J].Numer. Methods Partial Differential Equations, 2018,34(1):145-166.
[11] PANI A K. An H~1-Galerkin mixed finite element methods for parabolic partial differential equatios[J].SIAM J. Numer. Anal., 1998, 35(2):712-727.
[12] GUO L, CHEN H. H~1-Galerkin mixed finite element method for the regularized long wave equation[J].Computing, 2006, 77:205-221.
[13] PANI A K, SINHA R K, OTTA A K. An H~1-Galerkin mixed method for second order hyperbolic equations[J]. Int. J. Numer. Anal. Model., 2004, 1(2):111-130.
[14] LIU Y, LI H. H~1-Galerkin mixed finite element methods for pseudo-hyperbolic equations[J]. Appl.Math. Comput., 2009, 212(2):446-457.
[15] SHI D Y, SHI Y H. The lowest order H~1-Galerkin mixed finite element method for semi-linear pseudo-hyperbolic equation[J]. J. Syst. Sci. Math. SciS., 2015, 35(5):514-526.
[16] PANI A K, FAIRWEATHWR G. An H~1-Galerkin mixed finite element method for an evolution equation with a positive-type memory term[J]. SIAM J. Numer. Anal., 2002, 40(4):1475-1490.
[17] PANI A K. H~1-Galerkin mixed finite element methods for parabolic partial integro-differential equations[J]. IMA J. Numer. Anal., 2002, 22:231-252.
[18] LIN Q, LIN J F. Finite Element Methods:Accuracy and Improvement[M]. Beijing:Science Press,2006.
[2] RACHFORD H H, JR. Two-level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations[J]. SIAM J. Numer. Anal., 1973, 10(6):1010-1026.
[3] LUSKIN M. A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions[J]. SIAM J. Numer. Anal., 1979, 16(2):284-299.
[4] THOMEE V. Galerkin Finite Element Methods for Parabolic Problems[M]. Sweden:Springer Series in Computational Mathematics, 2000.
[5]董自明,李宏,赵智慧.一类抛物方程的降基连续时空有限元方法[J].应用数学,2017, 30(3):706-714.
[6]杨晓侠,石东洋,张芳.一类四阶抛物方程一个低阶非协调混合元方法的超收敛分析[J].应用数学,2016,29(2):370-380.
[7]白秀琴,张厚超,杨楠.一类变系数四阶抛物方程一个低阶非协调混合元方法的超收敛分析[J].应用数学,2017, 30(1):209-220.
[8] SHI D, WANG J. Unconditional super convergence analysis of conforming finite element for nonlinear parabolic equation[J]. Appl. Math. Comput., 2017, 294:216-226.
[9] SHI D, WANG J, YAN F. Unconditional superconvergence analysis for nonlinear parabolic equation with EQ_1~(Tot)nonconforming finite element[J]. J. Sci. Comput., 2017, 70(1):85-111.
[10] SHI D, WANG J. Unconditional superconvergence analysis of an H~1-Galerkin mixed finite element method for nonlinear Sobolev equations[J].Numer. Methods Partial Differential Equations, 2018,34(1):145-166.
[11] PANI A K. An H~1-Galerkin mixed finite element methods for parabolic partial differential equatios[J].SIAM J. Numer. Anal., 1998, 35(2):712-727.
[12] GUO L, CHEN H. H~1-Galerkin mixed finite element method for the regularized long wave equation[J].Computing, 2006, 77:205-221.
[13] PANI A K, SINHA R K, OTTA A K. An H~1-Galerkin mixed method for second order hyperbolic equations[J]. Int. J. Numer. Anal. Model., 2004, 1(2):111-130.
[14] LIU Y, LI H. H~1-Galerkin mixed finite element methods for pseudo-hyperbolic equations[J]. Appl.Math. Comput., 2009, 212(2):446-457.
[15] SHI D Y, SHI Y H. The lowest order H~1-Galerkin mixed finite element method for semi-linear pseudo-hyperbolic equation[J]. J. Syst. Sci. Math. SciS., 2015, 35(5):514-526.
[16] PANI A K, FAIRWEATHWR G. An H~1-Galerkin mixed finite element method for an evolution equation with a positive-type memory term[J]. SIAM J. Numer. Anal., 2002, 40(4):1475-1490.
[17] PANI A K. H~1-Galerkin mixed finite element methods for parabolic partial integro-differential equations[J]. IMA J. Numer. Anal., 2002, 22:231-252.
[18] LIN Q, LIN J F. Finite Element Methods:Accuracy and Improvement[M]. Beijing:Science Press,2006.