三类非线性方程的超收敛分析及外推研究
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摘要
本文主要研究具有很强实际应用背景的三类非线性方程：非线性粘弹性方程,一类广义神经传播方程和非线性Sobolev方程.分别用双线性元、Hermite型矩形元及一个非协调元解逼近时的收敛性、超逼近性、超收敛性及外推格式.其中对前两个单元来说,主要利用积分恒等式及其渐进展开式,通过构造辅助问题,提出相应地适当外推格式以提高有限元的逼近精度.而对于非协调元来说,主要借助于单元的特殊性质：例如精确解与其有限元插值之差在能量意义下正交、相容性误差比插值误差高一阶、导数转嫁等技巧导出了与使用协调元时完全相同的超收敛结果.研究表明直接利用单元上插值算子的性质对某些单元来说不仅可以简化证明过程,比使用传统的Ritz投影减少工作量,有时还可以得到比以往文献更好的结果,从而进一步扩展了有限元方法的使用范围.
This paper studies the convergence, superclose, superconvergence and extrapolation schemes of the approximate solutions of three types nonlinear equations with strong ap-plied backgrounds:nonlinear viscoelasticity type equations, a class of generalized neure conductive equations and nonlinear Sobolev equations with bilinear element, Hermite-type rectangular element and a noncforming element respectively. For the former two elements, by use of the integral identity and its asymptotic expansion, and constructing an auxiliary problem, corresonding the suitable extrapolation schemes so as to improve the accuracy of the finite element approximations. For the nonconforming element, by means of some special properties of the element:for example, the difference between the exact solution and its finite element interpolation is orthogonal in the sense of energy norm, compati-bility error is one order higher than the interpolation error, the derivative transformation techniques and so on, the same superconvergence results are derived. The study of this paper shows that using the element interpolation operator can not only simpify the proof directly, but also can reduce the computing cost comparing with the traditional Ritz pro-jection, and sometimes can get better results than the previous literature, which further extends the application of finite element methods.
This paper studies the convergence, superclose, superconvergence and extrapolation schemes of the approximate solutions of three types nonlinear equations with strong ap-plied backgrounds:nonlinear viscoelasticity type equations, a class of generalized neure conductive equations and nonlinear Sobolev equations with bilinear element, Hermite-type rectangular element and a noncforming element respectively. For the former two elements, by use of the integral identity and its asymptotic expansion, and constructing an auxiliary problem, corresonding the suitable extrapolation schemes so as to improve the accuracy of the finite element approximations. For the nonconforming element, by means of some special properties of the element:for example, the difference between the exact solution and its finite element interpolation is orthogonal in the sense of energy norm, compati-bility error is one order higher than the interpolation error, the derivative transformation techniques and so on, the same superconvergence results are derived. The study of this paper shows that using the element interpolation operator can not only simpify the proof directly, but also can reduce the computing cost comparing with the traditional Ritz pro-jection, and sometimes can get better results than the previous literature, which further extends the application of finite element methods.
引文
[1]Webb G F. Existence and asymptotic behavior for a strongly damped nonlinear wave equation[J]. Can J,1980,32(3):631-643.
[2]Q.LIN, S.H.Zhang. A direct global superconvergence analysis for Sobolev and vis-coelasticity type equations[J]. Appl. Math.,1997,42(4):23-24.
[3]Q.LIN, S.H.Zhang, N.N.Yan. A symptotic error expansion and defent correction for Sobolev and viscoelasticity type equations[J]. J. Comput. Math.,1998,16(1):51-62.
[4]D.Y.Shi, Y.C.Peng, S.C.Chen. Superconvergence of a nonconforming finite element ap-proximation to viscoelasticity type equations on anisotropic meshes[J]. Numer. Math. J. Chinese. Univ.,2006,15(4):375-384.
[5]D.Y.Jin, T.Liu, S.H.Zhang. Global superconvergence analysis of Wilson element for Sobolev and viscoelasticity type equations[J]. J. Syst. Sci. Complex.,2004,17(4): 452-463.
[6]史艳华,石东洋.粘弹性方程ACM有限元的超收敛分析和外推(英文)[J].应用数学,2009,22(3)：534-541.
[7]石东洋,关宏波.粘弹性方程的非协调变网格有限元方法[J].高校应用数学学报,2008,23(4)：452-458.
[8]Y.P.Chen, Y.Q.Huang. The superconvergence of mixed finite element methods for nonlinear hyperbolic equations[J]. Commun. Nonlinear. Sci. Numer. Simul.,1998, 3(3):155-158.
[9]Douglas J, Wang J. Superconvergence for mixed finite element methods on rectangular domains[J]. Calcolo,1989,26:121-134.
[10]石东洋,梁慧.一个新的非常规Hermite型各向异性矩形元的超收敛分析及外推[J].计算数学,2005,27(4)：369-382.
[11]Q.Lin, J.F.Lin. Finite Element Methods:Accuracy and Improvement[M]. Beijing: Science Press,2006.
[12]Q.Lin, J.M.Zhou. Superconvergence in higher-order Galerkin finite element meth-ods[J]. Comput. Methods. Appl. Mech. Engrg.,2007,196:3779-3784.
[13]M.X.Li, Q.Lin, S.H.Zhang. Superconvergence of finite element method for the Signorini problem[J]. J. Comput. Appl. Math.,2008,222:284-292.
[14]M.X.Li, Q.Lin, S.H.Zhang. Extrapolation and superconvergence of the Steklov eigen-value problem[J]. Adv. Comput. Math.,2010,33(1):25-44.
[15]C V. Pao. An mixed initial boundary value problem arising in neurophysicology[J]. J.Math. Anal.& Appl.,1975,52(1):105-119.
[16]程极济,林克椿.生物物理学[M].北京：人民教育出版社,1982.
[17]崔霞.广义神经传播方程的A.D.I.有限元分析[J].应用数学学报,1999,22(4)：628-633.
[18]那顺布和.一类神经传播方程的特征差分方法和最佳阶L2误差估计[J].生物数学学报,2009,24(3)：470-478.
[19]马戈,石东洋.广义神经传播方程的非协调混合有限元方法[J].数学的实践与认识,2010,40(4)：217-223.
[20]石东洋,郝颖.广义神经传播方程的一个各向异性非协调有限元超收敛分析[J].生物数学学报,2009,24(2)：279-286.
[21]万维明,刘亚成.神经传播方程初边值问题解的长时间行为[J].应用数学学报,1999,22(2)：311-315.
[22]王波.一类神经传导方程的变网格有限元方法及数值分析[J].生物数学学报,2006,21(1)：119-128.
[23]梅茗.高维广义神经传播方程Cauchy冲问题整体光滑解[J].应用数学学报,1991,14(4)：450-461.
[24]林群,严宁宁.高效有限元的构造与分析[M].保定：河北大学出版社,1996.
[25]D.Y.Shi, H.Q.Zhu. The superconvergence analysis of an anisotropic finite element[J]. J.Syst. Sci. & Complexity,2005,18(4):478-487.
[26]Q.Lin, J.F.Lin. Extrapolation of the bilinear element approximation for the Poisson equation on anisotropic meshes[J].Numer.Meth.for PDEs,2007,23(5):960-967.
[27]D.Y.Shi, S.P.Mao and S.C.Chen. An anisotropic nonconforming finite element with some superconvergence results [J]. Comput. Math.,2005,23(3):261-274.
[28]A.H.Zhou. Ananlysis of some high accuracy finite element methods for hyperbolic problem [J]. SIAM J. Numer. Anal.,2001,39(3):1014-1028.
[29]程极济,林克椿.生物物理学[M].北京：人民教育出版社,1982.
[30]张志跃.一类广义神经传播方程的有限元方法及其数值分析[J].数学物理学报,2001,21(A1)：647-654.
[31]石东洋,周家全.广义神经传播方程一个新的H1-Galerkin非协调混合有限元格式[J].河南师范大学学报,2010,38(5)：1-6.
[32]崔霞.广义神经传播方程的A.D.I.有限元分析[J].应用数学学报,1999,22(4)：628-633.
[33]史艳华,石东洋.粘弹性方程Hermite型有限元新的超收敛分析和外推[J].河南师范大学学报,2009,37(3)：148-151.
[34]Q.Lin, A.Zhou and H.Chen Superclose and extrapolation of the tetrahedral linear finite elements for poisson equation in three-dimensional rectangular field [J]. Math. Prac.& Theo.,2009,39(13):215-220.
[35]Zenisek A, Vanmaele M.The interpolation theorem for narrow qudrilateeral isopara-metric finite element [J]. Numer. Math.,1995,72(1):123-141.
[36]Thomee V, J.Xu, N.Zhang. Superconvergence of the gradient in piecewise linear finite element approximation to a parabolic problem [J]. SIAM J. Numer.Anal.,1989,26: 553-573.
[37]Apel Th, Nicaise S and Schpoberl L. Crouzeix-Raviart type finte elements on anisotropic meshes [J]. Numer. Math.,2001,89(2):1193-223.
[38]Z.W.Jing, H.Z.Chen. Error estimates of mixed finite element methods for Sobolev equation [J]. J. Northeast Math.,2001,17(3):301-314.
[39]Q.Lin, H.X.He. Asymptotic error expansion and Richardson extrapolation of eigen-value approximations for second order elliptic problems by the mixed finite element method [J]. Appl. Numer. Math.,2009,59:1884-1893.
[40]M.X.Li, Q.Lin and S.H.Zhang. Extrapolation and superconvergence of the Steklov eigenvalue problem [J]. Adv. Comput. Math.,2010,33:25-44.
[41]Q.Lin, Tobiska L and A.Zhou. Superconvergence and extrapolation of nonconform low order finite elements applied to the poisson equation [J]. IMA J. Numer. Anal.,2005, 25:160-181.
[42]S.C.Chen,D.Y.Shi and Y.C.Zhao. Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes [J]. IMA J. Numer. Anal.,2004,24(1):77-95.
[43]D.Y.Shi and H.Liang. Superconvergence analysis Wilson element on anisotropic meshes [J]. Appl. Math. & Mech.,2007,28(1):119-125.
[44]金大永,刘棠,张书华Sobolev型方程Wilson元解的高精度分析[J].数学的实践与认识,2003,33(8)：84-89.
[45]石东洋,任金城,郝晓斌Sobolev型方程各向异性网格下Wilson元的高精度分析[J].高等学校计算数学学报,2009,31(2)：169-180.
[46]Girault V and Raviart P A. Finite element method for Navier-Stokes equations:The-ory and Algorithms [M]. Springer-Verlag,1986.
[2]Q.LIN, S.H.Zhang. A direct global superconvergence analysis for Sobolev and vis-coelasticity type equations[J]. Appl. Math.,1997,42(4):23-24.
[3]Q.LIN, S.H.Zhang, N.N.Yan. A symptotic error expansion and defent correction for Sobolev and viscoelasticity type equations[J]. J. Comput. Math.,1998,16(1):51-62.
[4]D.Y.Shi, Y.C.Peng, S.C.Chen. Superconvergence of a nonconforming finite element ap-proximation to viscoelasticity type equations on anisotropic meshes[J]. Numer. Math. J. Chinese. Univ.,2006,15(4):375-384.
[5]D.Y.Jin, T.Liu, S.H.Zhang. Global superconvergence analysis of Wilson element for Sobolev and viscoelasticity type equations[J]. J. Syst. Sci. Complex.,2004,17(4): 452-463.
[6]史艳华,石东洋.粘弹性方程ACM有限元的超收敛分析和外推(英文)[J].应用数学,2009,22(3)：534-541.
[7]石东洋,关宏波.粘弹性方程的非协调变网格有限元方法[J].高校应用数学学报,2008,23(4)：452-458.
[8]Y.P.Chen, Y.Q.Huang. The superconvergence of mixed finite element methods for nonlinear hyperbolic equations[J]. Commun. Nonlinear. Sci. Numer. Simul.,1998, 3(3):155-158.
[9]Douglas J, Wang J. Superconvergence for mixed finite element methods on rectangular domains[J]. Calcolo,1989,26:121-134.
[10]石东洋,梁慧.一个新的非常规Hermite型各向异性矩形元的超收敛分析及外推[J].计算数学,2005,27(4)：369-382.
[11]Q.Lin, J.F.Lin. Finite Element Methods:Accuracy and Improvement[M]. Beijing: Science Press,2006.
[12]Q.Lin, J.M.Zhou. Superconvergence in higher-order Galerkin finite element meth-ods[J]. Comput. Methods. Appl. Mech. Engrg.,2007,196:3779-3784.
[13]M.X.Li, Q.Lin, S.H.Zhang. Superconvergence of finite element method for the Signorini problem[J]. J. Comput. Appl. Math.,2008,222:284-292.
[14]M.X.Li, Q.Lin, S.H.Zhang. Extrapolation and superconvergence of the Steklov eigen-value problem[J]. Adv. Comput. Math.,2010,33(1):25-44.
[15]C V. Pao. An mixed initial boundary value problem arising in neurophysicology[J]. J.Math. Anal.& Appl.,1975,52(1):105-119.
[16]程极济,林克椿.生物物理学[M].北京：人民教育出版社,1982.
[17]崔霞.广义神经传播方程的A.D.I.有限元分析[J].应用数学学报,1999,22(4)：628-633.
[18]那顺布和.一类神经传播方程的特征差分方法和最佳阶L2误差估计[J].生物数学学报,2009,24(3)：470-478.
[19]马戈,石东洋.广义神经传播方程的非协调混合有限元方法[J].数学的实践与认识,2010,40(4)：217-223.
[20]石东洋,郝颖.广义神经传播方程的一个各向异性非协调有限元超收敛分析[J].生物数学学报,2009,24(2)：279-286.
[21]万维明,刘亚成.神经传播方程初边值问题解的长时间行为[J].应用数学学报,1999,22(2)：311-315.
[22]王波.一类神经传导方程的变网格有限元方法及数值分析[J].生物数学学报,2006,21(1)：119-128.
[23]梅茗.高维广义神经传播方程Cauchy冲问题整体光滑解[J].应用数学学报,1991,14(4)：450-461.
[24]林群,严宁宁.高效有限元的构造与分析[M].保定：河北大学出版社,1996.
[25]D.Y.Shi, H.Q.Zhu. The superconvergence analysis of an anisotropic finite element[J]. J.Syst. Sci. & Complexity,2005,18(4):478-487.
[26]Q.Lin, J.F.Lin. Extrapolation of the bilinear element approximation for the Poisson equation on anisotropic meshes[J].Numer.Meth.for PDEs,2007,23(5):960-967.
[27]D.Y.Shi, S.P.Mao and S.C.Chen. An anisotropic nonconforming finite element with some superconvergence results [J]. Comput. Math.,2005,23(3):261-274.
[28]A.H.Zhou. Ananlysis of some high accuracy finite element methods for hyperbolic problem [J]. SIAM J. Numer. Anal.,2001,39(3):1014-1028.
[29]程极济,林克椿.生物物理学[M].北京：人民教育出版社,1982.
[30]张志跃.一类广义神经传播方程的有限元方法及其数值分析[J].数学物理学报,2001,21(A1)：647-654.
[31]石东洋,周家全.广义神经传播方程一个新的H1-Galerkin非协调混合有限元格式[J].河南师范大学学报,2010,38(5)：1-6.
[32]崔霞.广义神经传播方程的A.D.I.有限元分析[J].应用数学学报,1999,22(4)：628-633.
[33]史艳华,石东洋.粘弹性方程Hermite型有限元新的超收敛分析和外推[J].河南师范大学学报,2009,37(3)：148-151.
[34]Q.Lin, A.Zhou and H.Chen Superclose and extrapolation of the tetrahedral linear finite elements for poisson equation in three-dimensional rectangular field [J]. Math. Prac.& Theo.,2009,39(13):215-220.
[35]Zenisek A, Vanmaele M.The interpolation theorem for narrow qudrilateeral isopara-metric finite element [J]. Numer. Math.,1995,72(1):123-141.
[36]Thomee V, J.Xu, N.Zhang. Superconvergence of the gradient in piecewise linear finite element approximation to a parabolic problem [J]. SIAM J. Numer.Anal.,1989,26: 553-573.
[37]Apel Th, Nicaise S and Schpoberl L. Crouzeix-Raviart type finte elements on anisotropic meshes [J]. Numer. Math.,2001,89(2):1193-223.
[38]Z.W.Jing, H.Z.Chen. Error estimates of mixed finite element methods for Sobolev equation [J]. J. Northeast Math.,2001,17(3):301-314.
[39]Q.Lin, H.X.He. Asymptotic error expansion and Richardson extrapolation of eigen-value approximations for second order elliptic problems by the mixed finite element method [J]. Appl. Numer. Math.,2009,59:1884-1893.
[40]M.X.Li, Q.Lin and S.H.Zhang. Extrapolation and superconvergence of the Steklov eigenvalue problem [J]. Adv. Comput. Math.,2010,33:25-44.
[41]Q.Lin, Tobiska L and A.Zhou. Superconvergence and extrapolation of nonconform low order finite elements applied to the poisson equation [J]. IMA J. Numer. Anal.,2005, 25:160-181.
[42]S.C.Chen,D.Y.Shi and Y.C.Zhao. Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes [J]. IMA J. Numer. Anal.,2004,24(1):77-95.
[43]D.Y.Shi and H.Liang. Superconvergence analysis Wilson element on anisotropic meshes [J]. Appl. Math. & Mech.,2007,28(1):119-125.
[44]金大永,刘棠,张书华Sobolev型方程Wilson元解的高精度分析[J].数学的实践与认识,2003,33(8)：84-89.
[45]石东洋,任金城,郝晓斌Sobolev型方程各向异性网格下Wilson元的高精度分析[J].高等学校计算数学学报,2009,31(2)：169-180.
[46]Girault V and Raviart P A. Finite element method for Navier-Stokes equations:The-ory and Algorithms [M]. Springer-Verlag,1986.
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