非线性抛物积分微分方程非常规Hermite型矩形元的高精度分析
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摘要
在半离散格式下讨论一类非线性抛物积分微分方程非常规Hermite型矩形元逼近,利用插值理论、高精度分析和对时间t的导数转移技巧,得到了H~1模意义下O(h~3)阶的超逼近性.进一步地,运用插值后处理技术,得到整体超收敛结果.与此同时,借助于构造一个合适的外推格式,得到了更高精度O(h~4)阶的外推解.
An unconventional Hermite-type rectangular element approximation is discussed for a class of nonlinear parabolic integro-differential equations under a semi-discrete scheme. The superclose property with order O (h~3) in H~1 norm is obtained by means of the interpolation theory,a high-accuracy analysis and the derivative transfer techniques for the time t. Furthermore,the global superconvergence result is derived with the interpolated postprocessing technique. At the same time,the high-accuracy extrapolation solution with order O (h~4) is deduced through constructing a suitable extrapolation scheme.
An unconventional Hermite-type rectangular element approximation is discussed for a class of nonlinear parabolic integro-differential equations under a semi-discrete scheme. The superclose property with order O (h~3) in H~1 norm is obtained by means of the interpolation theory,a high-accuracy analysis and the derivative transfer techniques for the time t. Furthermore,the global superconvergence result is derived with the interpolated postprocessing technique. At the same time,the high-accuracy extrapolation solution with order O (h~4) is deduced through constructing a suitable extrapolation scheme.
引文
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[11]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(自然科学版),2009,37(3):148-151.SHI Y H,SHI D Y.A new superconvergence analysis and extrapolation of Hermite-type finite element method for viscoelasticity equation[J].Journal of Henan Normal University(Natural Science),2009,37(3):148-151.
[12]王萍莉,史艳华,石东洋.广义神经传播方程的超收敛分析及外推[J].西北师范大学学报(自然科学版),2012,48(1):22-32.WANG P L,SHI Y H,SHI D Y.Superconvergence analysis and extrapolations for the generalized nerve conductive equations[J].Journal of Northwest Normal University(Natural Science),2012,48(1):22-32.
[13]王芬玲,石东伟.非线性双曲型方程Hermite型矩形元的高精度分析[J].山东大学学报(理学版),2012,47(10):89-96.WANG F L,SHI D W.High analysis of Hermite-type rectangular element for nonlinear hyperbolic equation[J].Journal of Shandong University(Natural Science),2012,47(10):89-96.
[14]毛凤梅,刁群,王俊俊.一类非线性伪双曲方程Hermite型有限元的超收敛分析及外推[J].郑州大学学报(理学版),2015,47(1):6-9;23.MAO F M,DIAO Q,WANG J J.A new superconvergence analysis and extrapolation of Hermite-type finite element for nonlinear pseudo-hyperbolic equation[J].Journal of Zhengzhou University(Natural Science Edition),2015,47(1):6-9;23.
[15]穆静静,周树克.半线性粘弹性方程非常规Hermite型矩形元的高精度分析[J].河北师范大学学报(自然科学版),2016,40(1):17-23.MU J J,ZHOU S K.High accuracy analysis of unconventional Hermite-type finite element for semi-linear viscoelasticity equations[J].Journal of Hebei Normal University(Natural Science),2016,40(1):17-23.
[2]张铁.抛物型积分-微分方程有限元近似的超收敛性质[J].高等学校计算数学学报,2001,23(3):193-201.ZHANG T.Superconvergence of finite element approximations to integro-differential equations of parabolic type[J].Numerical Mathematies A Journal of Chinese Universities,2001,23(3):193-201.
[3]李宏,王焕清.半线性抛物型积分微分方程的间断时空有限元方法[J].计算数学,2006,28(3):293-308.LI H,WANG H Q.The space-time discontinuous finite element method for a semi-linear parabolic Integro-differential equation[J].Mathematica Numerica Sinica,2006,28(3):293-308.
[4]SHI D Y,ZHANG B Y.High accuracy analysis of anisotropic finite element method for nonlinear parabolic integrodifferential equations[J].Mathematica Applicata,2008,21(3):436-442.
[5]SHI D Y,WANG H H.An H1-Galerkin nonconforming mixed finite element method for integro-differential equation of parabolic type[J].Journal of Mathematical Research&Exposition,2009,29(5):871-881.
[6]石东洋,裴丽芳.非线性抛物型积分微分方程非协调三角形Carey元的收敛性分析[J].系统科学与数学,2009,29(6):854-864.SHI D Y,PEI L F.Convergence analysis of the nonconforming triangular Carey element for a kind of nonlinear parabolic integro-differential problems[J].Journal of Systems Science and Mathematical Sciences,2009,29(6):854-864.
[7]孙淑珍,石翔宇.抛物型积分微分方程双线性元方法的新估计[J].郑州大学学报(理学版),2016,48(4):6-9.SUN S Z,SHI X Y.New error estimates of bilinear finite element method to parabolic type integro-differential equation[J].Journal of Zhengzhou University(Natural Science Edition),2016,48(4):6-9.
[8]张铁.偏微分-积分方程的有限元方法[M].北京:科学出版社,2009:54.
[9]THOME V.Galerkin finite element methods for parabolic problems[M].Berlin:Spring,1997:231-232.
[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.SHI D Y,LIANG H.Superconvergence analysis and extrapolation of a new unconventional Hermite-type anisotropic rectangular element[J].Mathematica Numerica Sinica,2005,27(4):369-382.
[11]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(自然科学版),2009,37(3):148-151.SHI Y H,SHI D Y.A new superconvergence analysis and extrapolation of Hermite-type finite element method for viscoelasticity equation[J].Journal of Henan Normal University(Natural Science),2009,37(3):148-151.
[12]王萍莉,史艳华,石东洋.广义神经传播方程的超收敛分析及外推[J].西北师范大学学报(自然科学版),2012,48(1):22-32.WANG P L,SHI Y H,SHI D Y.Superconvergence analysis and extrapolations for the generalized nerve conductive equations[J].Journal of Northwest Normal University(Natural Science),2012,48(1):22-32.
[13]王芬玲,石东伟.非线性双曲型方程Hermite型矩形元的高精度分析[J].山东大学学报(理学版),2012,47(10):89-96.WANG F L,SHI D W.High analysis of Hermite-type rectangular element for nonlinear hyperbolic equation[J].Journal of Shandong University(Natural Science),2012,47(10):89-96.
[14]毛凤梅,刁群,王俊俊.一类非线性伪双曲方程Hermite型有限元的超收敛分析及外推[J].郑州大学学报(理学版),2015,47(1):6-9;23.MAO F M,DIAO Q,WANG J J.A new superconvergence analysis and extrapolation of Hermite-type finite element for nonlinear pseudo-hyperbolic equation[J].Journal of Zhengzhou University(Natural Science Edition),2015,47(1):6-9;23.
[15]穆静静,周树克.半线性粘弹性方程非常规Hermite型矩形元的高精度分析[J].河北师范大学学报(自然科学版),2016,40(1):17-23.MU J J,ZHOU S K.High accuracy analysis of unconventional Hermite-type finite element for semi-linear viscoelasticity equations[J].Journal of Hebei Normal University(Natural Science),2016,40(1):17-23.