非协调各向异性有限元方法研究
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摘要
本文针对不同类型的偏微分方程(包括广义神经传播方程、Navier-Stokes方程、二阶椭圆方程、非线性sine-Gordon方程及非线性抛物积分微分方程等),分别从非协调元、变网格及各向异性网格等不同角度,对非协调单元的构造,收敛性分析、超逼近和超收敛性及数值计算等方面进行了深入系统的探讨.
首先在各向异性网格下,利用变网格思想,先用Crouzeix-Raviart型非协调三角形元对变系数的非线性广义神经传播方程的Crank-Nicolson离散格式作了分析,利用该单元的Riesz投影与插值算子是一致的特性,导出了相应变量的最优误差估计.接着,在矩形网格下,利用带约束的旋转Q1元,通过变网格法又对非定Navier-Stokes问题进行了收敛性分析,给出了速度的H1-模及压力L2-模的最优误差估计.
其次,考虑各向异性任意四边形网格下的EQrot1元.作为一个尝试,我们通过一些新的估计技巧和方法将[65]中的结果推广到各向异性非平行四边形网格上去.针对二阶椭圆方程,导出了最优误差估计.同时通过数值计算验证了理论分析的正确性.
最后,研究了广义非线性sine-Gordon方程和非线性抛物积分微分方程的非协调Quasi-Wilson有限元方法.利用单元相容误差比插值误差高两阶的特殊性质,并借助于双线性元的高精度分析,采取与以往文献不同的新技巧,分别在半离散和全离散格式下给出了相应未知量的L2-模最优误差估计和离散H1-模意义下的超逼近性质.进一步地,通过插值后处理得到了H1-模的整体超收敛结果及非线性抛物积分微分方程的向后欧拉全离散格式下的最优误差估计和超逼近性.
In this thesis, we consider some kinds of partial differential equations (includ-ing general neural transmission equations-. Navier-Stokes equations、second order elliptic equations、nonlinear sine-Gordon equations and nolinear parabolic integro-differential equation, etc) and study the nonconforming finite elements, moving grids and anisotropic meshes from different points of view and give the comprehensive investigations on the construction of nonconforming elements、convergence analy-sis、superclose、superconvergence and numerical computations, etc.
Firstly, on anisotropic meshes, by use of moving grids ideas, we use a Crouzeix-Raviart type nonconforming triangle element to approximate the variable coeffi-cient、nonlinear general neural transmission equations and analyze a Crank-Nicolson discrete scheme. The optimal error estimates are derived by the consistent property of the Riesz projection and the interpolation operators about the element. Then, the constrained rotated Q1element on rectangular meshes is used for the nonsta-tionary Navier-Stokes equations and the convergence analysis are provided with moving grides methods, the optimal error estimates in H1-norm for the velocity and.L2-norm for the pressure are derived.
Secondly, we investigate the EQrat1element on anisotropic arbitrary quadrilat-eral grids, as an attempt, we promote the results in [65] up to anisotropic nonparallel quadrilateral grids based on some new techniques and approaches and derive the optimal error estimates for second order elliptic problems. At the same time, the numerical experiments are carried out to verify the theoretical analysis.
Finally, the nonconforming Quasi-Wilson finite element approximations for the general nonlinear sine-Gordon equation and the nonlinear parabolic integro-differential equation are discussed. Based on the special feature of the element, i.e., the consistency error estimate is of order O(h3) in the energy norm, which is two order higher than that of interpolation error estimate, and high accuracy analysis of the bilinear element, the optimal error estimates in L2-norm and superclose proper-ties in broken H1-norm of the corresponding unknown functions are derived for the semi-discrete and fully-discrete schemes with new approaches different from that in the previous literature, respectively. Moreover, the global superconvergence results of broken H1-norm are obtained through the postprocessing techniques and the su-perclose properties and the optimal estimates for the Backward-Euler fully-discrete scheme are derived for the nonlinear parabolic integro-differential equations.
首先在各向异性网格下,利用变网格思想,先用Crouzeix-Raviart型非协调三角形元对变系数的非线性广义神经传播方程的Crank-Nicolson离散格式作了分析,利用该单元的Riesz投影与插值算子是一致的特性,导出了相应变量的最优误差估计.接着,在矩形网格下,利用带约束的旋转Q1元,通过变网格法又对非定Navier-Stokes问题进行了收敛性分析,给出了速度的H1-模及压力L2-模的最优误差估计.
其次,考虑各向异性任意四边形网格下的EQrot1元.作为一个尝试,我们通过一些新的估计技巧和方法将[65]中的结果推广到各向异性非平行四边形网格上去.针对二阶椭圆方程,导出了最优误差估计.同时通过数值计算验证了理论分析的正确性.
最后,研究了广义非线性sine-Gordon方程和非线性抛物积分微分方程的非协调Quasi-Wilson有限元方法.利用单元相容误差比插值误差高两阶的特殊性质,并借助于双线性元的高精度分析,采取与以往文献不同的新技巧,分别在半离散和全离散格式下给出了相应未知量的L2-模最优误差估计和离散H1-模意义下的超逼近性质.进一步地,通过插值后处理得到了H1-模的整体超收敛结果及非线性抛物积分微分方程的向后欧拉全离散格式下的最优误差估计和超逼近性.
In this thesis, we consider some kinds of partial differential equations (includ-ing general neural transmission equations-. Navier-Stokes equations、second order elliptic equations、nonlinear sine-Gordon equations and nolinear parabolic integro-differential equation, etc) and study the nonconforming finite elements, moving grids and anisotropic meshes from different points of view and give the comprehensive investigations on the construction of nonconforming elements、convergence analy-sis、superclose、superconvergence and numerical computations, etc.
Firstly, on anisotropic meshes, by use of moving grids ideas, we use a Crouzeix-Raviart type nonconforming triangle element to approximate the variable coeffi-cient、nonlinear general neural transmission equations and analyze a Crank-Nicolson discrete scheme. The optimal error estimates are derived by the consistent property of the Riesz projection and the interpolation operators about the element. Then, the constrained rotated Q1element on rectangular meshes is used for the nonsta-tionary Navier-Stokes equations and the convergence analysis are provided with moving grides methods, the optimal error estimates in H1-norm for the velocity and.L2-norm for the pressure are derived.
Secondly, we investigate the EQrat1element on anisotropic arbitrary quadrilat-eral grids, as an attempt, we promote the results in [65] up to anisotropic nonparallel quadrilateral grids based on some new techniques and approaches and derive the optimal error estimates for second order elliptic problems. At the same time, the numerical experiments are carried out to verify the theoretical analysis.
Finally, the nonconforming Quasi-Wilson finite element approximations for the general nonlinear sine-Gordon equation and the nonlinear parabolic integro-differential equation are discussed. Based on the special feature of the element, i.e., the consistency error estimate is of order O(h3) in the energy norm, which is two order higher than that of interpolation error estimate, and high accuracy analysis of the bilinear element, the optimal error estimates in L2-norm and superclose proper-ties in broken H1-norm of the corresponding unknown functions are derived for the semi-discrete and fully-discrete schemes with new approaches different from that in the previous literature, respectively. Moreover, the global superconvergence results of broken H1-norm are obtained through the postprocessing techniques and the su-perclose properties and the optimal estimates for the Backward-Euler fully-discrete scheme are derived for the nonlinear parabolic integro-differential equations.
引文
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