发展方程的高精度有限元方法研究

发布时间：2018-04-08 17:38

本文选题：发展方程　切入点：非协调元　出处：《郑州大学》2016年博士论文

【摘要】：本文主要针对几类发展方程(诸如非线性Schr¨odinger方程、Benjamin-BonaMahony(BBM)方程、非定常不可压缩Navier-Stokes方程、Cahn-Hilliard(CH)方程以及对流占优扩散方程),分别从非协调有限元方法、协调和非协调混合元方法出发,对其收敛性、超逼近和超收敛等进行了深入系统的研究.首先,将一个非协调四边形单元(改进类Wilson元)运用于求解非线性Schr-¨odinger方程.通过该单元相容误差在能量模意义下可以达到O(h3)阶,比其插值误差高两阶这一特殊性质,对于半离散以及两种全离散格式(Backward-Euler(B-E)和Crank-Nicolson(C-N)格式),在广义矩形网格下导出了最优误差估计和超逼近性质.进一步地,通过插值后处理技术,在矩形网格下得到了整体超收敛结果.最后给出数值算例来验证理论的正确性和方法的有效性.其次,研究了BBM方程的低阶非协调有限元方法及混合元新格式.一方面,利用EQrot1元的两个特殊性质(插值算子等价于Ritz投影算子,且相容误差为O(h2)阶,比插值误差高一阶),分别在半离散和两种全离散格式(B-E和C-N格式)下导出其能量模意义下的超逼近和整体超收敛结果.另一方面,对该方程构造一种新的非协调混合元格式,借助于零阶Raviart-Thomas(R-T)元的高精度特性,同样在半离散以及全离散格式下得到了相关变量的超逼近和超收敛结果,并给出数值例子检验理论分析的正确性.再次,针对非定常不可压缩Navier-Stokes方程提出了一个低阶非协调混合元方法.运用带约束的旋转Q1(CQrot1)元以及分片常数(Q0)元分别逼近速度u和压力p,并得到了能量模下u以及L2模下p的超逼近和超收敛结果.同时还进行了数值实验,所得数值结果与理论分析相吻合.而后,对CH方程建立一个新的非协调混合元分析框架.利用一类非协调元的特殊性质(相容误差为O(h2)阶,比插值误差高一阶)和插值后处理技术,分别在半离散和B-E全离散格式下得到了原始变量u和辅助变量p在能量模意义下的超逼近和整体超收敛结果.并给出数值例子来验证该方法的有效性.最后,针对对流占优扩散方程,构造了一个新的特征混合元格式.并给出关于原始变量u以及辅助变量p的收敛性分析,最终结合数值例子来验证此方法的有效性.
[Abstract]:In this paper, several kinds of evolution equations (such as the nonlinear Schr odinger equation Benjamin-Bona Mahony BBM) equation, the unsteady incompressible Navier-Stokes equation Cahn-Hilliardsch equation and the convection-dominated diffusion equation are studied respectively from the nonconforming finite element method, the concordant and the non-conforming mixed element method.The convergence, superapproximation and superconvergence are studied systematically.Firstly, a nonconforming quadrilateral element (modified Wilson element) is applied to solve the nonlinear odinger equation.According to the special property that the compatible error of the element can reach OH3) order in the sense of energy mode, the error is two orders higher than its interpolation error.For semi-discrete and two full discrete schemes, Backward-Eulerian B-E and Crank-Nicolsonian C-N schemes, the optimal error estimates and superapproximation properties are derived under generalized rectangular meshes.Furthermore, the global superconvergence results are obtained under rectangular meshes by interpolation postprocessing technique.Finally, numerical examples are given to verify the correctness of the theory and the validity of the method.Secondly, the low order nonconforming finite element method for BBM equation and the new mixed element scheme are studied.On the one hand, by using two special properties of EQrot1 element (interpolation operator is equivalent to Ritz projection operator, and the consistent error is OH2),The results of superapproximation and global superconvergence in the sense of energy modules are derived under semi-discrete and two kinds of full discrete schemes (B-E and C-N schemes respectively), which are higher than interpolation errors.On the other hand, a new nonconforming mixed element scheme is constructed for the equation. By virtue of the high accuracy property of the zero order Raviart-Thomas-R-T element, the superapproximation and superconvergence results of the related variables are also obtained in semi-discrete and fully discrete schemes.Numerical examples are given to verify the correctness of the theoretical analysis.Thirdly, a low order nonconforming mixed element method is proposed for unsteady incompressible Navier-Stokes equations.The velocity u and the pressure p are approximated by the rotating Q 1 C Q rot1 element and the piecewise constant Q 0) element respectively. The superapproximation and superconvergence results of u and L 2 modes under the energy mode are obtained.At the same time, numerical experiments are carried out, and the numerical results are in good agreement with the theoretical analysis.Then, a new nonconforming mixed element analysis framework is established for Ch equation.By using the special properties of a class of nonconforming elements (the compatible error is OH2, which is one order higher than the interpolation error) and the interpolation post-processing technique,The results of superapproximation and global superconvergence of the original variable u and the auxiliary variable p in the sense of energy module are obtained in semi-discrete and B-E full discrete schemes respectively.A numerical example is given to verify the effectiveness of the method.Finally, a new characteristic mixed element scheme is constructed for convection-dominated diffusion equations.The convergence analysis of the original variable u and the auxiliary variable p is given. Finally, numerical examples are used to verify the validity of the method.
【学位授予单位】：郑州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

【相似文献】