非协调有限元的构造及其应用
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摘要
本文构造了几个新的非协调有限元,系统地研究了它们的收敛性质并讨论了它们的一些应用.这些非协调元包括:Quasi-Carey元,Quasi-Wilson元,高次Wilson元以及二阶非协调混合元.
与协调元相比,非协调元具有许多优势.一般来说,对某些问题,它们构造简单,同时还具有很好的收敛效果,例如Morley元和Wilson元.另外,相对于协调混合元,非协调混合元更容易构造地使其满足离散inf-sup条件,因此非协调元的研究得到科学工作者和工程师的广泛关注.根据第二Strang引理,非协调元的误差包括两部分,一部分为插值误差,另一部分为相容误差,许多情况下相容误差的阶低于或等于插值误差的阶.对于二阶椭圆问题,本文构造的Quasi-Carey元相容误差为O(h~2),比插值误差高一阶;在矩形网格上,本文证明了传统的Quasi-Wilson元的相容误差为O(h~3),比插值误差高两阶,同时还给出了一个新的Quasi-Wilson元,它的相容误差在任意四边形网格上为O(h~3);经过仔细分析,我们首次证明了在各向异性网格上,高次Wilson元插值误差为O(h~3),比相容误差O(h~2)高一阶.
作为应用,在第二章,我们得到了各向异性网格上Quasi-Carey元关于Sobolev方程O(h~2)阶的整体超收敛和后验误差估计,根据误差渐近展开得到了O(h~4)阶的整体外推结果.第三章,我们根据Quasi-Wilson元的特殊收敛性,把它应用到对流扩散方程,得到了与双线性元和P_1~(mod)元相同的O(h~(3/2))阶的最优收敛阶.第四章首先分析了高次Wilson元在各向异性网格上的收敛性质,并给出数值试验说明了理论分析的有效性.接着导出了高次Wilson元的整体超收敛性质,并在此基础上给出了解的后验误差估计.第五章把Quasi-Carey元和修正的高次Wilson元应用到Maxwell方程的有限元格式,得到了Crouzeix-Raviart型三角形非协调元,Carey元以及高次Wilson元达不到的最优收敛结果.第六章我们用两个新的具有O(h~2)阶的非协调元格式离散不可压Navier-Stokes方程,得到了速度的H~1-模和压力的L~2-模的O(h~2)阶的误差估计以及速度的L~2-模的O(h~3)阶的误差估计,同时还给出了数值算例来验证误差分析的有效性.
Several new nonconforming finite elements are constructed,the convergence analysis of these elements are discussed and their application are presented in this thesis systematically.These new nonconforming elements include:the Quasi-Carey element,the Quasi-Wilson element,the higher order Wilson element and the second order nonconforming mixed finite element.
Compared with the conforming finite element methods,the finite element methods of nonconforming have many advantages.Generally speaking,nonconforming elements have fewer degrees of freedom for its simpler structure and good convergence properties,such as the Morley element and the Wilson element.In addition, the nonconforming mixed finite element methods are usually much easier to be constructed to satisfy the discrete inf-sup condition than the conforming ones. Therefore,nonconforming finite element methods have drawn increasing attention from scientists and engineers.As we know,according to the second Strang lemma, the error of every nonconforming element consists of two parts,one arises from the interpolation error and the other is the consistency error due to nonconformity of the element.In most cases,the order of the consistency error is lower than or equal to that of the interpolation error.But,in this paper,one can see that for the second order elliptic problems the consistency error of the Quasi-Carey element is of order O(h~2),one order higher than that of its interpolation error O(h).We proved that the consistency error of the traditional Quasi-Wilson element is of order O(h~3),two order higher than that of its interpolation error.At the same time,a new QuasiWilson element for arbitrary quadrilateral meshes possessing consistency error with order O(h~3) is presented.After a careful analysis,we first show that the interpolation error of the higher order Wilson element is of order O(h~3) on anisotropic meshes,one order higher than that of its consistency error.
As application,in Chapter 2,we investigated the approximation of higher accuracy of the anisotropic nonconforming Quasi-Carey element for the Sobolev type equations.The superclose and global superconvergence with order O(h~2) are obtained. Moveover,by virtue of the extrapolation,we improved the approximate accuracy of the related approximate solution and derive a posteriori error estimate of higher accuracy of order O(h~4).In Chapter 3,based on the special convergence of the Quasi-Wilson element,we applied it to convection-diffusion equations and obtained the optimal convergence order O(h~(3/2)) as the bilinear element and the p_1~(mod) element.In Chapter 4,after analysing the error estimates of the higher order Wilson element on anisotropic meshes with a numerical test,the superclose properties of this element are proved.Then the interpolation postprocessing technique is used to obtain the global superconvergence and the posterior error estimate of higher accuracy.In Chapter 5,we applied the Quasi-Carey element and the modification higher order Wilson element to Maxwell's equations on the finite element scheme, the optimal convergence results are obtained.But the similar optimal convergence results can not be obtained for the nonconforming linear triangular Crouzeix-Raviart element,the Carey element and the higher order Wilson element.In Chapter 6,the new nonconforming mixed finite element schemes with second order convergence behavior are proposed for the stationary Navier-Stokes equations,the convergence analysis is presented and the error estimates of both H~1-norm of order O(h~2) and L~2-norm of order O(h~3) with respect to velocity as well as the L~2-norm of order O(h~2) for the pressure are derived.At the same time,the numerical results are presented to illustrate the error analysis.
与协调元相比,非协调元具有许多优势.一般来说,对某些问题,它们构造简单,同时还具有很好的收敛效果,例如Morley元和Wilson元.另外,相对于协调混合元,非协调混合元更容易构造地使其满足离散inf-sup条件,因此非协调元的研究得到科学工作者和工程师的广泛关注.根据第二Strang引理,非协调元的误差包括两部分,一部分为插值误差,另一部分为相容误差,许多情况下相容误差的阶低于或等于插值误差的阶.对于二阶椭圆问题,本文构造的Quasi-Carey元相容误差为O(h~2),比插值误差高一阶;在矩形网格上,本文证明了传统的Quasi-Wilson元的相容误差为O(h~3),比插值误差高两阶,同时还给出了一个新的Quasi-Wilson元,它的相容误差在任意四边形网格上为O(h~3);经过仔细分析,我们首次证明了在各向异性网格上,高次Wilson元插值误差为O(h~3),比相容误差O(h~2)高一阶.
作为应用,在第二章,我们得到了各向异性网格上Quasi-Carey元关于Sobolev方程O(h~2)阶的整体超收敛和后验误差估计,根据误差渐近展开得到了O(h~4)阶的整体外推结果.第三章,我们根据Quasi-Wilson元的特殊收敛性,把它应用到对流扩散方程,得到了与双线性元和P_1~(mod)元相同的O(h~(3/2))阶的最优收敛阶.第四章首先分析了高次Wilson元在各向异性网格上的收敛性质,并给出数值试验说明了理论分析的有效性.接着导出了高次Wilson元的整体超收敛性质,并在此基础上给出了解的后验误差估计.第五章把Quasi-Carey元和修正的高次Wilson元应用到Maxwell方程的有限元格式,得到了Crouzeix-Raviart型三角形非协调元,Carey元以及高次Wilson元达不到的最优收敛结果.第六章我们用两个新的具有O(h~2)阶的非协调元格式离散不可压Navier-Stokes方程,得到了速度的H~1-模和压力的L~2-模的O(h~2)阶的误差估计以及速度的L~2-模的O(h~3)阶的误差估计,同时还给出了数值算例来验证误差分析的有效性.
Several new nonconforming finite elements are constructed,the convergence analysis of these elements are discussed and their application are presented in this thesis systematically.These new nonconforming elements include:the Quasi-Carey element,the Quasi-Wilson element,the higher order Wilson element and the second order nonconforming mixed finite element.
Compared with the conforming finite element methods,the finite element methods of nonconforming have many advantages.Generally speaking,nonconforming elements have fewer degrees of freedom for its simpler structure and good convergence properties,such as the Morley element and the Wilson element.In addition, the nonconforming mixed finite element methods are usually much easier to be constructed to satisfy the discrete inf-sup condition than the conforming ones. Therefore,nonconforming finite element methods have drawn increasing attention from scientists and engineers.As we know,according to the second Strang lemma, the error of every nonconforming element consists of two parts,one arises from the interpolation error and the other is the consistency error due to nonconformity of the element.In most cases,the order of the consistency error is lower than or equal to that of the interpolation error.But,in this paper,one can see that for the second order elliptic problems the consistency error of the Quasi-Carey element is of order O(h~2),one order higher than that of its interpolation error O(h).We proved that the consistency error of the traditional Quasi-Wilson element is of order O(h~3),two order higher than that of its interpolation error.At the same time,a new QuasiWilson element for arbitrary quadrilateral meshes possessing consistency error with order O(h~3) is presented.After a careful analysis,we first show that the interpolation error of the higher order Wilson element is of order O(h~3) on anisotropic meshes,one order higher than that of its consistency error.
As application,in Chapter 2,we investigated the approximation of higher accuracy of the anisotropic nonconforming Quasi-Carey element for the Sobolev type equations.The superclose and global superconvergence with order O(h~2) are obtained. Moveover,by virtue of the extrapolation,we improved the approximate accuracy of the related approximate solution and derive a posteriori error estimate of higher accuracy of order O(h~4).In Chapter 3,based on the special convergence of the Quasi-Wilson element,we applied it to convection-diffusion equations and obtained the optimal convergence order O(h~(3/2)) as the bilinear element and the p_1~(mod) element.In Chapter 4,after analysing the error estimates of the higher order Wilson element on anisotropic meshes with a numerical test,the superclose properties of this element are proved.Then the interpolation postprocessing technique is used to obtain the global superconvergence and the posterior error estimate of higher accuracy.In Chapter 5,we applied the Quasi-Carey element and the modification higher order Wilson element to Maxwell's equations on the finite element scheme, the optimal convergence results are obtained.But the similar optimal convergence results can not be obtained for the nonconforming linear triangular Crouzeix-Raviart element,the Carey element and the higher order Wilson element.In Chapter 6,the new nonconforming mixed finite element schemes with second order convergence behavior are proposed for the stationary Navier-Stokes equations,the convergence analysis is presented and the error estimates of both H~1-norm of order O(h~2) and L~2-norm of order O(h~3) with respect to velocity as well as the L~2-norm of order O(h~2) for the pressure are derived.At the same time,the numerical results are presented to illustrate the error analysis.
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