Stokes问题的变网格非协调有限元法

众所周知,由于LBB条件的限制,用非协调元格式求解速度—压力型的Stokes问题具有构造简单,计算经济和误差阶匹配等优点而在实际计算中经常被采用。用非协调格式处理Stokes问题首先是由Crouzeix-Raviart提出来的,他们采用分片线性三中点三角元这一非协调元作为速度逼近空间,用分片常数有限元空间作为压力逼近空间(即C—R格式),得     

一类四阶抛物方程一个低阶非协调混合元方法的超收敛分析

对一类四阶抛物方程利用EQ_1~(rot)元和零阶Raviart-Thomas元提出一个低阶非协调混合元逼近格式.首先证明半离散格式逼近解的存在唯一性.其次,基于上述两个单元的高精度分析,利用对时间变量的导数转移技巧并借助插值后处理技术,在半离散格式下得到了原始变量u,中间变量v=—△u的H~1-模意义下以及流量=—▽u的L~2-模意义下O(h~2)阶的超逼近性质和超收敛结果.最后,证明向后Euler全离散格式逼近解的存在唯一性,并通过采用一个新的分裂技巧,导出u和v在H~1-模意义下以及在L~2-模意义下关于h的无条件的O(h~2+τ)阶的超逼近性质和超收敛结果.这里,h及τ分别表示空间剖分参数和时间步长.     

非线性四阶双曲方程扩展的非协调混合元方法的超收敛分析及外推

对一类非线性四阶双曲方程,利用EQ_1~(rot)元及零阶Raviart-Thomas元建立一个新的扩展的非协调混合元逼近格式.首先证明了逼近解的存在唯一性.其次,基于EQ_1~(rot)元特殊性质,再利用零阶Raviart-Thomas元的高精度分析结果和插值后处理技术,在半离散格式下导出了原始变量u和中间变量v=-?u在H~1模及中间变量q=?u,σ=-?(?u)在(L~2)~2模意义下具有O(h~2)阶的超逼近性质和超收敛结果.最后,利用EQ_1~(rot)元的渐近展开式,构造一个新的合适的外推格式,得到相关变量O(h~3)阶的外推解.     

三维区域上Qrot1元的渐近展开及外推

本文研究了正方体区域上Qrot1非协调元渐近展开式.利用林群、吕涛等提出的有限元误差渐近展开法,获得了正方体区域上Qrot1非协调元特征值的误差渐近展开式.理论分析和数值实验结果表明三维Qrot1非协渊元特征值外推公式是有效的,可以把特征值的精度从二阶提高到四阶.     

三维有限元本征值的外推

通过二维和三维积分恒等式,探讨泊松方程本征值问题三角线元和四面体线元Richardson外推的可行性.理论分析表明,如果剖分为均匀一致和拟一致,外推均可将解的精度提高二阶.     

用有限元方法求解界面特征值问题

用有限元方法计算椭圆型界面特征值问题,实验数据显示近似特征值的变化规律:界面特征值问题中系数的间断性对协调和非协调Crouzeix-Raviart有限元特征值的收敛性并无影响,而且对协调有限元特征值外推以后得到高精度的解,相应的外推值还提供特征值下界;Crouzeix-Raviart元特征值提供特征值下界,这对一般有界区域如"镂空"型区域也成立.另外,还展示近似特征函数的图形.     

非线性抛物方程的EQ_1~(rot)非协调混合元方法的超收敛分析

基于EQ_1~(rot)非协调矩形元及零阶R-T元对非线性抛物方程构造了一个新的混合元格式.利用EQ_1~(rot)元所具有的两个特殊性质:(I)插值算子与其投影算子是一致的;(II)当所考虑问题的精确解属于H~3(ΩΩ)时,其相容误差可以达到O(h~2)(h是剖分参数),比插值误差高一阶.同时借助关于这两个单元的高精度分析、平均值技巧和插值后处理技术,得到了关于原始变量以及通量的超逼近和整体超收敛结果.     

三维重调和方程Adini元特征值展开式及其应用

讨论了重调和方程三维Adini元的特征值的渐进展开,通过展开式指出其特征值是下界逼近,并指出收敛阶为O(h~2),并用数值实验验证我们的理论分析.     

对流扩散方程的稳定化流线扩散法最小二乘非协调有限元分析

将最小二乘法和稳定化的流线扩散法相结合,研究了对流扩散方程的非协调有限元格式,用矩形EQ_1~(rot)元和零阶R-T元分别来逼近位移和应力,利用单元本身的特殊性质,证明了离散格式解的存在惟一性,得到了位移H~1-模和应力H(div)-模的最优误差估计.     

EFK方程一个新的低阶非协调混合有限元方法的高精度分析

对Extended Fisher-Kolmogorov(EFK)方程,利用EQ_1~(rot)元和零阶RaviartThomas(R-T)元建立了一个新的非协调混合元逼近格式.首先,证明了半离散格式逼近解的一个先验估计并证明了逼近解的存在唯一性.在半离散格式下,利用上述两种元的高精度分析结果以及这个先验估计,在不需要有限元解u_h属于L~∞的条件下,得到了原始变量u和中间变量v=-?u的H~1-模以及流量p=u的(L~2)~2-模意义下O(h~2)阶的超逼近性质.同时,借助插值后处理技术,证明了上述变量的具有O(h~2)阶的整体超收敛结果.其次,建立了一个新的线性化向后Euler全离散格式并证明了其逼近解的存在唯一性.另一方面,通过对相容误差和非线性项采取与传统误差分析不同的新的分裂技巧,分别导出了以往文献中尚未涉及的关于u和v在H~1-模以及p在(L~2)~2-模意义下具有O(h~2+τ)阶的超逼近性质,进一步地,借助插值后处理技术,得到了上述变量的整体超收敛结果.这里h和τ分别表示空间剖分参数和时间步长.最后,给出了一个数值算例,计算结果验证了理论分析的正确性.     

Eigenvalue approximation from below using non-conforming finite elements

This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators,with emphasis on obtaining lower bounds. In addition,this article also contains some new materials for eigenvalue approximations of the Laplace operator,which include:1) the proof of the fact that the non-conforming Crouzeix-Raviart element approximates eigenvalues associated with smooth eigenfunctions from below;2) the proof of the fact that the non-conforming EQ rot1 element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements;3) the explanation of the phenomena that numerical eigenvalues λ 1,h and λ 3,h of the non-conforming Q rot1 element approximate the true eigenvalues from below for the L-shaped domain. Finally,we list several unsolved problems.     

非定常线性化Navier-Stokes方程的子格粘性非协调有限元方法

本文结合子格粘性法的思想,空间采用非协调Crouzeix-Raviart元逼近,时间采用Crank-Nicolson差分离散,对非定常线性化Navier-Stokes方程建立了全离散的子格粘性非协调有限元格式.对稳定性和误差估计作出了详细的分析, 得出了最优的误差估计.最后, 通过数值算例进一步验证了该方法的稳定性和收敛性.     

基于Crouzeix-Raviart元的界面浸入有限元方法及其收敛性分析

本文对具间断系数的二阶椭圆界面问题提出一种浸入有限元方法(theimmersed finite element method), 即在界面单元上采用依赖于界面的线性多项式空间离散, 而在非界面单元上采用Crouzeix-Raviart非协调元离散. 论证表明, 该方法具有对界面问题解的最优L²-模和H¹-模收敛精度.     

Approximation of functionally graded plates with non-conforming finite elements

In this paper rectangular plates made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGM plates are determined using a variational formulation arising from the Reissner–Mindlin theory. To approximate the problem a simple locking-free Discontinuous Galerkin finite element of non-conforming type is used, choosing a piecewise linear non-conforming approximation for both rotations and transversal displacement. Several numerical simulations are carried out in order to show the capability of the proposed element to capture the properties of plates of various gradings, subjected to thermo-mechanical loads.     

Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.     

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.     

AN ERROR ANALYSIS METHOD SPP-BEAM AND A CONSTRUCTION GUIDELINE OF NONCONFORMING FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC PROBLEMS

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems,we present a side-patchwise projection based error analysis method(SPP-BEAM for short).Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method.In addition,it is universal enough to admit generalizations.Then,we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces.As an application,we use the theory to design a P3 second order triangular H2 non-conforming element by enriching two P4 bubble functions and,another P4 second order triangular H2 nonconforming finite element,and a P3 second order tetrahedral H2 non-conforming element by enriching eight P4 bubble functions,adding some more degrees of freedom.     

A posteriori error estimators for the Stokes equations II non-conforming discretizations

Summary We present an a posteriori error estimator for the non-conforming Crouzeix-Raviart discretization of the Stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation. The error estimator yields global upper and local lower bounds for the error of the finite element solution. It can easily be generalized to the stationary, incompressible Navier-Stokes equations and to other non-conforming finite element methods. Numerical examples show the efficiency of the proposed error estimator.     

各向异性网格下抛物方程一个新的非协调混合元收敛性分析

本文将 Crouzeix-Raviart 型非协调线性三角形元应用到抛物方程,建立了一个新的混合元格式.在抛弃传统有限元分析的必要工具 Ritz 投影算子的前提下,直接利用单元的插值性质和导数转移技巧, 分别得到了各向异性剖分下关于原始变量u 的H^-1-模和积分意义下L²-模以及通量p=-▽u 在L²-模下的最优阶误差估计.数值结果与我们的理论分析是相吻合的.     

三维区域上Q_1~(rot)元的渐近展开及外推

本文研究了正方体区域上Q₁^rot非协调元渐近展开式.利用林群、吕涛等提出的有限元误差渐近展开法,获得了正方体区域上Q₁^rot非协调元特征值的误差渐近展开式.理论分析和数值实验结果表明三维Q₁^rot非协调元特征值外推公式是有效的,可以把特征值的精度从二阶提高到四阶.