四阶椭圆方程的一个十六参数能量正交四面体元(英文)

In this paper,the 16-parameter nonconforming tetrahedral element which has an energy-orthogonal shape function space is presented for the discretization of fourth order elliptic partial differential operators in three spatial dimensions.The newly constructed element is proved to be convergent for a model biharmonic equation.     

粘性不可压缩对流占优Oseen方程的非协调子格稳定化有限元法分析

<正>1引言本文考虑粘性不可压缩对流占优Oseen方程,(?)(1)其中:Ω(?)R~d(d=2)为具有Lipschitz连续边界的有界开集,β∈W~(1,∞)(Ω)且▽·β=0,μ、σ为常数,f∈L~2(Ω).当采用通常的混合有限元方法(MFEM)求解时,一般会遇到以下两个困难:·为保证速度和压力数值解稳定,要求有限元空间满足inf-sup(or Babuska-Brezzi)条件.·当对流占优,即0μ《||β||_(L~∞(Ω))时,数值解会产生伪振荡.     

Numerical simulation of the wave equation with discontinuous coefficients by nonconforming finite elements

The goal of this article is to apply the mortar finite element method to the numerical simulation of (electromagnetic and/or acoustic) waves propagating in an inhomogeneous support. This approach allows us to use meshes well adapted to the local physical parameters of the media without any conformity constraints. A complete mathematical study is supplied providing the expected optimal convergence rate. Numerical performances of such a technique, as well as its advantages, are also discussed. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 637–656, 1999     

High accuracy nonconforming finite elements for fourth order problems

The approach of nonconforming finite element method admits users to solve the partial differential equations with lower complexity,but the accuracy is usually low.In this paper,we present a family of highaccuracy nonconforming finite element methods for fourth order problems in arbitrary dimensions.The finite element methods are given in a unified way with respect to the dimension.This is an effort to reveal the balance between the accuracy and the complexity of finite element methods.     

二阶双曲方程一个新的非协调混合元超收敛性分析

研究了一类二阶双曲型方程在新混合元格式下的非协调混合有限元方法.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,运用高精度分析和对时间t的导数转移技巧,借助于插值后处理技术,分别导出了关于原始变量u的H~1-模和通量=-▽u在L~2-模下的O(h~2)阶超逼近性质和整体超收敛结果.进一步,给出了一些数值算例验证了理论分析的正确性.     

Error estimates using the cell discretization method for second-order hyperbolic equations

The cell discretization algorithm provides approximate solutions to second-order hyperbolic equations with coefficients independent of time. We obtain error estimates that show general convergence for homogeneous problems using semi-discrete approximations. A polynomial implementation of the algorithm is described that is a nonconforming extension of the finite element method that can also produce the continuous approximations of an h–p finite element method. Numerical tests are made that confirm the theoretical estimates. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 531–548, 1997     

抛物型方程一个新的非协调混合元超收敛性分析及外推

本文研究了抛物型方程在新混合元格式下的非协调混合有限元方法. 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,运用高精度分析和对时间t的导数转移技巧,借助于插值后处理技术,分别导出了关于原始变量u的H¹-模和通量p=▽u在L²-模下的O（h²）阶超逼近性质和整体超收敛. 进一步,通过构造合适的辅助问题,运用Richardson 外推格式,得到了具有更高精度O（h³）阶的外推结果. 最后,给出了一些数值结果验证了理论分析的正确性.     

精确有限元法

本文提出构造有限单元的新方法——精确有限元法.它可以求解在任意边界条件下任意变系数正定或非正定偏微分方程。文中给出它的收敛性证明和计算偏微分方程的一般格式。用精确元法所得到的单元是一个非协调元,单元之间的相容条件容易处理.与相同自由度普通有限元相比,由精确元法所得到的解的高阶导数具有较高的收敛精度.文末给出数值算例,所得到的结果均收敛于精确解,并有较好的数值精度.     

New mixed finite elements for plane elasticity and Stokes equations

We consider mixed finite elements for the plane elasticity system and the Stokes equation. For the unmodified Hellinger-Reissner formulation of elasticity in which the stress and displacement fields are the primary unknowns, we derive two new nonconforming mixed finite elements of triangle type. Both elements use piecewise rigid motions to approximate the displacement and piecewise polynomial functions to approximate the stress, where no vertex degrees of freedom are involved. The two stress finite element ...     

Stokes型积分-微分方程的Crouzeix-Raviart型非协调三角形各向异性有限元方法

在半离散格式下.研究了Stokes型积分一微分方程的Crouzeix-Raviart型非协调三角形各向异性有限元方法,在不需要传统Ritz-Volterra投影下,通过辅助空间等新的技巧得到了与传统有限元方法相同的误差估计.     

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.     

一类偏积分微分方程二阶差分全离散格式

本给出了数值求解一类偏积分微分方程的二阶全离散差分格式．采用了Crank-Nicolson格式；积分项的离散利用了Lubieh的二阶卷积积分公式；给出了稳定性的证明，误差估计及收敛性的结果．     

A first-second order splitting method for a third-order partial differential equation

A splitting of a third order partial differential equation into a first-order and a second-order one is proposed as the basis for a mixed finite element method to approximate its solution. A time-continuous numerical method is described and error estimates for its solution are demonstrated. Finally, a full discretization is described based on backward Euler finite differences in time, and error estimates for the resulting approximation are established. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 89–96, 1998     

拟线性抛物型积分微分方程的一个新最低阶混合元格式

对一类拟线性抛物型积分微分方程构造了一个新的最低阶三角形协调混合元格式,并直接利用单元插值的性质,给出了相应的收敛性分析和H~1-模及L~2-模意义下的最优误差估计.     

Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is approximated through the spectral projection of the covariance operator, which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises, the well-posedness of the SPDE is established under certain covariance operator-dependent conditions. These SPDEs with projected noises are then numerically approximated with the finite element method. A general error estimate framework is established for the finite element approximations. Based on this framework, optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained. It is shown that with the proposed approach, convergence order of white noise driven SPDEs is improved by half for one-dimensional problems, and by an infinitesimal factor for higher-dimensional problems.