混合有限元非嵌套网格型预处理法

1 引言 众所周知,二阶椭圆型问题混合有限元离散以后的矩阵是不定的,所以对混合法很难形成一种有效的区域分解法,在文[9]、[10]、[11]中提出了一些混合有限元方法的区域分解法,但在实际计算中有很多局限性。最近Chen对混合有限元法提出一种全新的解释并把它应用到多重网格法中,他的基本思想是混合有限元离散的代数系统实际上等价于某个非协调有限元离散的代数系统,这样可把一个不定问题转化为一个正定问题,本文将基于这种思想考虑混合有限元的区域分解法。 若按传统的Dryia-widlund两水平加性Schwarz方法,要求两层网格间具有嵌套关系,这样在应用中将带来很大的不便。本文将不要求粗网格嵌入细网格中,减少两层网格间的     

Navier—Stokes方程的变网格非协调有限元法

本文通过所谓的速度－压力型公式讨论了Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程的变网格非协调有限元逼近，得到了在模意义下的速度，压力误差估计，且在一定条件下，某些误差估计能达到最优。     

一类六参数非协调任意凸四边形单元

本文构造了一类六参数非协调四边形单元,证明由此产生的有限元对任意四边形网格通过Irons分片检查,其收敛效果同Wilson元相当且形状函数的选择不依赖于单元本身。类Wilson元及改进的Wilson任意四边形单元是其中的特例。     

任意区域的网格自动生成和任意高阶有限元方法

本文首先简要介绍非拟合网格有限元方法求解复杂区域上椭圆问题的发展现状.然后结合最近本文作者发展的非拟合网格有限元方法,针对二阶椭圆方程提出一种任意光滑区域上的任意高阶协调有限元方法.本文在带悬点的Cartesian网格上自动生成诱导网格,在诱导网格上构造协调的高阶有限元空间,采用Nitsche技术处理Dirichlet边界条件,并证明方法的适定性和hp先验误差估计.数值算例验证了本文的理论结果.     

拟协调元的精度分析

利用双参数有限元的框架，证明利用拟协调元方法构造的非协调三角形板元都具有一个非常特殊的性质。即相容误差比插值误差高一阶。这是常规有限元和一般非协调元所不具备的。     

非匹配网格上Stokes-Darcy 问题非协调元离散的两水平和多水平方法

本文讨论了非匹配网格上Stokes-Darcy 问题的两种低阶非协调元方法, 给出了误差估计, 对耦合的非协调元离散问题, 通过粗网格求得的界面条件, 我们提出了一个解耦的两水平算法. 并且我们将两水平方法推广到多水平情形, 其只需在一个很粗的网格上解一耦合问题, 然后在逐步加细的网格上求解解耦的问题, 理论分析和数值试验都说明方法的高效性.     

一类新的六参数非协调四边形单元

利用分析ｓｐｅｃｈｔ元的技巧，构造了一类新的非协调四边形单元，并证明由此产生的有限元对任意四边形网格收敛且效果同Ｗｉｌｓｏｎ元．ＱＰ６元是其中的特例     

电报方程H~1-Galerkin非协调混合有限元分析

主要研究一类电报方程的H~1-Galerkin非协调混合有限元方法,在任意四边形网格剖分下,其逼近空间分别取为类Wilson元与双线性Q_1元,在不需要满足LBB相容性条件及不采用传统的Ritz投影的情况下,得到了与常规有限元方法相同的L~2-模和H~1-模的误差估计,进一步拓展了H~1-Galerkin混合有限元和类Wilson元的应用范围.     

非协调元特征值渐近下界

利用有限元收敛速度下界的结果获得某些非协调元方法新的Aubin-Nitsche估计形式,然后再结合非协调元特征值的展开式获得不需要额外条件下非协调元特征值渐近下界的结果.     

有限元空间的嵌入性质和紧致性

本文将Sobolev嵌入定理和Rellich-Kondrachov紧致定理推广到多套函数有限元空间.特殊地,在非协调元,杂交元和拟协调元空间等情形建立了这两个定理.     

On the convergence of nonnested multigrid methods with nested spaces on coarse grids

Multigrid methods for discretized partial differential problems using nonnested conforming and nonconforming finite elements are here defined in the general setting. The coarse‐grid corrections of these multigrid methods make use of different finite element spaces from those on the finest grid. In general, the finite element spaces on the finest grid are nonnested, while the spaces are nested on the coarse grids. An abstract convergence theory is developed for these multigrid methods for differential problems without full elliptic regularity. This theory applies to multigrid methods of nonnested conforming and nonconforming finite elements with the coarse‐grid corrections established on nested conforming finite element spaces. Uniform convergence rates (independent of the number of grid levels) are obtained for both the V and W‐cycle methods with one smoothing on all coarse grids and with a sufficiently large number of smoothings solely on the finest grid. In some cases, these uniform rates are attained even with one smoothing on all grids. The present theory also applies to multigrid methods for discretized partial differential problems using mixed finite element methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 265–284, 2000     

非嵌套网格上的Morley元两水平加性Schwarz方法

1．引言设0是RZ中的有界多边形区域，其边界为Rfl．考虑下面的重调和Dirichlet问题：（1．1）的变分形式为：求。EHI（fi）使得对？／EL‘（m，问题（1．幻的唯一可解性可由冯（m上的M线性型的强制性和连续性以及La。Mlgram定理得出（of［4］）．令人一｛丸）是n的一个三角剖分，并且满足最小角条件，其中h是它的网格参数．设Vh为Money元空间［41．问题（1．2）的有限元离散问题为：求。eVh使得当有限元参数人很小时，这个方程组很大，而且矩阵A的条件数变得非常大，直接求解，存贮量及计算量都很大．如果B可逆，则方程组（1．4）等…     

MULTIGRID METHODS FOR THE GENERALIZED STOKES EQUATIONS BASED ON MIXED FINITE ELEMENT METHODS

1. IntroductionIn this paper, we consider the fOllowing generalized stationary Stokes equations:where fl is a bounded convex domain in R', u represents the velocity of fluid, p its pressure; Fand G are external fOrce and source terms. Note that the source…     

A generalized BPX multigrid framework covering nonnested V-cycle methods

More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or noninherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and it produces a variable V-cycle, or nonuniform convergence rate V-cycle methods, or other nonoptimal results in analysis thus far.

This paper completes a long-time effort in extending the BPX multigrid framework so that it truly covers the nonnested V-cycle. We will apply the extended BPX framework to the analysis of many V-cycle nonnested multigrid methods. Some of them were proven previously only for two-level and W-cycle iterations. Some numerical results are presented to support the theoretical analysis of this paper.

     

General presentations of algebras

For any finite dimensional basic associative algebra, we study the presentation spaces and their relation with the representation spaces. We prove two theorems about a general presentation, one on its subrepresentations and the other on its canonical decomposition. As a special case, we consider rigid presentations. We show how to complete a rigid presentation and study the number of nonisomorphic direct summands and different complements. Based on that, we construct a simplicial complex governing the canonical decompositions of rigid presentations and provide some examples.     

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.     

Nonconforming finite element methods on quadrilateral meshes

It is well known that it is comparatively difcult to design nonconforming fnite elements on quadrilateral meshes by using Gauss-Legendre points on each edge of triangulations.One reason lies in that these degrees of freedom associated with these Gauss-Legendre points are not all linearly independent for usual expected polynomial spaces,which explains why only several lower order nonconforming quadrilateral fnite elements can be found in literature.The present paper proposes two families of nonconforming fnite elements of any odd order and one family of nonconforming fnite elements of any even order on quadrilateral meshes.Degrees of freedom are given for these elements,which are proved to be well-defned for their corresponding shape function spaces in a unifying way.These elements generalize three lower order nonconforming fnite elements on quadrilaterals to any order.In addition,these nonconforming fnite element spaces are shown to be full spaces which is somehow not discussed for nonconforming fnite elements in literature before.     

On quasilinear spaces of convex bodies and intervals

Algebraic systems abstracting properties of convex bodies and intervals, with respect to addition and multiplication by scalars, known as quasilinear spaces, are studied axiomatically. We discuss special quasilinear spaces with group structure called quasivector spaces. We show that every quasivector space is a direct sum of a vector space and a symmetric quasivector space. A complete characterization of symmetric quasivector spaces in the finite dimensional case is given, which permits to reduce computation in quasilinear spaces to computation in familiar vector spaces.     

A note on finite topologies and switching functions

Finite topologies and switching functions are investigated. We associate switching functions to families of subsets of a finite set as done for instance by Adám (Truth Functions and the Problem of their Realizations by Two-terminal Graphs [Akademiai Kiadó, Budapest, 1968]); we consider then the special case where the families are (finite) topologies. We characterize switching functions which correspond to finite topologies, we associate certain functions, formed with the aid of subfunctions, to topologies on subspaces and on quotient spaces, and we use them to prove some theorems concerning these topologies and to reconstruct (in a certain weak sense) a topological space given the (quotient) spaces obtained by identifying one fixed point with each one of the others.     

An optimal order multigrid method for biharmonic,C 1 finite element equations

Summary In this paper, we study a special multigrid method for solving large linear systems which arise from discretizing biharmonic problems by the Hsieh-Clough-Tocher,C ¹ macro finite elements or several otherC ¹ finite elements. Since the multipleC ¹ finite element spaces considered are not nested, the nodal interpolation operator is used to transfer functions between consecutive levels in the multigrid method. This method converges with the optimal computational order.