Interpolation theory of anisotropic finite elements and applications

Interpolation theory is the foundation of finite element methods.In this paper,after reviewing some existed interpolation theorems of anisotropic finite element methods,we present a new way to analyse the interpolation error of anisotropic elements based on Newton's formula of polynomial interpolation as well as its applications.     

Nonconforming rotated Q_1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q1 element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

CONVERGENCE AND SUPERCONVERGENCE ANALYSIS OF LAGRANGE RECTANGULAR ELEMENTS WITH ANY ORDER ON ARBITRARY RECTANGULAR MESHES

This paper is to study the convergence and superconvergence of rectangular finite elements under anisotropic meshes. By using of the orthogonal expansion method, an anisotropic Lagrange interpolation is presented. The family of Lagrange rectangular elements with all the possible shape function spaces are considered, which cover the Intermediate families, Tensor-product families and Serendipity families. It is shown that the anisotropic interpolation error estimates hold for any order Sobolev norm. We extend the convergence and superconvergence result of rectangular finite elements to arbitrary rectangular meshes in a unified way.     

Anisotropic weak Hardy spaces and interpolation theorems

In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces.     

二阶问题Adini元的误差估计(英文)

The main aim of this paper is to have an accurate analysis on the famous Adini's element for the second order problems under to the anisotropic meshes.We firstly show that the interpolation of Adini's element satisfy the anisotropic property.Then the optimal error estimate is obtained without the regularity assumption on the meshes.     

二阶问题Adini元的误差估计

The main aim of this paper is to have an accurate analysis on the famous Adini＇s element for the second order problems under to the anisotropic meshes. We firstly show that the interpolation of Adini＇s element satisfy the anisotropic property. Then the optimal error estimate is obtained without the regularity assumption on the meshes.     

ON THE ANISOTROPIC ACCURACY ANALYSIS OF ACMES NONCONFORMING FINITE ELEMENT

The main aim of this paper is to study the superconvergence accuracy analysis of the famous ACM＇s nonconforming finite element for biharmonic equation under anisotropic meshes. By using some novel approaches and techniques, the optimal anisotropic interpolation error and consistency error estimates are obtained. The global error is of order O（h^2）. Lastly, some numerical tests are presented to verify the theoretical analysis.     

AN ANISOTROPIC NONCONFORMING FINITE ELEMENT WITH SOME SUPERCONVERGENCE RESULTS

The main aim of this paper is to study the error estimates of a nonconforming finite element with some superconvergence results under anisotropic meshes. The anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches and techniques, respectively. Furthermore, the superclose and a superconvergence estimate on the central points of elements are also obtained without the regularity assumption and quasi-uniform assumption requirement on the meshes. Finally, a numerical test is carried out, which coincides with our theoretical analysis.     

CONVERGENCE ANALYSIS OF MORLEY ELEMENT ON ANISOTROPIC MESHES

The main aim of this paper is to study tile convergence of a nonconforming triangular plate element-Morley element under anisotropic meshes. By a novel approach, an explicit bound for the interpolation error is derived for arbitrary triangular meshes （which even need not satisfy the maximal angle condition and the coordinate system condition ）, the optimal consistency error is obtained for a family of anisotropically graded finite element meshes.     

THE GENERALIZED MAXIMUM ANGLE CONDITION FOR THE (?)_1 ISOPARAMETRIC ELEMENT

We consider the quadrilateral(?)1 isoparametric element and establish an optimal errorestimate in H~1 norm for the interpolation operator under a weaker mesh condition whichadmits anisotropic quadrilaterals and allows the quadrilateral to become a regular trianglein the sense of maximum angle condition[5,11].     

ACM元各向异性分析的Newton方法

本文将一维Lagrange插值多项式的Newton表达式推广到二维非标准的Hermite插值,给出著名板元-ACM元插值多项式的Newton表达式,由此给出ACM元对四阶和二阶椭圆问题的各向异性插值误差估计,为复杂单元的各向异性分析开辟了新的途径.     

Anisotropic interpolation error estimates via orthogonal expansions

We prove anisotropic interpolation error estimates for quadrilateral and hexahedral elements with all possible shape function spaces, which cover the intermediate families, tensor product families and serendipity families. Moreover, we show that the anisotropic interpolation error estimates hold for derivatives of any order. This goal is accomplished by investigating an interpolation defined via orthogonal expansions.     

Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q ₁ element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q ₁ element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

各向异性网格下二阶椭圆问题的一个混合元形式

提出了二阶椭圆问题的一个混合变分形式,同时证明了Rariart-Thomas元的各向异性插值性质,并给出了单元的对二阶问题的最优误差估计。     

Anisotropic error bounds of Lagrange interpolation with any order in two and three dimensions

In this paper, using the Newton’s formula of Lagrange interpolation, we present a new proof of the anisotropic error bounds for Lagrange interpolation of any order on the triangle, rectangle, tetrahedron and cube in a unified way.     

Zienkiewicz元插值的非各向异性估计

本文证明著名的用于四阶椭圆问题尤其是板弯曲问题的Zienkiewicz元的插值误差在各向异性网格下不收敛, 从而表明, 发展各向异性有限元的理论和单元构造是必要的.     

Stokes问题的各向异性平行四边形有限元逼近

1 引言 Stokes问题是标准的混合问题,速度与压力同时计算,关于该问题有限元求解的文章很多(见文献[1-5])但大多都是基于对区域的正则剖分或拟一致剖分,即要求网格剖分满足hk/pK≤C,(A)K∈Jh,其中C>0为一常数,hk,pK分别为单元K的直径及内切园直径,在实际应用问题中,由于边界层或区域的拐角处需考虑物质的各向异性特征,此时对空间区域Q的剖分不再满足正则性或拟一致条件,而需要用各向异性网格剖分,才能更贴切地描述其真实情形.