A simple nonconforming quadrilateral finite element

We introduce and analyze a simple nonconforming quadrilateral finite element and then we derive optimal a priori error estimates for arbitrary regular quadrilaterals. The idea of extension to some non-conforming elements for three-dimensional hexahedrons is also presented.     

A quadrilateral nonconforming finite element for linear elasticity problem

In this paper, a four-parameter quadrilateral nonconforming finite element with DSP (double set parameters) is presented. Then we discuss the quadrilateral nonconforming finite element approximation to the linear elastic equations with pure displacement boundary. The optimal convergence rate of the method is established in the broken energy and -norms, and in particular, the convergence is uniform with respect to the Lamé parameter . Also the performance of the scheme does not deteriorate as the material becomes nearly incompressible. Lastly, a numerical test is carried out, which coincides with our theoretical analysis. The research is supported by NSF of China (No. 10471133) and the project of Creative Engineering of Henan Province of China.     

Analysis of a class of nonconforming finite elements for crystalline microstructures

An analysis is given for a class of nonconforming Lagrange-type finite elements which have been successfully utilized to approximate the solution of a variational problem modeling the deformation of martensitic crystals with microstructure. These elements were first proposed and analyzed in 1992 by Rannacher and Turek for the Stokes equation. Our analysis highlights the features of these elements which make them effective for the computation of microstructure. New results for superconvergence and numerical quadrature are also given.

     

Two lower order nonconforming quadrilateral elements for the Reissner-Mindlin plate

This paper generalizes two nonconforming rectangular elements of the Reissner-Mindlin plate to the quadrilateral mesh. The first quadrilateral element uses the usual conforming bilinear element to approximate both components of the rotation, and the modified nonconforming rotated Q ₁ element enriched with the intersected term on each element to approximate the displacement, whereas the second one uses the enriched modified nonconforming rotated Q ₁ element to approximate both the rotation and the displacement. Both elements employ a more complicated shear force space to overcome the shear force locking, which will be described in detail in the introduction. We prove that both methods converge at optimal rates uniformly in the plate thickness t and the mesh distortion parameter in both the H ¹-and the L ²-norms, and consequently they are locking free. This work was supported by the National Natural Science Foundation of China (Grant No. 10601003) and National Excellent Doctoral Dissertation of China (Grant No. 200718)     

On finite element subspaces over quadrilateral and hexahedral meshes for incompressible viscous flow problems

Summary Optimal rates of convergence for the approximate solution of the stationary Stokes equations are obtained for finite element schemes which use piecewise constants to approximate the pressure and piecewise linear or piecewise bilinear or trilinear polynomials to approximate the velocity over fairly general quadrilateral or hexahedral meshes.This research was supported by NSF Grant MC80-16532     

Approximation by quadrilateral finite elements

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order in and order in is that the given space of functions on the reference element contain all polynomial functions of total degree at most . In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree . We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

     

Conforming and nonconforming harmonic finite elements

ABSTRACT

Instead of using the full polynomial space, a conforming and a nonconforming finite element methods are designed where only harmonic polynomials (a much smaller space) are employed in the computation. The conforming quadratic harmonic polynomial finite element is defined only on a special triangular grid. The nonconforming quadratic harmonic finite element is defined on general triangular grids. The optimal order of convergence is proved for both finite element methods, and confirmed by numerical computations. In addition, numerical comparisons with the standard conforming and nonconforming finite elements are presented.     

Locking-free nonconforming finite elements for three-dimensional elasticity problem

Two locking-free nonconforming finite elements are presented for three-dimensional elasticity problem with pure displacement boundary condition. Convergence rate of the elements are uniformly optimal with respect to λ. The energy norm and L² norm errors are O(h²) and O(h³), respectively. Lastly, a numerical experiment is carried out, which coincides with the theoretical analysis.     

Mixed finite elements for incompressible magneto-hydrodynamics

We present a new mixed finite element discretization for three-dimensional stationary incompressible magneto-hydrodynamics. The fluid variables are discretized by standard inf–sup stable velocity–pressure pairs and the magnetic variables by a mixed approach using Nédélec's elements of the first kind. The resulting method is shown to be quasi-optimally convergent. To cite this article: A. Schneebeli, D. Schötzau, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Finite elements for incompressible flow

We describe in this paper a finite element method for the solution of viscous incompressible flow problems which incorporates an approximate form of the incompressibility condition automatically into the finite element basis. Several examples of such finite elements are presented and applied to a simple test problem.     

Stokes方程的一个新的非协调四边形单元格式

对于Stokes方程给出了一个新的非协调四边形单元格式.新单元具有构造简单,自由度较少等优势.特别指出的是,该单元在矩形网格下,还是一个Locking-free元,可用于平面弹性问题.尽管该单元不含协调部分,其相容误差估计较困难,通过采用新的技巧和方法得到了最优误差估计.     

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.     

特征值问题协调/非协调有限元后验误差分析

本文研究对称椭圆特征值问题的有限元后验误差估计,包括协调元和非协调元,具有下列特色:(1)对协调/非协调元建立了有限元特征函数uh的误差与相应的边值问题有限元解的误差在局部能量模意义下的恒等关系式,该边值问题的右端为有限元特征值λh与uh的乘积,有限元解恰好为uh.从而边值问题有限元解在能量模意义下的局部后验误差指示子,包括残差型和重构型后验误差指示子,成为有限元特征函数在能量模意义下的局部后验误差指示子.(2)讨论了协调有限元特征函数的基于插值后处理的梯度重构型后验误差估计,对有限元特征函数的导数得到了最大模意义下的渐近准确局部后验误差指示子.     

A mixed finite element for the stokes problem using quadrilateral elements

The object of this paper is to complete the results obtained in [3] by showing that the new mixed finite element that we have constructed in [3] also works for quadrilateral elements and to compare this method with the standard finite volume method. Estimates of optimal order are derived for both the new mixed finite element and an associated finite volume method.     

A mixed finite element for the stokes problem using quadrilateral elements

The object of this paper is to complete the results obtained in [3] by showing that the new mixed finite element that we have constructed in [3] also works for quadrilateral elements and to compare this method with the standard finite volume method. Estimates of optimal order are derived for both the new mixed finite element and an associated finite volume method.     

Optimality of multilevel preconditioning for nonconforming P1 finite elements

We prove the optimality of hierarchical and BPX-type preconditioners for finite element discretizations with nonconforming P1 finite elements. Such preconditioners were proposed about 15 years ago, and until now only suboptimal estimates of their preconditioning properties have been available. The main new tool is an improved Bernstein type inequality for an associated subdivision process generated by the prolongations which allows us to give an asymptotically optimal upper bound for the spectrum of the preconditioned systems.     

New robust nonconforming finite elements of higher order

We show that existing quadrilateral nonconforming finite elements of higher order exhibit a reduction in the order of approximation if the sequence of meshes is still shape-regular but consists no longer of asymptotically affine equivalent mesh cells. We study second order nonconforming finite elements as members of a new family of higher order approaches which prevent this order reduction. We present a new approach based on the enrichment of the original polynomial space on the reference element by means of nonconforming cell bubble functions which can be removed at the end by static condensation. Optimal estimates of the approximation and consistency error are shown in the case of a Poisson problem which imply an optimal order of the discretization error. Moreover, we discuss the known nonparametric approach to prevent the order reduction in the case of higher order elements, where the basis functions are defined as polynomials on the original mesh cell. Regarding the efficient treatment of the resulting linear discrete systems, we analyze numerically the convergence of the corresponding geometrical multigrid solvers which are based on the canonical full order grid transfer operators. Based on several benchmark configurations, for scalar Poisson problems as well as for the incompressible Navier-Stokes equations (representing the desired application field of these nonconforming finite elements), we demonstrate the high numerical accuracy, flexibility and efficiency of the discussed new approaches which have been successfully implemented in the FeatFlow software (www.featflow.de). The presented results show that the proposed FEM-multigrid combinations (together with discontinuous pressure approximations) appear to be very advantageous candidates for efficient simulation tools, particularly for incompressible flow problems.     

On the error bounds of nonconforming finite elements

We prove that the error estimates of a large class of nonconforming finite elements are dominated by their approximation errors, which means that the well-known Cea's lemma is still valid for these nonconforming finite element methods. Furthermore, we derive the error estimates in both energy and L2 norms under the regularity assumption u ∈ H1+s(Ω) with any s 0. The extensions to other related problems are possible.     

The interpolation theorem for narrow quadrilateral isoparametric finite elements

Summary. The interpolation theorem for convex quadrilateral isoparametric finite elements is proved in the case when the condition is not satisfied, where is the diameter of the element and is the radius of an inscribed circle in . The interpolation error is in the -norm and in the -norm provided that the interpolated function belongs to . In the case when the long sides of the quadrilateral are parallel the constants appearing in the estimates are evaluated. Received September 1993 / Revised version received March 6, 1995