A LOCKING-FREE ANISOTROPIC NONCONFORMING FINITE ELEMENT FOR PLANAR LINEAR ELASTICITY PROBLEM

The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value prob-lem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L2-norms are independent of the Lame parameterλ.     

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.     

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.     

A new cubic nonconforming finite element on rectangles

A new nonconforming rectangle element with cubic convergence for the energy norm is introduced. The degrees of freedom (DOFs) are defined by the 12 values at the three Gauss points on each of the four edges. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are 11. The nonconforming element consists of . We count the corresponding dimension for Dirichlet and Neumann boundary value problems of second‐order elliptic problems. We also present the optimal error estimates in both broken energy and norms. Finally, numerical examples match our theoretical results very well. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 691–705, 2015     

A new quadratic nonconforming finite element on rectangles

A new quadratic nonconforming finite element on rectangles (or parallelograms) is introduced. The nonconforming element consists of P₂ ⊕ Span{x²y,xy²} on a rectangle and eight degrees of freedom. Our element is essentially of seven degrees of freedom since the degree of freedom associated with the integration on rectangle is essentially of bubble‐function nature. Global basis functions are constructed for both Dirichlet and Neumann type of problems; accordingly the corresponding dimensions are counted. The local and global interpolation operators are defined. Error estimates of optimal order are derived in both broken energy and L²(Ω) norms for second‐order of elliptic problems. Brief numerical results are also shown to confirm the optimality of the presented quadratic nonconforming element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A new a priori error estimate of nonconforming finite element methods

This paper is devoted to a new error analysis of nonconforming finite element methods.Compared with the classic error analysis in literature,only weak continuity,the F-E-M-Test for nonconforming finite element spaces,and basic Hm regularity for exact solutions of 2m-th order elliptic problems under consideration are assumed.The analysis is motivated by ideas from a posteriori error estimates and projection average operators.One main ingredient is a novel decomposition for some key average terms on(n.1)-dimensional faces by introducing a piecewise constant projection,which defines the generalization to more general nonconforming finite elements of the results in literature.The analysis and results herein are conjectured to apply for all nonconforming finite elements in literature.     

Robust a posteriori error estimation for the nonconforming Fortin-Soulie finite element approximation

We obtain a computable a posteriori error bound on the broken energy norm of the error in the Fortin-Soulie finite element approximation of a linear second order elliptic problem with variable permeability. This bound is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms. The estimator is completely free of unknown constants and provides a guaranteed numerical bound on the error.

     

Second-order locking-free nonconforming elements for planar linear elasticity

In this paper, we present two nonconforming finite elements for the pure displacement planar elasticity problem. Both of them are locking-free and have two order of convergence. Some numerical results attest the validity of our theoretical analysis.     

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.     

A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges

The solution of the Stokes problem in three-dimensional domainswith edges has anisotropic singular behaviour which is treatednumerically by using anisotropic finite element meshes. Thevelocity is approximated by Crouzeix–Raviart (nonconformingP₁ ) elements and the pressure by piecewise constants. Thismethod is stable for general meshes (without minimal or maximalangle condition). Denoting by N_e the number of elements in themesh, the interpolation and consistency errors are of the optimalorder h N_e^–1/3 which is proved for tensor product meshes.As a by-product, we analyse also nonconforming prismatic elementswith P₁ [oplus ] span {x₃²} as the local space for the velocitywhere x₃ is the direction of the edge.     

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.     

Morley type non‐C0 nonconforming rectangular plate finite elements on anisotropic meshes

In this article, two Morley type non‐C⁰ nonconforming rectangular finite elements are discussed to numerically solve the fourth order plate bending problem under anisotropic meshes. The optimal anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches. Some numerical tests are given to confirm the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A new rotated nonconforming pyramid element

In this paper, a new nonparametric nonconforming pyramid finite element is introduced. This element takes the five face mean values as the degrees of the freedom and the finite element space is a subspace of P₂. Different from the other nonparametric elements, the basis functions of this new element can be expressed explicitly without solving linear systems locally, which can be achieved by introducing a new reference pyramid. To evaluate the integration, a class of new quadrature formulae with only two/three equally weighted points on pyramid are constructed. We present the error estimation in the presence of quadrature formulae. Numerical results are shown to confirm the optimality of the convergence order for the second order elliptic problems.     

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.     

A low order anisotropic nonconforming characteristic finite element method for a convection-dominated transport problem

A low order anisotropic nonconforming rectangular finite element method for the convection-diffusion problem with a modified characteristic finite element scheme is studied in this paper. The O(h²) order error estimate in L²-norm with respect to the space, one order higher than the expanded characteristic-mixed finite element scheme with order O(h), and the same as the conforming case for a modified characteristic finite element scheme under regular meshes, is obtained by use of some distinct properties of the interpolation operator and the mean value technique, instead of the so-called elliptic projection, which is an indispensable tool in the convergence analysis of the previous literature. Lastly, some numerical results of the element are provided to verify our theoretical analysis.     

A mixed nonconforming finite element for linear elasticity

This article considers a mixed finite element method for linear elasticity. It is based on a modified mixed formulation that enforces the continuity of the stress weakly by adding a jump term of the approximated stress on interior edges. The symmetric stress are approximated by nonconforming linear elements and the displacement by piecewise constants. We establish ??(h) error bound in the (broken) L² norm for the divergence of the stress and ??(h) error bound in the L² norm for both the displacement and the stress tensor. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Projection stabilized nonconforming finite element methods for the stokes problem

In this article we study a projection‐stabilized nonconforming finite element discretization of the Stokes problem. We present a priori error analysis and give a recovery‐based a posteriori error estimator for the considered problem. Numerical results illustrate the theoretical performance of the error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 218–240, 2017     

A posterior error analysis for the nonconforming discretization of Stokes eigenvalue problem

In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.     

Reliable a posteriori error control for nonconforming finite element approximation of Stokes flow

We derive computable a posteriori error estimates for the lowest order nonconforming Crouzeix-Raviart element applied to the approximation of incompressible Stokes flow. The estimator provides an explicit upper bound that is free of any unknown constants, provided that a reasonable lower bound for the inf-sup constant of the underlying problem is available. In addition, it is shown that the estimator provides an equivalent lower bound on the error up to a generic constant.

     

Composite penalty method of a low order anisotropic nonconforming finite element for the Stokes problem

Composite penalty method of a low order anisotropic nonconforming quadrilateral finite element for the Stokes problem is presented. This method with a large penalty parameter can achieve the same accuracy as the stand method with a small penalty parameter and the convergence rate of this method is two times as that of the standard method under the condition of the same order penalty parameter. The superconvergence for velocity is established as well. The results of this paper are also valid to the most of the known nonconforming finite element methods.