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 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 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.     

A nonconforming mixed finite element for second-order elliptic problems

In a recent work, Hiptmair [Mathematisches Institut, M9404, 1994] has constructed and analyzed a family of nonconforming mixed finite elements for second-order elliptic problems. However, his analysis does not work on the lowest order elements. In this article, we show that it is possible to construct a nonconforming mixed finite element for the lowest order case. We prove the convergence and give estimates of optimal order for this finite element. Our proof is based on the use of the properties of the so-called nonconforming bubble function to control the consistency terms introduced by the nonconforming approximation. We further establish an equivalence between this mixed finite element and the nonconforming piecewise quadratic finite element of Fortin and Soulie [J. Numer. Methods Eng., 19, 505–520, 1983]. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 445–457, 1997     

A locking-free anisotropic nonconforming rectangular finite element approximation for the planar elasticity problem

This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.     

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 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.     

Three‐dimensional quadratic nonconforming brick element

A new nonconforming brick element with quadratic convergence for the energy norm is introduced. The nonconforming element consists of on a cube [?1,1]³, and 14 degree of freedom (DOF). Two types of DOF are introduced. One consists of the value at the eight vertices and six face‐centroids and the other consists of the value at the eight vertices and the integration value of six faces. Error estimates of optimal order are derived in both broken energy and norms for second‐order elliptic problems. If a genuine hexahedron, which is not a parallelepiped, is included in the partition, the proposed element is also convergent, but with a lower order. Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 158–174, 2014     

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.

     

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     

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.

     

A coupling of nonconforming and mixed finite element methods for Biot's consolidation model

In this article, we develop a nonconforming mixed finite element method to solve Biot's consolidation model. In particular, this work has been motivated to overcome nonphysical oscillations in the pressure variable, which is known as locking in poroelasticity. The method is based on a coupling of a nonconforming finite element method for the displacement of the solid phase with a standard mixed finite element method for the pressure and velocity of the fluid phase. The discrete Korn's inequality has been achieved by adding a jump term to the discrete variational formulation. We prove a rigorous proof of a‐priori error estimates for both semidiscrete and fully‐discrete schemes. Optimal error estimates have been derived. In particular, optimality in the pressure, measured in different norms, has been proved for both cases when the constrained specific storage coefficient c₀ is strictly positive and when c₀ is nonnegative. Numerical results illustrate the accuracy of the method and also show the effectiveness of the method to overcome the nonphysical pressure oscillations. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations

A new numerical method for computing the divergence-free part of the solution of the time-harmonic Maxwell equations is studied in this paper. It is based on a discretization that uses the locally divergence-free Crouzeix-Raviart nonconforming vector fields and includes a consistency term involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive ) in both the energy norm and the norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.     

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 robust nonconforming -element

Finite element methods for some elliptic fourth order singular perturbation problems are discussed. We show that if such problems are discretized by the nonconforming Morley method, in a regime close to second order elliptic equations, then the error deteriorates. In fact, a counterexample is given to show that the Morley method diverges for the reduced second order equation. As an alternative to the Morley element we propose to use a nonconforming -element which is -conforming. We show that the new finite element method converges in the energy norm uniformly in the perturbation parameter.

     

Quadratic finite element approximation of the Signorini problem

Applying high order finite elements to unilateral contact variational inequalities may provide more accurate computed solutions, compared with linear finite elements. Up to now, there was no significant progress in the mathematical study of their performances. The main question is involved with the modeling of the nonpenetration Signorini condition on the discrete solution along the contact region. In this work we describe two nonconforming quadratic finite element approximations of the Poisson-Signorini problem, responding to the crucial practical concern of easy implementation, and we present the numerical analysis of their efficiency. By means of Falk's Lemma we prove optimal and quasi-optimal convergence rates according to the regularity of the exact solution.

     

双曲型方程的非协调变网格有限元方法

采用变网格的思想讨论了双曲型方程在各向异性网格下的Crouzeix-Raviart型非协调有限元逼近.在不需要引入传统分析中Riesz投影的情况下,得到了相应最优误差估计.     

Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations

We study a discontinuous Galerkin finite element method (DGFEM) for the Stokes equations with a weak stabilization of the viscous term. We prove that, as the stabilization parameter γ tends to infinity, the solution converges at speed γ^?1 to the solution of some stable and well‐known nonconforming finite element methods (NCFEM) for the Stokes equations. In addition, we show that an a posteriori error estimator for the DGFEM‐solution based on the reconstruction of a locally conservative H(div, Ω)‐tensor tends at the same speed to a classical a posteriori error estimator for the NCFEM‐solution. These results can be used to affirm the robustness of the DGFEM‐method and also underline the close relationship between the two approaches. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

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