A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.     

Two-sided bounds of eigenvalues of second-and fourth-order elliptic operators

This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which give at least asymptotically lower bounds of the exact eigenvalues. We present some new numerical experiments for the plate bending problem on a rectangular domain. The main result is that if we know an estimate from below by nonconforming FEM, then by using a postprocessing procedure we can obtain two-sided bounds of the first (essential) eigenvalue. For the other eigenvalues λ_l, l = 2, 3, …, we prove and give conditions when this method is applicable. Finally, the numerical results presented and discussed in the paper illustrate the efficiency of our method.     

A remark on the optimality of adaptive finite element methods

We show that the standard assumption on the smallness of the marking parameter θ in adaptive finite element methods can be avoided for the proof of the optimality of the algorithm. To this end we propose a new technique based on comparison of the solutions of different finite element spaces obtained by different refinements of a given mesh. We consider conforming and nonconforming low-order finite elements on triangular and tetrahedral meshes.     

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     

On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1

We present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H ¹(Ω), indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. We consider a polygonal domain Ω in two or three space dimensions and its shape-regular simplicial triangulation. The nonconforming approximations of H ¹(Ω) consist of functions from H ¹ on each element such that the mean values of their traces on interelement boundaries coincide. The key idea is to extend the proof of the discrete Poincaré–Friedrichs inequalities for piecewise constant functions used in the finite volume method. The results have applications in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.     

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     

Patch‐wise local projection stabilized finite element methods for convection–diffusion–reaction problems

In this article, we develop patch‐wise local projection‐stabilized conforming and nonconforming finite element methods for the convection–diffusion–reaction problems. It is a composition of the standard Galerkin finite element method, the patch‐wise local projection stabilization, and weakly imposed Dirichlet boundary conditions on the discrete solution. In this paper, a priori error analysis is established with respect to a patch‐wise local projection norm for the conforming and the nonconforming finite element methods. The numerical experiments confirm the efficiency of the proposed stabilization technique and validate the theoretical convergence rates.     

An adaptive least squares mixed finite element method for the stress‐displacement formulation of linear elasticity

A least‐squares mixed finite element method for linear elasticity, based on a stress‐displacement formulation, is investigated in terms of computational efficiency. For the stress approximation quadratic Raviart‐Thomas elements are used and these are coupled with the quadratic nonconforming finite element spaces of Fortin and Soulie for approximating the displacement. The local evaluation of the least‐squares functional serves as an a posteriori error estimator to be used in an adaptive refinement algorithm. We present computational results for a benchmark test problem of planar elasticity including nearly incompressible material parameters in order to verify the effectiveness of our adaptive strategy. For comparison, conforming quadratic finite elements are also used for the displacement approximation showing convergence orders similar to the nonconforming case, which are, however, not independent of the Lamé parameters. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

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.     

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM

Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conforming, nonconforming, and mixed low order finite element methods in a model situation: the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.

     

Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H¹ norm best approximation error estimates for H² functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.     

Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.     

关于解椭圆型问题的两个子区域不重叠区域分解算法

关于解椭圆型问题的两个子区域不重叠区域分解算法顾金生，胡显承（清华大学）ＯＮＴＨＥＤＯＭＡＩＮＤＥＣＯＭＰＯＳＩＴＩＯＮＭＥＴＨＯＤＳＦＯＲＥＬＬＩＰＴＩＣＰＲＯＢＬＥＭＳＷＩＴＨＴＷＯＳＵＢＳＴＲＵＣＴＵＲＥＳ￥ＧｕＪｉｎ－ｓｈｅｎｇ；ＨｕＸｉａｎ...     

基于非协调元的区域分解法──强重迭情形

基于非协调元的区域分解法──强重迭情形黄建国（上海交通大学应用数学系）ＡＤＯＭＡＩＮＤＥＣＯＭＰＯＳＩＴＩＯＮＭＥＴＨＯＤＦＯＲＮＯＮＣＯＮＦＯＲＭＩＮＧＦＩＮＩＴＥＥＬＥＭＥＮＴ──ＴＨＥＣＡＳＥＯＦＳＴＲＯＮＧＯＶＥＲＬＡＰ￥ＨｕａｎｇＪｉａｎ...     

A unified framework for a posteriori error estimation for the Stokes problem

In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on $[H^1_0(\Omega )]^d$ -conforming velocity reconstruction and $\underline{\varvec{H}}(\mathrm{div},\Omega )$ -conforming, locally conservative flux (stress) reconstruction. It?gives guaranteed, fully computable global upper bounds as well as local lower bounds on the energy error. In order to apply this framework to a given numerical method, two simple conditions need to be checked. We show how to do this for various conforming and conforming stabilized finite element methods, the discontinuous Galerkin method, the Crouzeix–Raviart nonconforming finite element method, the mixed finite element method, and a general class of finite volume methods. The tools developed and used include a new simple equilibration on dual meshes and the solution of local Poisson-type Neumann problems by the mixed finite element method. Numerical experiments illustrate the theoretical developments.     

Two-level stabilized nonconforming finite element method for the Stokes equations

In this article, we present a new two-level stabilized nonconforming finite elements method for the two dimensional Stokes problem. This method is based on a local Gauss integration technique and the mixed nonconforming finite element of the NCP ₁ — P ₁ pair (nonconforming linear element for the velocity, conforming linear element for the pressure). The two-level stabilized finite element method involves solving a small stabilized Stokes problem on a coarse mesh with mesh size H and a large stabilized Stokes problem on a fine mesh size h = H/3. Numerical results are presented to show the convergence performance of this combined algorithm.     

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.     

A nonconforming combined method for solving Laplace's boundary value problems with singularities

Summary For solving Laplace's boundary value problems with singularities, a nonconforming combined approach of the Ritz-Galerkin method and the finite element method is presented. In this approach, singular functions are chosen to be admissible functions in the part of a solution domain where there exist singularities; and piecewise linear functions are chosen to be admissible functions in the rest of the solution domain. In addition, the admissible functions used here are constrained to be continuous only at the element nodes on the common boundary of both methods. This method is nonconforming; however, the nonconforming effect does not result in larger errors of numerical solutions as long as a suitable coupling strategy is used.In this paper, we will develop such an approach by using a new coupling strategy, which is described as follows: IfL+1=O(|lnh|), the average errors of numerical solutions and their generalized derivatives are stillO(h), whereh is the maximal boundary length of quasiuniform triangular elements in the finite element method, andL+1 is the total number of singular admissible functions in the Ritz-Galerkin method. The coupling relation,L+1=O(|lnh|), is significant because only a few singular functions are required for a good approximation of solutions.This material is from Chapter 5 in my Ph.D. thesis: Numerical Methods for Elliptic Boundary Value Problems with Singularities. Part I: Boundary Methods for Solving Elliptic Problems with Singularities. Part II: Nonconforming Combinations for Solving Elliptic Problems with Singularities, the Department of Mathematics and Applied Mathematics, University of Toronto, May 1986     

Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Part I: L2-error estimates

Superconvergence properties in the L₂ norm are derived for the recovered gradients of piecewise quadratic finite element approximations on triangular partitions for two-dimensional elliptic problems and systems, including the case of linear elasticity. The analysis covers problems defined on polygonal domains, where the solutions have low regularity. The effects of numerical integration are treated.     

On the relationship between finite volume and finite element methods applied to the Stokes equations

We investigate the relationship between finite volume and finite element approximations for the lower‐order elements, both conforming and nonconforming for the Stokes equations. These elements include conforming, linear velocity‐constant pressure on triangles, conforming bilinear velocity‐constant pressure on rectangles and their macro‐element versions, and nonconforming linear velocity‐constant pressure on triangles and nonconforming rotated bilinear velocity‐constant pressure on rectangles. By applying the relationship between the two methods, we obtain the convergence finite volume solutions for the Stokes equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 440–453, 2001.