A High Accuracy Post-processing Algorithm for the Eigenvalues of Elliptic Operators

In a very recent paper (Hu et al., The lower bounds for eigenvalues of elliptic operators by nonconforming finite element methods, Preprint, 2010), we prove that the eigenvalues by the nonconforming finite element methods are smaller than the exact ones for the elliptic operators. It is well-known that the conforming finite element methods produce the eigenvalues above to the exact ones. In this paper, we combine these two aspects and derive a new post-processing algorithm to approximate the eigenvalues of elliptic operators. We implement this algorithm and find that it actually yields very high accuracy approximation on very coarser mesh. The numerical results demonstrate that the high accuracy herein is of two fold: the much higher accuracy approximation on the very coarser mesh and the much higher convergence rate than a single lower/upper bound approximation. Moreover, we propose some acceleration technique for the algorithm of the discrete eigenvalue problem based on the solution of the discrete eigenvalue problem which yields the upper bound of the eigenvalue. With this acceleration technique we only need several iterations (two iterations in our example) to find the numerical solution of the discrete eigenvalue problem which produces the lower bound of the eigenvalue. Therefore we only need to solve essentially one discrete eigenvalue problem.     

The Lower/Upper Bound Property of Approximate Eigenvalues by Nonconforming Finite Element Methods for Elliptic Operators

This paper is a complement of the work (Hu et al. in arXiv:1112.1145v1[math.NA], 2011), where a general theory is proposed to analyze the lower bound property of discrete eigenvalues of elliptic operators by nonconforming finite element methods. One main purpose of this paper is to propose a novel approach to analyze the lower bound property of discrete eigenvalues produced by the Crouzeix–Raviart element when exact eigenfunctions are smooth. In particular, under some conditions on the triangular mesh, it is proved that the Crouzeix–Raviart element method of the Laplace operator yields eigenvalues below exact ones. Such a theoretical result explains most of numerical results in the literature and also partially answers the problem of Boffi (Acta Numerica 1–120, 2010). This approach can be applied to the Crouzeix–Raviart element of the Stokes eigenvalue problem and the Morley element of the buckling eigenvalue problem of a plate. As a second main purpose, a new identity of the error of eigenvalues is introduced to study the upper bound property of eigenvalues by nonconforming finite element methods, which is successfully used to explain why eigenvalues produced by the rotated $Q_1$ element of second order elliptic operators (when eigenfunctions are smooth), the Adini element (when eigenfunctions are singular) and the new Zienkiewicz-type element of fourth order elliptic operators, are above exact ones.     

A Penalized Crouzeix–Raviart Element Method for Second Order Elliptic Eigenvalue Problems

In this paper we propose a penalized Crouzeix–Raviart element method for eigenvalue problems of second order elliptic operators. The key idea is to add a penalty term to tune the local approximation property and the global continuity property of discrete eigenfunctions. The feature of this method is that by adjusting the penalty parameter, some of the resulted discrete eigenvalues are upper bounds of exact ones, and the others are lower bounds, and consequently a large portion of them can be reliable and approximate eigenvalues with high accuracy. Furthermore, we design an algorithm to select a penalty parameter which meets the condition. Finally we provide numerical tests to demonstrate the performance of the proposed method.     

Nonconforming immersed finite element spaces for elliptic interface problems

In this paper, we use a unified framework introduced in Chen and Zou (1998) to study two nonconforming immersed finite element (IFE) spaces with integral-value degrees of freedom. The shape functions on interface elements are piecewise polynomials defined on sub-elements separated either by the actual interface or its line approximation. In this unified framework, we use the invertibility of the well known Sherman–Morison systems to prove the existence and uniqueness of IFE shape functions on each interface element in either a rectangular or triangular mesh. Furthermore, we develop a multi-edge expansion for piecewise functions and a group of identities for nonconforming IFE functions which enable us to show the optimal approximation capability of these IFE spaces.     

A Numerical Method to Verify the Elliptic Eigenvalue Problems Including a Uniqueness Property

We propose a numerical method to enclose the eigenvalues and eigenfunctions of second-order elliptic operators with local uniqueness. We numerically construct a set containing eigenpairs which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space by using a finite element approximation and constructive error estimates. We then prove the local uniqueness separately of eigenvalues and eigenfunctions. This local uniqueness assures the simplicity of the eigenvalue. Numerical examples are presented. Received: November 2, 1998; revised June 5, 1999     

非光滑区域上椭圆型特征值问题的间断有限元方法应用

本文针对非光滑区域上椭圆特征值特征值问题利用间断有限元方法（DG）近似．利用大量的数值算例发现,DG方法对非光滑区域（凹角,裂缝等问题）上Laplace特征值问题的近似比协调有限元、非协调元（如C—R元）,甚至比有限元校正格式有着更好的效果．     

A Class of Domain Decomposition Preconditioners for <Emphasis Type="Italic">hp</Emphasis>-Discontinuous Galerkin Finite Element Methods

In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented.     

An optimal order numerical quadrature approximation of a planar isoparametric eigenvalue problem on triangular finite element meshes

Abstract This paper deals with a finite element numerical quadrature method. It is applied for a class of second-order self-adjoint elliptic operators defined on a bounded domain in the plane. Isoparametric finite element transformations and triangular Lagrange finite elements are used.We establish the rate of convergence for approximate eigenvalues and eigenfunctions of second-order elliptic eigenvalue problems, obtained by a numerical quadrature finite element approximation. Thus the relationship between possible quadrature formulas and the optimal and almost optimal precision of the method is established. The emphasis of the paper is on the error analysis of the approximate eigenpairs. Numerical results confirming the theory are presented.     

On a posteriori estimates of inverse operators for linear parabolic initial-boundary value problems

We present numerically verified a posteriori estimates of the norms of inverse operators for linear parabolic differential equations. In case that the corresponding elliptic operator is not coercive, existing methods for a priori estimates of the inverse operators are not accurate and, usually, exponentially increase in time variable. We propose a new technique for obtaining the estimates of the inverse operator by using the finite dimensional approximation and error estimates. It enables us to obtain very sharp bounds compared with a priori estimates. We will give some numerical examples which confirm the actual effectiveness of our method.     

A posteriori error analysis of nonconforming finite-element discretization for the Stokes equations

In this paper, we extend the unifying theory for a posteriori error analysis of the nonconforming finite-element methods to the Stokes problems. We present explicit residual-based computable error indicators, we prove its reliability and efficiency based on two assumptions concerning both the weak continuity and the weak orthogonality of the nonconforming finite-element spaces, respectively, and we apply the unified framework to various nonconforming finite elements from the literature.     

A discrete de Rham complex with enhanced smoothness

Abstract Discrete de Rham complexes are fundamental tools in the construction of stable elements for some finite element methods. The purpose of this paper is to discuss a new discrete de Rham complex in three space dimensions, where the finite element spaces have extra smoothness compared to the standard requirements. The motivation for this construction is to produce discretizations which have uniform stability properties for certain families of singular perturbation problem. In particular, we show how the spaces constructed here lead to discretizations of Stokes type systems which have uniform convergence properties as the Stokes flow approaches a Darcy flow. Keywords: Discrete exact sequences, nonconforming finite elements, Darcy–Stokes flow, uniform error estimates. Mathematics Subject Classification (1991): Primary 65N12, 65N15, 65N30     

Coupling finite elements and auxiliary sources

We consider second-order scalar elliptic boundary value problems on unbounded domains, which model, for instance, electrostatic fields. We propose a discretization that relies on a Trefftz approximation by multipole auxiliary sources in some parts of the domain and on standard mesh-based primal Lagrangian finite elements in other parts. Several approaches are developed and, based on variational saddle point theory, rigorously analyzed to couple both discretizations across the common interface:1. Least-squares-based coupling using techniques from PDE-constrained optimization.2. Coupling through Dirichlet-to-Neumann operators.3. Three-field variational formulation in the spirit of mortar finite element methods.We compare these approaches in a series of numerical experiments.     

Higher order dual Lagrange multiplier spaces¶for mortar finite element discretizations

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements. In particular, we focus on dual Lagrange multiplier spaces. These non-standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We construct locally supported and continuous dual basis functions for quadratic finite elements starting from the discontinuous quadratic dual basis functions for the Lagrange multiplier space. In particular, we compare different dual Lagrange multiplier spaces and piecewise linear and quadratic finite elements. The optimality of the associated mortar method is shown. Numerical results illustrate the performance of our approach. Received: July 2002 / Accepted: November 2002     

Superconvergence and Extrapolation Analysis of a Nonconforming Mixed Finite Element Approximation for Time-Harmonic Maxwell’s Equations

In this paper, a nonconforming mixed finite element approximating to the three-dimensional time-harmonic Maxwell’s equations is presented. On a uniform rectangular prism mesh, superclose property is achieved for electric field E and magnetic filed H with the boundary condition E×n=0 by means of the asymptotic expansion. Applying postprocessing operators, a superconvergence result is stated for the discretization error of the postprocessed discrete solution to the solution itself. To our best knowledge, this is the first global superconvergence analysis of nonconforming mixed finite elements for the Maxwell’s equations. Furthermore, the approximation accuracy will be improved by extrapolation method.     

Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

In this paper, we will introduce composite finite elements for solving elliptic boundary value problems with discontinuous coefficients. The focus is on problems where the geometry of the interfaces between the smooth regions of the coefficients is very complicated. On the other hand, efficient numerical methods such as, e.g., multigrid methods, wavelets, extrapolation, are based on a multi-scale discretization of the problem. In standard finite element methods, the grids have to resolve the structure of the discontinuous coefficients. Thus, straightforward coarse scale discretizations of problems with complicated coefficient jumps are not obvious. In this paper, we define composite finite elements for problems with discontinuous coefficients. These finite elements allow the coarsening of finite element spaces independently of the structure of the discontinuous coefficients. Thus, the multigrid method can be applied to solve the linear system on the fine scale. We focus on the construction of the composite finite elements and the efficient, hierarchical realization of the intergrid transfer operators. Finally, we present some numerical results for the multigrid method based on the composite finite elements (CFE–MG).     

Coupling of Nonconforming Finite Elements and Boundary Elements I: A Priori Estimates

Nonconforming finite element methods are sometimes considered as a variational crime and so we may regard its coupling with boundary element methods. In this paper, the symmetric coupling of nonconforming finite elements and boundary elements is established and a priori error estimates are shown. The coupling involves a further continuous layer on the interface in order to separate the nonconformity in the domain from its boundary data which are required to be continuous. Numerical examples prove the new scheme useful in practice. A posteriori error control and adaptive algorithms will be studied in the forthcoming Part II. Received: November 26, 1997; revised February 10, 1999     

A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator

This paper is the first in a series of two in which we discuss some theoretical and practical aspects of a feedback finite element method for solving systems of linear second-order elliptic partial differential equations (with particular interest in classical linear elasticity). In this first part we introduce some nonstandard finite element spaces, which, though based on the usual square bilinear elements, permit local mesh refinement. The algebraic structure of these spaces and their approximation properties are analyzed. An “equivalent estimator” for the H¹ finite element error is developed. In the second paper we shall discuss the asymptotic properties of the estimator and computational experience.     

On boundary-hybrid finite element methods for the Laplace equation

About two decades ago, I. Babu ka, J.T. Oden and J.K. Lee introduced finite element methods that calculate the normal derivative of the solution along the mesh interfaces and recover the solution via local Neumann problems. These methods for the treatment of the homogeneous Laplace equation were called ‘boundary-hybrid methods’. The approach was revisited in [12] for general symmetric and positive definite elliptic equations with homogeneous boundary conditions. The new approximation is nonconforming and lends itself well for an a posteriori error estimator for conforming finite element approximations. Numerical tests presented in [12] corroborated that the error estimates are accurate and cheap for conforming approximations. This paper provides the iterative solution methods and Galerkin discretization methods on which the numerical approximations in [12] were based.     

A new local stabilized nonconforming finite element method for the Stokes equations

In this paper, we propose and study a new local stabilized nonconforming finite method based on two local Gauss integrations for the two-dimensional Stokes equations. The nonconforming method uses the lowest equal-order pair of mixed finite elements (i.e., NCP ₁–P ₁). After a stability condition is shown for this stabilized method, its optimal-order error estimates are obtained. In addition, numerical experiments to confirm the theoretical results are presented. Compared with some classical, closely related mixed finite element pairs, the results of the present NCP ₁–P ₁ mixed finite element pair show its better performance than others.     

A fixed-point approximation method for some noncoercive variational problems

This paper deals with the finite-element approximation of some variational problems, namely, linear elliptic boundary value problems, variational inequalities, and quasi-variational inequalities with noncoercive operators. To prove optimal L^∞-error estimates, we introduce a simple and direct argument combining continuous piecewise linear finite elements with the Banach fixedpoint theorem