Strong uniform consistency and asymptotic normality of a kernel based error density estimator in functional autoregressive models

Estimating the innovation probability density is an important issue in any regression analysis. This paper focuses on functional autoregressive models. A residual-based kernel estimator is proposed for the innovation density. Asymptotic properties of this estimator depend on the average prediction error of the functional autoregressive function. Sufficient conditions are studied to provide strong uniform consistency and asymptotic normality of the kernel density estimator.     

Efficient estimation of the error distribution in a varying coefficient regression model

It is shown that the weighted residual-based estimator of Schick, Zhu, and Du (2017) is efficient in some special cases and can be made to be efficient by adding a stochastic correction term. The efficiency is shown by deriving the efficient influence function and establishing a uniform stochastic expansion with this influence function. The correction term relies on estimators of the score function for the errors and other characteristics of the model.     

A posteriori error estimates for a discontinuous Galerkin approximation of Steklov eigenvalue problems

We derive a residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem. Moreover, we prove the reliability and efficiency of the error estimator. Numerical results are provided to verify our theoretical findings.     

An hp-Efficient Residual-Based A Posteriori Error Estimator for Maxwell's Equations

We present a residual-based a posteriori error estimator for Maxwell's equations in the electric field formulation. The error estimator is formulated in terms of the residual of the considered problem and we state upper and lower bounds in terms of the energy error of the computed solution for the estimator. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

求解椭圆边值问题惩罚形式的间断有限元方法

本文提出了一个新的求解二阶椭圆边值问题的惩罚形式间断有限元方法并给出了稳定性和收敛性分析. 特别地,本文建立了间断有限元解的基于余量的后验误差估计,给出了求解间断有限元方程的自适应算法.       

Truncated Estimator of Asymptotic Covariance Matrix in Partially Linear Models with Heteroscedastic Errors

A partially linear regression model with heteroscedastic and/or serially correlated errors is studied here. It is well known that in order to apply the semiparametric least squares estimation （SLSE） to make statistical inference a consistent estimator of the asymptotic covariance matrix is needed. The traditional residual-based estimator of the asymptotic covariance matrix is not consistent when the errors are heteroscedastic and/or serially correlated. In this paper we propose a new estimator by truncating, which is an extension of the procedure in White. This estimator is shown to be consistent when the truncating parameter converges to infinity with some rate.     

简化摩擦接触问题的对称弱超内罚间断Galerkin方法的先验和后验误差估计

本文讨论了简化摩擦接触问题的一类对称弱超内罚间断Galerkin方法.首先,在能量范数意义下得到最优先验误差估计.进一步,我们推导了一类残量型后验误差估计子,并证明了它的可靠性和有效性.     

A posteriori error estimators for the steady incompressible Navier–Stokes equations

A residual-based a posteriori error estimator for finite element discretizations of the steady incompressible Navier–Stokes equations in the primitive variable formulation is discussed. Though the estimator is similar to existing ones, an alternate derivation is presented, involving an abstract estimate that may prove of some intrinsic value. The estimator is particularized to Hood–Taylor and modified Hood–Taylor finite elements and proved to be a global upper bound (up to a positive multiplicative constant) of the true error. Numerical examples are provided to illustrate the performance of the resulting mesh adaptation process. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 561–574, 1997     

A posteriori error analysis of a quadratic finite volume method for nonlinear elliptic problems

In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results.     

Robustness in a posteriori error analysis for FEM flow models

Summary. We derive a residual-based a posteriori error estimator for a stabilized finite element discretization of certain incompressible Oseen-like equations. We focus our attention on the behaviour of the effectivity index and we carry on a numerical study of its sensitiveness to the problem and mesh parameters. We also consider a scalar reaction-convection-diffusion problem and a divergence-free projection problem in order to investigate the effects on the robustness of our a posteriori error estimator of the reaction-convection-diffusion phenomena and, separately, of the incompressibility constraint. Received March 21, 2001 / Revised version received July 30, 2001 / Published online October 17, 2001     

A posteriori error estimates for nonconforming finite element methods

Summary. Computable a posteriori error bounds for a large class of nonconforming finite element methods are provided for a model Poisson-problem in two and three space dimensions. Besides a refined residual-based a posteriori error estimate, an averaging estimator is established and an -estimate is included. The a posteriori error estimates are reliable and efficient; the proof of reliability relies on a Helmholtz decomposition. Received March 4, 1997 / Revised version received September 4, 2001 / Published online December 18, 2001     

Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problems with Dirac delta source terms

Two residual-based a posteriori error estimators of the nonconforming Crouzeix-Raviart element are derived for elliptic problems with Dirac delta source terms.One estimator is shown to be reliable and efficient,which yields global upper and lower bounds for the error in piecewise W1,p seminorm.The other one is proved to give a global upper bound of the error in Lp-norm.By taking the two estimators as refinement indicators,adaptive algorithms are suggested,which are experimentally shown to attain optimal convergence orders.     

Residual-based a posteriori error estimate for a mixed Reißner-Mindlin plate finite element method

Reliable and efficient residual-based a posteriori error estimates are established for the stabilised locking-free finite element methods for the Reissner-Mindlin plate model. The error is estimated by a computable error estimator from above and below up to multiplicative constants that do neither depend on the mesh-size nor on the plate's thickness and are uniform for a wide range of stabilisation parameter. The error is controlled in norms that are known to converge to zero in a quasi-optimal way. An adaptive algorithm is suggested and run for improving the convergence rates in three numerical examples for thicknesses 0.1, .001 and .001.     

Residual error estimate for BEM-based FEM on polygonal meshes

A conforming finite element method on polygonal meshes is reviewed which handles hanging nodes naturally. Trial functions are defined to fulfil the homogeneous PDE locally and they are treated by means of local boundary integral equations. Using a quasi-interpolation operator of Clément type a residual-based error estimate is obtained. This a posteriori estimator can be used to rate the accuracy of the approximation over polygonal elements or it can be applied to an adaptive BEM-based FEM. The numerical experiments confirm our results and show optimal convergence for the adaptive strategy on general meshes.     

A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations

A unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the Herrmann formulation within the Hellinger-Reissner principle. This quasi-optimally converging extension of PEERS (Plane Elasticity Element with Reduced Symmetry) is called Dual-Mixed Hybrid formulation (DMH). Explicit residual-based a posteriori error estimates for DMH are introduced and are mathematically shown to be locking-free, reliable, and efficient. The estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation. Numerical evidence supports that the adaptive scheme leads to optimal convergence for Lamé and Stokes benchmark problems with singularities.     

A Priori and a Posteriori Error Analysis of a Wavelet-Based Stabilization for the Mixed Finite Element Method

We use Galerkin least-squares terms and biorthogonal wavelet bases to develop a new stabilized dual-mixed finite element method for second-order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces the trace of the solution on the boundary as a new unknown that acts also as a Lagrange multiplier. We show that the resulting stabilized dual-mixed variational formulation and the associated discrete scheme defined with Raviart–Thomas spaces are well-posed and derive the usual a priori error estimates and the corresponding rate of convergence. Furthermore, a reliable and efficient residual-based a posteriori error estimator and a reliable and quasi-efficient one are provided.     

A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

In this paper, we develop an a posteriori error analysis of a mixed finite element method for a fluid–solid interaction problem posed in the plane. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the normal displacements of the solid and the fluid. The coupling of primal and dual-mixed finite element methods is applied to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the elastic body. The finite element subspaces consider continuous piecewise linear elements for the pressure and a Lagrange multiplier defined on the interface, and PEERS for the stress and rotation in the solid domain. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Clément interpolant and Raviart–Thomas operator are the main tools for proving the reliability of the estimator. Then, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, some numerical results confirming the reliability and efficiency of the estimator are reported.     

A new a posteriori error estimate for the Morley element

In this paper, we present a posteriori error analysis for the nonconforming Morley element of the fourth order elliptic equation. We propose a new residual-based a posteriori error estimator and prove its reliability and efficiency. These results refine those of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) by dropping two edge jump terms in both the energy norm of the error and the estimator, and those of Wang and Zhang (Local a priori and a posteriori error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) by showing the efficiency in the sense of Verfürth (A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner, New York, 1996). Moreover, the normal component in the estimators of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) and Wang and Zhang (Local a priori and a posteriori error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) is dropped, and therefore only the tangential component of the stress on each edge comes into the estimator. In addition, we generalize these results to three dimensional case.     

RCV-based error density estimation in the ultrahigh dimensional additive model

In this paper, we mainly study how to estimate the error density in the ultrahigh dimensional sparse additive model, where the number of variables is larger than the sample size. First, a smoothing method based on B-splines is applied to the estimation of regression functions. Second, an improved two-stage refitted cross-validation (RCV) procedure by random splitting technique is used to obtain the residuals of the model, and then the residual-based kernel method is applied to estimate the error density function. Under suitable sparse conditions, the large sample properties of the estimator including the weak and strong consistency, as well as normality and the law of the iterated logarithm are obtained. Especially, the relationship between the sparsity and the convergence rate of the kernel density estimator is given. The methodology is illustrated by simulations and a real data example, which suggests that the proposed method performs well.

     

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.