On the local error and the local truncation error of linear multistep methods

Two different measures of the local accuracy of a linear multistep method — the local error and the local trunction error — appear in the literature. It is shown that the principal parts of these errors are not identical for general linear multistep methods, but that they are so for a sub-class which contains all methods of Adams type. It is sometimes argued that local error is the more natural measure; this view is challenged.     

On the spectrum of stiffness matrices arising from isogeometric analysis

We study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems. Motivated by the applicative interest in the fast solution of the related linear systems, we are looking for a spectral characterization of the involved matrices. In particular, we investigate non-singularity, conditioning (extremal behavior), spectral distribution in the Weyl sense, as well as clustering of the eigenvalues to a certain (compact) subset of \(\mathbb C\) . All the analysis is related to the notion of symbol in the Toeplitz setting and is carried out both for the cases of 1D and 2D problems.     

An a posteriori error estimator for anisotropic refinement

Summary. Besides an algorithm for local refinement, an a posteriori error estimator is the basic tool of every adaptive finite element method. Using information generated by such an error estimator the refinement of the grid is controlled. For 2nd order elliptic problems we present an error estimator for anisotropically refined grids, like -d cuboidal and 3-d prismatic grids, that gives correct information about the size of the error; additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way. Numerical examples are presented for 2-d rectangular and 3-d prismatic grids. Received March 15, 1994 / Revised version received June 3, 1994     

On using the Lagrange coefficients for a posteriori error estimation

A posteriori error estimation of the objective functional is considered by means of a differential presentation of a finite-difference scheme and adjoint equations. The local approximation error is presented as a Taylor series remainder in the Lagrangian form. The field of the Lagrange coefficients is determined by a high-accuracy finite-difference template affecting the computation results. The feasibility of using the Lagrange coefficients for refining the solution and for estimating its uncertainty is considered.     

Partition of unity refinement for local approximation

In this article, we propose a Partition of Unity Refinement (PUR) method to improve the local approximations of elliptic boundary value problems in regions of interest. The PUR method only needs to refine the local meshes and hanging nodes generate no difficulty. The mesh qualities such as uniformity or quasi‐uniformity are kept. The advantages of the PUR include its effectiveness and relatively easy implementation. In this article, we present the basic ideas and implementation of the PUR method on triangular meshes. Numerical results for elliptic Dirichlet boundary value problem on an L‐shaped domain are shown to demonstrate the effectiveness of the proposed method. The extensions of the PUR method to multilevel and higher dimension are straightforward. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 803–817, 2011     

On optimal global error bounds obtained by scaled local error estimates

Summary In the second section a general method of obtaining optimal global error bounds by scaling local error estimates is developed. It is reduced to the solution of a fixpoint problem. In Sect. 3 we show more concrete error estimates reflecting a singularity of order . It is shown that under general circumstances an optimal global error bound is achieved by an (asymptotically) geometric mesh for the local error estimates. In the fourth section we specialize this to the best approximation ofg(x)x by piecewise polynomials with variable knots and degrees having a total numberN of parameters. This generalizes the result of R. DeVore and the author forg(x)=1. In the last section this problem is studied for the functione ^–x on (0, ). The exact asymptotic behaviour of the approximation withN parameters is determined toe ^qoN , whereq _o=0.895486 ....     

On residual-based a posteriori error estimation in hp-FEM

A family , [0,1], of residual-based error indicators for the hp-version of the finite element method is presented and analyzed. Upper and lower bounds for the error indicators are established. To do so, the well-known Clément/Scott–Zhang interpolation operator is generalized to the hp-context and new polynomial inverse estimates are presented. An hp-adaptive strategy is proposed. Numerical examples illustrate the performance of the error indicators and the adaptive strategy.     

Adaptive local grid refinement algorithms for finite-element collocation

An adaptive grid refinement procedure allows accurate solutions to advection-dominated, time-dependent flows using finite-element collocation. The technique relies on a data structure that is readily amenable to parallel computing. The paper discusses computational aspects of the method.     

A local error analysis of the boundary-concentrated hp-FEM

^** Email: teibner{at}mathematik.tu-chemnitz.de^*** Email: melenk{at}tuwien.ac.at The boundary-concentrated finite-element method (FEM) is a variantof the hp-version of the FEM that is particularly suited forthe numerical treatment of elliptic boundary value problemswith smooth coefficients and boundary conditions with low regularityor non-smooth geometries. In this paper, we consider the caseof the discretization of a Dirichlet problem with the exactsolution u H¹⁺() and investigate the local error in variousnorms. For 2D problems, we show that the error measured in thesenorms is O(N^––ß), where N denotes thedimension of the underlying finite-element space and ß> 0. Furthermore, we present a new Gauss–Lobatto-basedinterpolation operator that is adapted to the case of non-uniformpolynomial degree distributions.     

Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements

We derive new trace inequalities for NURBS-mapped domains. In addition to Sobolev-type inequalities, we derive discrete trace inequalities for use in NURBS-based isogeometric analysis. All dependencies on shape, size, polynomial degree, and the NURBS weighting function are precisely specified in our analysis, and explicit values are provided for all bounding constants appearing in our estimates. As hexahedral finite elements are special cases of NURBS, our results specialize to parametric hexahedral finite elements, and our analysis also generalizes to T-spline-based isogeometric analysis. We compare the bounding constants appearing in our explicit trace inequalities with numerically computed optimal bounding constants, and we discuss application of our results to a Laplace problem. We finish this paper with a brief exploration of so-called patch-wise trace inequalities for isogeometric analysis.     

On local analysis

We extend to Gaussian distributions a result providing smoothed analysis estimates for condition numbers given as relativized distances to ill-posedness. We also introduce a notion of local analysis meant to capture the behavior of these condition numbers around a point.     

失效率的E-Bayes估计和多层Bayes估计

提出了一种可靠性参数的估计方法—E-Bayes估计法,对寿命服从指数分布的产品,在无失效数据情形,给出了失效率的E-Bayes估计的定义、E-Bayes估计和多层Bayes估计,并在此基础上给出了E-Bayes估计的性质.最后,结合发动机的实际问题进行了计算,结果表明E-Bayes估计法可行且便于应用.     

E-Bayesian estimation and hierarchical Bayesian estimation of failure rate

This paper introduces a new method, E-Bayesian estimation method, to estimate failure rate. The method is suitable for the censored or truncated data with small sample sizes and high reliability. The definition and properties of E-Bayesian estimation are given. A real data set is discussed, which shows that the method is both efficiency and easy to operate.     

A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization

We present in this Note fully computable a posteriori error estimates allowing for accurate error control in the conforming finite element discretization of pure diffusion problems. The derived estimates are based on the local conservativity of the conforming finite element method on a dual grid associated with simplex vertices rather than directly on the Galerkin orthogonality. To cite this article: M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Hierarchical estimation as a basis for hierarchical forecasting

Email: l.w.g.strijbosch{at}uvt.nl Received on 6 November 2006. Accepted on 29 October 2007. In inventory management, hierarchical forecasting (HF) is ahot issue: families of items are formed for which total demandis forecasted; total forecast then is broken up to produce forecastsfor the individual items. Since HF is a complicated procedure,analytical results are hard to obtain; consequently, most literatureis based on simulations and case studies. This paper succeedsin following a more theoretical approach by simplifying theproblem: we consider estimation instead of forecasting. So,from a random sample we estimate both the total demand and thefraction of this total that individual items take; multiplyingthese two quantities gives a new estimate of the individualdemand. Then, our research question is: Can aggregation of items,followed by fractioning, lead to more accurate estimates ofindividual demand? We consider two simple situations that canbe analysed fully theoretically. Thirdly, a more practical situationis investigated by means of simulation.     

Efficient and reliable hierarchical error estimates for an elliptic obstacle problem

We present and analyze novel hierarchical a posteriori error estimates for a self-adjoint elliptic obstacle problem. Under a suitable saturation assumption, we prove the efficiency and reliability of our hierarchical estimates. The proof is based upon some new observations on the efficiency of some hierarchical error indicators. These new observations allow us to remove an additional regularity condition on the underlying grid required in the previous analysis. Numerical computations confirm our theoretical findings.     

Anisotropic mesh refinement for finite element methods based on error reduction

In this paper, we propose an anisotropic adaptive refinement algorithm based on the finite element methods for the numerical solution of partial differential equations. In 2-D, for a given triangular grid and finite element approximating space V, we obtain information on location and direction of refinement by estimating the reduction of the error if a single degree of freedom is added to V. For our model problem the algorithm fits highly stretched triangles along an interior layer, reducing the number of degrees of freedom that a standard h-type isotropic refinement algorithm would use.     

A new methodology for anisotropic mesh refinement based upon error gradients

We introduce a new strategy for controlling the use of anisotropic mesh refinement based upon the gradients of an a posteriori approximation of the error in a computed finite element solution. The efficiency of this strategy is demonstrated using a simple anisotropic mesh adaption algorithm and the quality of a number of potential a posteriori error estimates is considered.