Numerical comparison of three a posteriori error estimators for nonconforming finite element method

Mesh refinement for anisotropic diffusion in magnetized plasmas

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

An adaptive algorithm for the Stokes problem using continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio

An automatic adaptive refinement procedure for the analysis of shell structures using the nine-node degenerated solid shell element is suggested. The basic adaptive refinement principle and the effects of singularities and boundary layers on the convergence rate of the nine-node element used are discussed. A new stress recovery procedure based on the patch convective co-ordinate system concept is developed for the construction of a continuous smoothed stress field over the shell domains. The stress recovery procedure is easy to implement, requires a modest computational effort and needs only local patch information. It can be applied to shells with non-uniform thickness as well as to multi-layered shell structures. The smoothed recovered stress obtained is then used with the Zienkiewicz and Zhu error estimator for a posteriori error estimation during the adaptive refinement analysis. Numerical results which are in good agreement with theoretical predictions are obtained and they indicate that the current adaptive refinement procedure can eliminate the effect of singularities inside the problem domains so that a near-optimal convergence rate is achieved in all the numerical examples. This also indicates that the stress recovery procedure can produce an accurate stress field and as a result the error estimator can reflect the error distribution of the finite element solution. Even though in the current study only one type of element is used in the analysis, the whole adaptive refinement scheme can be readily applied to any other types of degenerated solid element. © 1997 John Wiley & Sons, Ltd.

In this paper, we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Stokes problem. In particular, we derive computable upper bounds on ...

This letter proposes to automate generation of reduced-order models used for accelerated S-parameter computation by applying a posteriori model error estimators. So far, a posteriori error estimators were used in Reduced Basis Method (RBM) and Proper Orthogonal Decomposition (POD) to select frequency points at which basis vectors are generated. This letter shows how a posteriori error estimators can be applied to automatically select the order of the reduced model in second-order Model Order Reduction (MOR) methods. Three different error estimators are investigated and compared in order to arrive at a new MOR scheme that is fast, reliable, and fully automated. The effectiveness of the proposed approach is verified by very high accuracy of the computed scattering parameters ( S-parameters) for an example of a waveguide filter over a prescribed frequency band.

Adaptive stochastic Galerkin FEM

This paper establishes a posteriori error analysis for the Stokes equations discretized by an interior penalty type method using $H(\mathrm{div})$ finite elements. The a posteriori error estimator is then employed for designing two grid refinement strategies; one is locally based and the other is globally based. The locally based refinement technique is believed to be able to capture local singularities in the numerical solution. The numerical formulations for the Stokes problem make use of $H(\mathrm{div})$ conforming elements of Raviart–Thomas type. Therefore, the finite element solution features a full satisfaction of the continuity equation (mass conservation). The result of this paper provides a rigorous analysis for the method's reliability and efficiency. In particular, an $H^1$ norm a posteriori error estimator is obtained, together with upper and lower bound estimates. Numerical results are presented to verify the new theory of a posteriori error estimators.

A simplified stochastic Hookean dumbbells model arising from viscoelastic flows is considered, the convective terms being disregarded. A finite element discretization in space is proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates, using an implicit function theorem and regularity results obtained in [Bonito et al., J. Evol. Equ. 6 (2006) 381–398] for the solution of the continuous problem. A posteriori error estimates are also derived. Numerical results with small time steps and a large number of realizations confirm the convergence rate with respect to the mesh size.

A novel approach to implicit residual‐type error estimation in mesh‐free methods and an adaptive refinement strategy are presented. This allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual‐type estimators circumventing the need of flux‐equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh‐free methods. The adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision. Copyright © 2008 John Wiley & Sons, Ltd.

A new h-version adaptive refinement scheme is developed for the analysis of Thin-walled structures in elastostatic range. The basic concepts of adaptivity for the finite element method are discussed and a brief review of the related theories and techniques is presented. Focus is devoted to three essential ingredients for the whole adaptive refinement scheme: automatic mesh generation, error estimation on the FE solution and adaptive refinement scheme design.

The numerical study of a microscale model for lithium-ion batteries

In many scientific problems, adaptive finite element methods has been widely used to improve the accuracy of numerical solutions. The general idea is to refine or adjust the mesh such that the errors are “equally” distributed over the computational mesh, with the aim of achieving a better accurate solution using an optimal number of degrees of freedom. By using the information from the approximated solution and the known data, the a posteriori error estimator provides the information about the size and the distribution of the error of the finite element approximation. There is a large numerical analysis literature on adaptive finite element methods, and various types of a posteriori estimates have been proposed for different problems, see e.g. [1]. The a posterior error estimate and adaptive finite element method were first introduced by [2]. Since the later 1980s, much research work on a posteriori error estimate has been developed including the residual type a posteriori error estimate [8], recovery type a posteriori error estimate [16], a posteriori error estimate based on hierarchic basis [5, 4], and so on. For the literature, the readers are referred to the books [1, 3, 12, 14], the papers [6, 13, 15], and the references cited therein.KeywordsError EstimatorPosteriori ErrorPosteriori Error EstimateInitial MeshElement Stiffness MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this work, we combine an hp–adaptive strategy with a posteriori error estimates for variational inequalities, which are given by contact problems. The a posteriori error estimates are obtained using a general approach based on the saddle point formulation of contact problems and making use of a yposteriori error estimates for variational equations. Error estimates are presented for obstacle problems and Signorini problems with friction.Numerical experiments confirm the reliability of the error estimates for finite elements of higher order. The use of the hp–adaptive strategy leads to meshes with the same characteristics as geometric meshes and to exponential convergence. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented which generates C1 surfaces in the limit. The scheme satisfies important requirements for practical applications in computer graphics and engineering. These requirements include the necessity to generate smooth surfaces with local creases and cusps. The scheme can be applied to open nets in which case it generates boundary curves that allow a C0‐join of several subdivision patches. Due to the local support of the scheme, adaptive refinement strategies can be applied. We present a simple device to preserve the consistency of such adaptively refined nets.