A reliable, efficient and localized error estimator for a discontinuous Galerkin method for the Signorini problem

An approach to model contact-friction interactions between beams within assemblies of fibers is presented in this paper in order to simulate the mechanical behaviour of entangled structures at the scale of individual fibers using the finite element method. The determination of contact elements associating pairs of material particles is based on the detection of proximity zones between beams and on the construction of intermediate geometries approximating the actual contact zone, and allowing to consider contact along zones of non-zero lengths. The penalty method for contact is improved by adjusting the penalty parameter for each contact zone, thus stabilizing contact algorithms and allowing to handle high numbers of contact elements. Applications to samples of textile materials involving few hundreds of fibers are presented to demonstrate the abilities of the method. The presented examples are related to the simulation of woven fabrics – computation of the initial configuration and application of test loadings – and the identification of the transverse mechanical behaviour of a twisted textile yarn.

We present an adaptive refinement strategy for the h-version of the boundary element method with weakly singular operators on surfaces. The model problem deals with the exterior Stokes problem, and thus considers vector functions. Our error indicators are computed by local projections onto ID subspaces defined by mesh refinement. These indicators measure the error separately for the vector components and allow for component-independent adaption. Assuming a saturation condition, the indicators give rise to an efficient and reliable error estimator. Also we describe how to deal with meshes containing quadrilaterals which are not shape regular. The theoretical results are underlined by numerical experiments. To justify the saturation assumption, in the Appendix we prove optimal lower a priori error estimates for edge singularities on uniform and graded meshes.

A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to π 2 ${\frac{\pi }{2}}$ . The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allow the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks.

Stable multiscale bases and local error estimation for elliptic problems

Model order reduction (MOR) is usually equipped with two stages: the offline stage and the online stage.For many intrusive MOR methods that are based on projection, the offline stage is usually realized via a greedy algorithm.The greedy algorithm is used to properly select important parameter samples that contribute most to the solution space.The offline computational time is basically the runtime of the greedy algorithm.For large-scale systems, the offline computation is expensive and the runtime is longer than several hours even if it is run on a high-performance server.Sometimes, the system is not very large, for example, the number of degrees of freedom is only O(10 5 ), but the system structure is complicated such that the greedy algorithm still takes long time to converge.Once the reduced-order model is obtained, the online stage of querying the reduced-order model for simulation is very fast and in real-time.This work concerns a strategy of significantly accelerating the greedy algorithm.We propose multifidelity error estimators and replace the high-fidelity error estimator in the greedy algorithm.In fact, a bi-fidelity error estimation has been used in the adaptive greedy algorithm proposed in [1] without being formally defined.To further improve the convergence speed of the greedy algorithm, we propose multi-fidelity error estimation built upon the bi-fidelity error estimation.Here, we use a more efficient high-fidelity error estimator than the two different high-fidelity error estimators used in [1].Although the proposed multi-fidelity error estimation is dependent on the original high-fidelity error estimator, the concept of using multi-fidelity error estimation is generic and can be extended to develop multi-fidelity error estimation associated with other high-fidelity error estimators.The multi-fidelity error estimator is then used to adaptively update the training set of the greedy algorithm, keeping the offline computational costs on a possibly lowest level.Consequently, the computational complexity at each iteration of the greedy algorithm is reduced and the algorithm converges more than 3 times faster without incurring noticeable accuracy loss.

In this work, we consider linear elliptic problems posed in long domains, i.e. the domains whose size in one coordinate direction is much greater than the size in the other directions. If the variation of the coefficients and right‐hand side along the emphasized direction is small, the original problem can be reduced to a lower‐dimensional one that is supposed to be much easier to solve. The a‐posteriori estimation of the error stemming from the model reduction constitutes the goal of the present work. For general coefficient matrix and right‐hand side of the equation, the reliable and efficient error estimator is derived that provides a guaranteed upper bound for the modelling error, exhibits the optimal asymptotics as the size of the domain tends to infinity and correctly indicates the local error distribution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Directional, anisotropic features like layers in the solution of partial differential equations can be resolved favorably by using anisotropic finite element meshes. An adaptive algorithm for such meshes includes the ingredients Error estimation and Information extraction/Mesh refinement. Related articles on a posteriori error estimation on anisotropic meshes revealed that reliable error estimation requires an anisotropic mesh that is aligned with the anisotropic solution. To obtain anisotropic meshes the so‐called Hessian strategy is used, which provides information such as the stretching direction and stretching ratio of the anisotropic elements. This article combines the analysis of anisotropic information extraction/mesh refinement and error estimation (for several estimators). It shows that the Hessian strategy leads to well‐aligned anisotropic meshes and, consequently, reliable error estimation. The underlying heuristic assumptions are given in a stringent yet general form. Numerical examples strengthen the exposition. Hence the analysis provides further insight into a particular aspect of anisotropic error estimation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 625–648, 2002; DOI 10.1002/num.10023

In this article, we propose a residual based reliable and efficient error estimator for the new dual mixed finite element method of the elasticity problem in a polygonal domain, introduced by M. Farhloul and M. Fortin. With the help of a specific generalized Helmholtz decomposition of the error on the strain tensor and the classical decomposition of the error on the gradient of the displacements, we show that our global error estimator is reliable. Efficiency of our estimator follows by using classical inverse estimates. The lower and upper error bounds obtained are uniform with respect to the Lamé coefficient λ, in particular avoiding locking phenomena. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the $L^2$-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier--Stokes equations. The novel error estimators only take the $\mathrm{curl}$ of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the Taylor--Hood and mini finite element methods.

The estimation of the classifier error plays a key role in accuracy-based learning classifier systems. In this paper we study the current definition of the classifier error in XCSF and discuss the limitations of the algorithm that is currently used to compute the classifier error estimate from online experience. Subsequently, we introduce a new definition for the classifier error and apply the Bayes Linear Analysis framework to find a more accurate and reliable error estimate. This results in two incremental error estimate update algorithms that we compare empirically to the performance of the currently applied approach. Our results suggest that the new estimation algorithms can improve the generalization capabilities of XCSF, especially when the action-set subsumption operator is used.

Local and global error estimations in linear structural dynamics

Automatic time step control algorithms for structural dynamics

An adaptive version of Glimm's scheme

A-priori and a-posteriori error estimates for discontinuous Galerkin method of the Maxwell eigenvalue problem

Aerosol coagulation is a significant process regarding the dynamics of aerosols in the atmosphere. This process leads to an evolution of the size distribution of aerosols over time induced by particle collisions, and is described by Smoluchowski equation [1].Accurate numerical simulations of this process are computationally demanding by typical discretization methods [2]. We investigate the effectiveness of the reduced basis method [3] to provide accurate but less demanding simulations, by constructing a reduced order subspace included within the reference high-dimensional space used to provide high fidelity simulations.We also provide residual-based a posteriori error estimates [4] which enable certification of results up to a given error tolerance. We obtain efficient online model and error estimates as their online computational cost only scale with the reduced dimension, and not the dimension of the high-fidelity model.By careful design of reduced subspaces we ensure that some properties of the high fidelity operator, such as mass conservation [5], are also preserved by reduced order models.[1] M. V. Smoluchowski. Drei vortage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen. Physik, 17, 557–585, 1916[2] E. Debry, and B. Sportisse. Solving aerosol coagulation with size-binning methods. Applied Numerical Mathematics, 57(9), 1008–1020, 2007[3] A. Quarteroni, et al. Reduced Basis Methods for Partial Differential Equations. Springer Cham. 2015[4] M. Grepl, and A. Patera, A Posteriori Error Bounds for Reduced-Basis Approximations of Parametrized Parabolic Partial Differential EquationsESAIM: Mathematical Modelling and Numerical Analysis, 39(1), 157-181, 2005[5] F. Filbet, and P. Laurençot. Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation. Archiv der Mathematik, 83(6), 558-567, 2004