A stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elements

Marco Picasso
2009
Article

Résumé

We propose a simple stopping criterion for the conjugate gradient (CG) algorithm in the framework of anisotropic, adaptive finite elements for elliptic problems. The goal of the adaptive algorithm is to find a triangulation such that the estimated relative error is close to a given tolerance TOL. We propose to stop the CG algorithm whenever the residual vector has Euclidian norm less than a small fraction of the estimated error. This stopping criterion is based on a posteriori error estimates between the true solution u and the computed solution u(h)(n) (the superscript n stands for the CG iteration number, the subscript It for the typical mesh size) and on heuristics to relate the error between u(h) and u(h)(n) to the residual vector.

Official source

https://infoscience.epfl.ch/record/159903?ln=fr

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Concepts associés

Chargement

Publications associées

Chargement

Adaptive finite elements for a linear parabolic problem

Marco Picasso

A posteriori error estimates for the heat equation in two space dimensions are presented. A classical discretization is used, Euler backward in time, and continuous, piecewise linear triangular finite elements in space. The error is bounded above and below by an explicit error estimator based on the residual. Numerical results are presented for uniform triangulations and constant time steps. The quality of our error estimator is discussed. An adaptive algorithm is then proposed. Successive Delaunay triangulations are generated, so that the estimated relative error is close to a preset tolerance. Again, numerical results demonstrate the efficiency of our approach. (C) 1998 Elsevier Science S.A. All rights reserved.

1998

Chargement

An adaptive algorithm is proposed for solving the Stokes problem with continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio. Anisotropic, a posteriori error estimates in the H-1 x L-2 norm are derived, the constant being independent from the mesh aspect ratio. The estimates involve the first derivatives of the velocity error and the velocity of a dual problem having the pressure error in the right side. An explicit error indicator is then proposed. The first derivatives of the velocity error are approximated using Zienkiewicz-Zhu error estimator (post-processing). The pressure error in the right side of the dual problem is approximated by smoothing. Numerical results on meshes with large aspect ratio show that the error indicator is robust whenever the pressure error in the dual problem is approximated with sufficient accuracy. Finally, an adaptive algorithm is proposed, with goal to build an anisotropic triangulation such that the relative estimated error is close to a preset tolerance. (c) 2004 IMACS. Published by Elsevier B.V. All rights reserved.

2005

Chargement

Adaptive finite element simulations of dendritic growth including fluid flow induced by shrinkage

Jacek Narski

A phase field model describing the solidification of a binary alloy is investigated. The location of the solid and liquid phases in the computational domain is described by introducing an order parameter, the phase-field, which varies smoothly from one in the solid to zero in the liquid through a slightly diffused interface. The solidification process of binary alloys is controlled by the local concentration of the alloy and the temperature. The concentration is altered by the existing flows in the melt. With temperature being a given constant, the model corresponds to coupling the phase-field equation, the concentration equation and the compressible Navier-Stokes equations. The main difficulty when solving numerically phase field models is due to the very rapid change of the phase field and the concentration field across the diffused interface, whose thickness has to be taken very small in comparison to the dimension of the computational domain in order to correctly capture the physics of the phase transformation. A high spatial resolution is therefore needed to describe the smooth transition. In this work, we present a physical model governing the solidification process. In order to reduce the number of grid points required for the reliable simulations, we introduce an adaptive algorithm that aims to build successive meshes with large aspect ratio such that the relative estimated error of the concentration and/or velocity in the H1-norm is close to a preset tolerance TOL. For this purpose, we introduce error indicators which measure the error of the concentration and the velocity in the directions of maximum and minimum stretching of the element. Finally, we apply our method to 2D and 3D simulations of the dendritic growth proving its efficiency.

EPFL2007

Chargement