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Lina Song - One of the best experts on this subject based on the ideXlab platform.
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Recovery-Based Error Estimator for Stabilized Finite Element Method for the Stationary Navier--Stokes Problem
SIAM Journal on Scientific Computing, 2016Co-Authors: Lina Song, Haiyan Su, Xinlong FengAbstract:A recovery-based Error Estimator is proposed and analyzed for stabilized $P_1/P_0$ (continuous linear velocity/constant pressure) finite element approximations to the stationary Navier--Stokes problem. We establish the reliability and efficiency of the Error Estimator. A crucial part of this work is the estimation for the nonlinear term of the Navier--Stokes problem. It turns out such a term can be bounded by the recovery-based Error Estimator. Numerical results are provided to illustrate the performance of the Error Estimator.
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recovery based Error Estimator for stabilized finite element methods for the stokes equation
Computer Methods in Applied Mechanics and Engineering, 2014Co-Authors: Lina SongAbstract:Abstract A recovery-based Error Estimator is proposed and analyzed for stabilized P 1 / P 0 (continuous linear velocity/constant pressure) finite element approximations to the Stokes equation. Reliability and efficiency of the Estimator are established for various stabilized methods. For several test problems, numerical results show that our Estimator is more accurate than the classical residual Error Estimator.
Simona Perotto - One of the best experts on this subject based on the ideXlab platform.
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an anisotropic zienkiewicz zhu type Error Estimator for 3d applications
International Journal for Numerical Methods in Engineering, 2011Co-Authors: Patrick E Farrell, Stefano Micheletti, Simona PerottoAbstract:We extend the anisotropic Zienkiewicz–Zhu a posteriori Error Estimator of (Proceedings of the ENUMATH-2009, Uppsala, Sweden, 29 June–3 July 2009) to three dimensions. Like the standard Zienkiewicz–Zhu Estimator, the proposed Estimator is designed to be independent of the problem at hand, is cheap to compute and easy to implement. In contrast to the standard Zienkiewicz–Zhu Estimator, the elementwise counterpart of the proposed Estimator explicitly takes into account the geometrical properties of the actual tetrahedron. Thus, in a wide variety of applications, the Estimator is able to detect the anisotropic features exhibited by the solution of the governing equations. A metric-based optimization procedure, rigorously addressed, drives the adaptation of the mesh. It is shown numerically to yield quasi-optimal triangulations, dictating the accuracy-vs-number of elements behaviour. Despite being heuristic to some extent, in practice the overall anisotropic adaptation procedure turns out to be effective. Copyright © 2010 John Wiley & Sons, Ltd.
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a recovery based Error Estimator for anisotropic mesh adaptation in cfd
SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada, 2010Co-Authors: Stefano Micheletti, Simona Perotto, Patrick E FarrellAbstract:We provide a unifying framework that generalizes the 2D and 3D settings proposed in [32] and [17], respectively. In these two works we propose a gradient recovery type a posteriori Error Estimator for finite element approximations on anisotropic meshes. The novelty is the inclusion of the geometrical features of the computational mesh (size, shape and orientation) in the Estimator itself. Moreover, we preserve the good properties of recovery based Error Estimators, in particular their computational cheapness and ease of implementation. A metric-based optimization procedure, relying on the Estimator, drives the anisotropic adaptation of the mesh. The focus of this work then moves to a goal-oriented framework. In particular, we extend the idea proposed in [32, 17] to the control of a goal functional. The preliminary results are promising, since it is shown numerically to yield quasi-optimal triangulations with respect to the Error-vs-number of elements behaviour.
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anisotropic adaptation via a zienkiewicz zhu Error Estimator for 2d elliptic problems
ENUMATH 2009 the 8th European Conference on Numerical Mathematics and Advanced Applications, 2010Co-Authors: Stefano Micheletti, Simona PerottoAbstract:We propose a Zienkiewicz–Zhu a posteriori Error Estimator in 2D, which shares the computational advantages typical of the original Estimator. The novelty is the inclusion of the geometrical features of the computational mesh, useful for an anisotropic mesh adaptation. The adapted triangulations are shown numerically to be quasi-optimal with respect to the Error-vs-number of elements behavior.
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reliability and efficiency of an anisotropic zienkiewicz zhu Error Estimator
Computer Methods in Applied Mechanics and Engineering, 2006Co-Authors: Stefano Micheletti, Simona PerottoAbstract:Abstract In this paper we study the efficiency and the reliability of an anisotropic a posteriori Error Estimator in the case of the Poisson problem supplied with mixed boundary conditions. The Error Estimator may be classified as a residual-based one, but its novelty is twofold: firstly, it employs anisotropic estimates of the interpolation Error for linear triangular finite elements and, secondly, it makes use of the Zienkiewicz–Zhu recovery procedure to approximate the gradient of the exact solution. Finally, we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy, and we include some numerical test cases to assess the robustness of the proposed numerical algorithm.
Marco Picasso - One of the best experts on this subject based on the ideXlab platform.
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an easily computable Error Estimator in space and time for the wave equation
Mathematical Modelling and Numerical Analysis, 2019Co-Authors: Olga Gorynina, Alexei Lozinski, Marco PicassoAbstract:We propose a cheaper version of a posteriori Error Estimator from Gorynina et al. (Numer. Anal. (2017)) for the linear second-order wave equation discretized by the Newmark scheme in time and by the finite element method in space. The new Estimator preserves all the properties of the previous one (reliability, optimality on smooth solutions and quasi-uniform meshes) but no longer requires an extra computation of the Laplacian of the discrete solution on each time step.
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an easily computable Error Estimator in space and time for the wave equation
arXiv: Numerical Analysis, 2017Co-Authors: Olga Gorynina, Alexei Lozinski, Marco PicassoAbstract:We propose a cheaper version of \textit{a posteriori} Error Estimator from arXiv:1707.00057 for the linear second-order wave equation discretized by the Newmark scheme in time and by the finite element method in space. The new Estimator preserves all the properties of the previous one (reliability, optimality on smooth solutions and quasi-uniform meshes) but no longer requires an extra computation of the Laplacian of the discrete solution on each time step.
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adaptive finite elements with large aspect ratio based on an anisotropic Error Estimator involving first order derivatives
Computer Methods in Applied Mechanics and Engineering, 2006Co-Authors: Marco PicassoAbstract:An anisotropic Error Estimator involving only first order derivatives is proposed for the Laplace problem and continuous, piecewise linear finite elements. Upper and lower bounds are presented, the involved constants being independent of the mesh aspect ratio provided the Error gradient is equidistributed in the directions of maximum and minimum stretching. An anisotropic adaptive algorithm is then proposed, with aim to equidistribute the Error gradient in the directions of maximum and minimum stretching. Numerical results in two and three space dimensions show that the effectivity index is aspect ratio independent on such adapted meshes.
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numerical study of the effectivity index for an anisotropic Error indicator based on zienkiewicz zhu Error Estimator
Communications in Numerical Methods in Engineering, 2002Co-Authors: Marco PicassoAbstract:The framework of Formaggia and Perotto (Numerische Mathematik 2001; 89:641-667) is considered to derive a new anisotropic Error indicator for a Laplace problem in the energy norm. The matrix containing the Error gradient is approached using a Zienkiewicz-Zhu Error Estimator. A numerical study of the effectivity index is proposed for anisotropic unstructured meshes, showing that our indicator is sharp. An anisotropic adaptive algorithm is implemented, aiming at controlling the estimated relative Error. Copyright (C) 2003 John Wiley Sons, Ltd.
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an anisotropic Error indicator based on zienkiewicz zhu Error Estimator application to elliptic and parabolic problems
SIAM Journal on Scientific Computing, 2002Co-Authors: Marco PicassoAbstract:The anisotropic Error indicator presented in [M. Picasso, Comm. Numer. Methods Engrg., 19 (2003), pp. 13--23.] in the frame of the Laplace equation is extended to elliptic and parabolic problems. Our Error indicator is derived using the anisotropic interpolation estimates of [L. Formaggia and S. Perotto, Numer. Math., 89 (2001), pp. 641--667; L. Formaggia and S. Perotto, Numer. Math., (2002), DOI 10.1007/s002110200415], together with a Zienkiewicz--Zhu Error Estimator to approach the Error gradient. A numerical study of the effectivity index is proposed for elliptic, diffusion-convection, and parabolic problems. An adaptive algorithm is implemented, aimed at controlling the relative estimated Error.
Stefano Micheletti - One of the best experts on this subject based on the ideXlab platform.
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an anisotropic zienkiewicz zhu type Error Estimator for 3d applications
International Journal for Numerical Methods in Engineering, 2011Co-Authors: Patrick E Farrell, Stefano Micheletti, Simona PerottoAbstract:We extend the anisotropic Zienkiewicz–Zhu a posteriori Error Estimator of (Proceedings of the ENUMATH-2009, Uppsala, Sweden, 29 June–3 July 2009) to three dimensions. Like the standard Zienkiewicz–Zhu Estimator, the proposed Estimator is designed to be independent of the problem at hand, is cheap to compute and easy to implement. In contrast to the standard Zienkiewicz–Zhu Estimator, the elementwise counterpart of the proposed Estimator explicitly takes into account the geometrical properties of the actual tetrahedron. Thus, in a wide variety of applications, the Estimator is able to detect the anisotropic features exhibited by the solution of the governing equations. A metric-based optimization procedure, rigorously addressed, drives the adaptation of the mesh. It is shown numerically to yield quasi-optimal triangulations, dictating the accuracy-vs-number of elements behaviour. Despite being heuristic to some extent, in practice the overall anisotropic adaptation procedure turns out to be effective. Copyright © 2010 John Wiley & Sons, Ltd.
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a recovery based Error Estimator for anisotropic mesh adaptation in cfd
SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada, 2010Co-Authors: Stefano Micheletti, Simona Perotto, Patrick E FarrellAbstract:We provide a unifying framework that generalizes the 2D and 3D settings proposed in [32] and [17], respectively. In these two works we propose a gradient recovery type a posteriori Error Estimator for finite element approximations on anisotropic meshes. The novelty is the inclusion of the geometrical features of the computational mesh (size, shape and orientation) in the Estimator itself. Moreover, we preserve the good properties of recovery based Error Estimators, in particular their computational cheapness and ease of implementation. A metric-based optimization procedure, relying on the Estimator, drives the anisotropic adaptation of the mesh. The focus of this work then moves to a goal-oriented framework. In particular, we extend the idea proposed in [32, 17] to the control of a goal functional. The preliminary results are promising, since it is shown numerically to yield quasi-optimal triangulations with respect to the Error-vs-number of elements behaviour.
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anisotropic adaptation via a zienkiewicz zhu Error Estimator for 2d elliptic problems
ENUMATH 2009 the 8th European Conference on Numerical Mathematics and Advanced Applications, 2010Co-Authors: Stefano Micheletti, Simona PerottoAbstract:We propose a Zienkiewicz–Zhu a posteriori Error Estimator in 2D, which shares the computational advantages typical of the original Estimator. The novelty is the inclusion of the geometrical features of the computational mesh, useful for an anisotropic mesh adaptation. The adapted triangulations are shown numerically to be quasi-optimal with respect to the Error-vs-number of elements behavior.
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reliability and efficiency of an anisotropic zienkiewicz zhu Error Estimator
Computer Methods in Applied Mechanics and Engineering, 2006Co-Authors: Stefano Micheletti, Simona PerottoAbstract:Abstract In this paper we study the efficiency and the reliability of an anisotropic a posteriori Error Estimator in the case of the Poisson problem supplied with mixed boundary conditions. The Error Estimator may be classified as a residual-based one, but its novelty is twofold: firstly, it employs anisotropic estimates of the interpolation Error for linear triangular finite elements and, secondly, it makes use of the Zienkiewicz–Zhu recovery procedure to approximate the gradient of the exact solution. Finally, we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy, and we include some numerical test cases to assess the robustness of the proposed numerical algorithm.
Zhimin Zhang - One of the best experts on this subject based on the ideXlab platform.
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A Robust Residual-Type a Posteriori Error Estimator for Convection---Diffusion Equations
Journal of Scientific Computing, 2014Co-Authors: Shaohong Du, Zhimin ZhangAbstract:In this paper, a new robust residual type a posteriori Error Estimator is developed and analyzed for convection---diffusion equations. A novel dual norm is introduced, under which the Error Estimator is proved to be robust with respect to the singularly perturbed parameter $$\varepsilon $$?. Both theoretical and numerical results showed that the Estimator performs better than the existing ones in literature.
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analysis of the superconvergent patch recovery technique and a posteriori Error Estimator in the finite element method ii
Computer Methods in Applied Mechanics and Engineering, 1998Co-Authors: Zhimin ZhangAbstract:Abstract This is the first in a series of two papers in which the patch recovery technique proposed by Zienkiewicz and Zhu is analyzed. In this work it is proved that the recovered derivative by the least-squares fitting is superconvergent for the two-point boundary value problems. Further, the a posteriori Error Estimator based on the recovery technique is shown to be asymptotically exact. This confirms the computational results obtained in [1,2].