Relevant bibliographies by topics / A posteriori error bound

Academic literature on the topic 'A posteriori error bound'

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Published: 6 September 2023

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Journal articles on the topic "A posteriori error bound"

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Liu, Jie, Tian Xia, and Wei Jiang. "A Posteriori Error Estimates with Computable Upper Bound for the Nonconforming RotatedQ1Finite Element Approximation of the Eigenvalue Problems." Mathematical Problems in Engineering 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/891278.

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This paper discusses the nonconforming rotatedQ1finite element computable upper bound a posteriori error estimate of the boundary value problem established by M. Ainsworth and obtains efficient computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem. We extend the a posteriori error estimate to the Steklov eigenvalue problem and also derive efficient computable upper bound a posteriori error indicators. Finally, through numerical experiments, we verify the validity of the a posteriori error estimate of the boundary value problem; meanwhile, the numerical results show that the a posteriori error indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective.
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Antonopoulou, Dimitra, and Michael Plexousakis. "A posteriori analysis for space-time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 2 (March 2019): 523–49. http://dx.doi.org/10.1051/m2an/2018059.

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This paper presents an a posteriori error analysis for the discontinuous in time space–time scheme proposed by Jamet for the heat equation in multi-dimensional, non-cylindrical domains Jamet (SIAM J. Numer. Anal. 15 (1978) 913–928). Using a Clément-type interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of two-dimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with space-time dependent coefficients but posed on a cylindrical domain. We formulate a discontinuous in time space–time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of Evans (American Mathematical Society (1998)) for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso (Comput. Meth. Appl. Mech. Eng. 167 (1998) 223–237), proposed for adaptive, Runge-Kutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.
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Cochez-Dhondt, Sarah, and Serge Nicaise. "A Posteriori Error Estimators Based on Equilibrated Fluxes." Computational Methods in Applied Mathematics 10, no. 1 (2010): 49–68. http://dx.doi.org/10.2478/cmam-2010-0002.

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Abstract We consider the conforming of finite element approximations of reactiondiffusion problems. We propose new a posteriori error estimators based on H(div)- conforming finite elements and equilibrated fluxes. It is shown that these estimators give rise to an upper bound where the constant is one in front of the indicator, up to higher order terms. Lower bounds can also be established with constants depending on the shape regularity of the mesh and the local variation of the coefficients. We further analyze the convergence of an adaptive algorithm. The reliability and efficiency of the proposed estimators are confirmed by various numerical tests.
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Mandic, Danilo P., and Jonathon A. Chambers. "Relationships Between the A Priori and A Posteriori Errors in Nonlinear Adaptive Neural Filters." Neural Computation 12, no. 6 (June 1, 2000): 1285–92. http://dx.doi.org/10.1162/089976600300015358.

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The lower bounds for the a posteriori prediction error of a nonlinear predictor realized as a neural network are provided. These are obtained for a priori adaptation and a posteriori error networks with sigmoid nonlinearities trained by gradient-descent learning algorithms. A contractivity condition is imposed on a nonlinear activation function of a neuron so that the a posteriori prediction error is smaller in magnitude than the corresponding a priori one. Furthermore, an upper bound is imposed on the learning rate η so that the approach is feasible. The analysis is undertaken for both feedforward and recurrent nonlinear predictors realized as neural networks.
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CREUSÉ, E., S. NICAISE, and G. KUNERT. "A POSTERIORI ERROR ESTIMATION FOR THE STOKES PROBLEM: ANISOTROPIC AND ISOTROPIC DISCRETIZATIONS." Mathematical Models and Methods in Applied Sciences 14, no. 09 (September 2004): 1297–341. http://dx.doi.org/10.1142/s0218202504003635.

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The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail. Our analysis covers two- and three-dimensional domains, conforming and non-conforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators.
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KNEZEVIC, DAVID J., NGOC-CUONG NGUYEN, and ANTHONY T. PATERA. "REDUCED BASIS APPROXIMATION ANDA POSTERIORIERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS." Mathematical Models and Methods in Applied Sciences 21, no. 07 (July 2011): 1415–42. http://dx.doi.org/10.1142/s0218202511005441.

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In this paper we present reduced basis (RB) approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold — to provide dimension reduction; an efficient proper orthogonal decomposition–Greedy sampling method for identification of optimal and numerically stable approximations — to yield rapid convergence; accurate (online) calculation of the solution-dependent stability factor by the successive constraint method — to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the RB approximation and associated outputs — to provide certainty in our predictions; and an offline–online computational decomposition strategy for our RB approximation and associated error bound — to minimize marginal cost and hence achieve high performance in the real-time and many-query contexts. The method is applied to a transient natural convection problem in a two-dimensional "complex" enclosure — a square with a small rectangle cutout — parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the RB approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.
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Korneev, V. G. "A Posteriori Error Control at Numerical Solution of Plate Bending Problem." Applied Mechanics and Materials 725-726 (January 2015): 674–80. http://dx.doi.org/10.4028/www.scientific.net/amm.725-726.674.

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The classical approach to a posteriori error control considered in this paper bears on the counter variational principles of Lagrange and Castigliano. Its efficient implementation for problems of mechanics of solids assumes obtaining equilibrated stresses/resultants which, at the same time, are sufficiently close to the exact values. Besides, it is important that computation of the error bound with the use of such stresses/resultants would be cheap in respect to the arithmetic work. Following these guide lines, we expand the preceding results for elliptic partial differential equations and theory elasticity equations upon the problem of thin plate bending. We obtain guaranteed a posteriori error bounds of simple forms for solutions by the finite element method and discuss the algorithms of linear complexity for their computation. The approach of the paper also allows to improve some known general a posteriori estimates by means of arbitrary not equilibrated stress fields.
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Buffa, Annalisa, and Eduardo M. Garau. "A posteriori error estimators for hierarchical B-spline discretizations." Mathematical Models and Methods in Applied Sciences 28, no. 08 (July 2018): 1453–80. http://dx.doi.org/10.1142/s0218202518500392.

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In this paper, we develop a function-based a posteriori error estimators for the solution of linear second-order elliptic problems considering hierarchical spline spaces for the Galerkin discretization. We obtain a global upper bound for the energy error over arbitrary hierarchical mesh configurations which simplifies the implementation of adaptive refinement strategies. The theory hinges on some weighted Poincaré-type inequalities where the B-spline basis functions are the weights appearing in the norms. Such inequalities are derived following the lines in [A. Veeser and R. Verfürth, Explicit upper bounds for dual norms of residuals, SIAM J. Numer. Anal. 47 (2009) 2387–2405], where the case of standard finite elements is considered. Additionally, we present numerical experiments that show the efficiency of the error estimators independently of the degree of the splines used for the discretization, together with an adaptive algorithm guided by these local estimators that yields optimal meshes and rates of convergence, exhibiting an excellent performance.
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Sabawi, Younis A. "Posteriori Error bound For Fullydiscrete Semilinear Parabolic Integro-Differential equations." Journal of Physics: Conference Series 1999, no. 1 (September 1, 2021): 012085. http://dx.doi.org/10.1088/1742-6596/1999/1/012085.

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GREPL, MARTIN A. "CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS." Mathematical Models and Methods in Applied Sciences 22, no. 03 (March 2012): 1150015. http://dx.doi.org/10.1142/s0218202511500151.

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We present reduced basis approximations and associated a posteriori error bounds for parabolic partial differential equations involving (i) a nonaffine dependence on the parameter and (ii ) a nonlinear dependence on the field variable. The method employs the Empirical Interpolation Method in order to construct "affine" coefficient-function approximations of the "nonaffine" (or nonlinear) parametrized functions. We consider linear time-invariant as well as linear time-varying nonaffine functions and introduce a new sampling approach to generate the function approximation space for the latter case. Our a posteriori error bounds take both error contributions explicitly into account — the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. We show that these bounds are rigorous upper bounds for the approximation error under certain conditions on the function interpolation, thus addressing the demand for certainty of the approximation. As regards efficiency, we develop an offline–online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for the many-query or real-time contexts. Numerical results are presented to confirm and test our approach.
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Dissertations / Theses on the topic "A posteriori error bound"

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Kunert, Gerd, Zoubida Mghazli, and Serge Nicaise. "A posteriori error estimation for a finite volume discretization on anisotropic meshes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601352.

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A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using \emph{anisotropic meshes} which can improve the accuracy of the discretization considerably. The main focus is on \emph{a posteriori} error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient \emph{a posteriori} error estimation is achieved for the finite volume method on anisotropic meshes.
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Rankin, Richard Andrew Robert. "Fully computable a posteriori error bounds for noncomforming and discontinuous galekin finite elemant approximation." Thesis, University of Strathclyde, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.501776.

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We obtain fully computable constant free a posteriori error bounds on the broken energy seminorm of the error in nonconforming and discontinuous Galerkin finite element approximations of a linear second ore elliptic problem on meshes omprised of triangular elements. We do this for nonconforming finite element approximations of uniform arbitrary order as well as for non-uniform order symmetric interior penalty Galerkin, non-symmetric interior penalty Galerkin and ncomplete interior penalty Galerkin finite element approximations.
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Merdon, Christian. "Aspects of guaranteed error control in computations for partial differential equations." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://dx.doi.org/10.18452/16818.

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Diese Arbeit behandelt garantierte Fehlerkontrolle für elliptische partielle Differentialgleichungen anhand des Poisson-Modellproblems, des Stokes-Problems und des Hindernisproblems. Hierzu werden garantierte obere Schranken für den Energiefehler zwischen exakter Lösung und diskreten Finite-Elemente-Approximationen erster Ordnung entwickelt. Ein verallgemeinerter Ansatz drückt den Energiefehler durch Dualnormen eines oder mehrerer Residuen aus. Hinzu kommen berechenbare Zusatzterme, wie Oszillationen der gegebenen Daten, mit expliziten Konstanten. Für die Abschätzung der Dualnormen der Residuen existieren viele verschiedene Techniken. Diese Arbeit beschäftigt sich vorrangig mit Equilibrierungsschätzern, basierend auf Raviart-Thomas-Elementen, welche effiziente garantierte obere Schranken ermöglichen. Diese Schätzer werden mit einem Postprocessing-Verfahren kombiniert, das deren Effizienz mit geringem zusätzlichen Rechenaufwand deutlich verbessert. Nichtkonforme Finite-Elemente-Methoden erzeugen zusätzlich ein Inkonsistenzresiduum, dessen Dualnorm mit Hilfe diverser konformer Approximationen abgeschätzt wird. Ein Nebenaspekt der Arbeit betrifft den expliziten residuen-basierten Fehlerschätzer, der für gewöhnlich optimale und leicht zu berechnende Verfeinerungsindikatoren für das adaptive Netzdesign liefert, aber nur schlechte garantierte obere Schranken. Eine neue Variante, die auf den equilibrierten Flüssen des Luce-Wohlmuth-Fehlerschätzers basiert, führt zu stark verbesserten Zuverlässigkeitskonstanten. Eine Vielzahl numerischer Experimente vergleicht alle implementierten Fehlerschätzer und zeigt, dass effiziente und garantierte Fehlerkontrolle in allen vorliegenden Modellproblemen möglich ist. Insbesondere zeigt ein Modellproblem, wie die Fehlerschätzer erweitert werden können, um auch auf Gebieten mit gekrümmten Rändern garantierte obere Schranken zu liefern.
This thesis studies guaranteed error control for elliptic partial differential equations on the basis of the Poisson model problem, the Stokes equations and the obstacle problem. The error control derives guaranteed upper bounds for the energy error between the exact solution and different finite element discretisations, namely conforming and nonconforming first-order approximations. The unified approach expresses the energy error by dual norms of one or more residuals plus computable extra terms, such as oscillations of the given data, with explicit constants. There exist various techniques for the estimation of the dual norms of such residuals. This thesis focuses on equilibration error estimators based on Raviart-Thomas finite elements, which permit efficient guaranteed upper bounds. The proposed postprocessing in this thesis considerably increases their efficiency at almost no additional computational costs. Nonconforming finite element methods also give rise to a nonconsistency residual that permits alternative treatment by conforming interpolations. A side aspect concerns the explicit residual-based error estimator that usually yields cheap and optimal refinement indicators for adaptive mesh refinement but not very sharp guaranteed upper bounds. A novel variant of the residual-based error estimator, based on the Luce-Wohlmuth equilibration design, leads to highly improved reliability constants. A large number of numerical experiments compares all implemented error estimators and provides evidence that efficient and guaranteed error control in the energy norm is indeed possible in all model problems under consideration. Particularly, one model problem demonstrates how to extend the error estimators for guaranteed error control on domains with curved boundary.
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Camacho, Fernando F. "A Posteriori Error Estimates for Surface Finite Element Methods." UKnowledge, 2014. http://uknowledge.uky.edu/math_etds/21.

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Problems involving the solution of partial differential equations over surfaces appear in many engineering and scientific applications. Some of those applications include crystal growth, fluid mechanics and computer graphics. Many times analytic solutions to such problems are not available. Numerical algorithms, such as Finite Element Methods, are used in practice to find approximate solutions in those cases. In this work we present L2 and pointwise a posteriori error estimates for Adaptive Surface Finite Elements solving the Laplace-Beltrami equation −△Γ u = f . The two sources of errors for Surface Finite Elements are a Galerkin error, and a geometric error that comes from replacing the original surface by a computational mesh. A posteriori error estimates on flat domains only have a Galerkin component. We use residual type error estimators to measure the Galerkin error. The geometric component of our error estimate becomes zero if we consider flat domains, but otherwise has the same order as the residual one. This is different from the available energy norm based error estimates on surfaces, where the importance of the geometric components diminishes asymptotically as the mesh is refined. We use our results to implement an Adaptive Surface Finite Element Method. An important tool for proving a posteriori error bounds for non smooth functions is the Scott-Zhang interpolant. A refined version of a standard Scott-Zhang interpolation bound is also proved during our analysis. This local version only requires the interpolated function to be in a Sobolev space defined over an element T instead of an element patch containing T. In the last section we extend our elliptic results to get estimates for the surface heat equation ut − △Γ u = f using the elliptic reconstruction technique.
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Ainsworth, Mark. "A posteriori error estimation in the finite element method." Thesis, Durham University, 1989. http://etheses.dur.ac.uk/6326/.

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The work broadly consists of two parts. In the first part we construct a framework for analyzing and developing a posteriori error estimators for use in the finite element solution of elliptic partial differential equations which have smooth solutions. The analysis makes use of complementary variational principles and the superconvergence phenomenon associated with the finite element method. The second part generalizes these results to the important case when the solution of the boundary value problem contains singularities. It is shown how the classical techniques may be easily modified to perform satisfactorily for the singular case.
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Köhler, Karoline Sophie. "On efficient a posteriori error analysis for variational inequalities." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2016. http://dx.doi.org/10.18452/17635.

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Effiziente und zuverlässige a posteriori Fehlerabschätzungen sind eine Hauptzutat für die effiziente numerische Berechnung von Lösungen zu Variationsungleichungen durch die Finite-Elemente-Methode. Die vorliegende Arbeit untersucht zuverlässige und effiziente Fehlerabschätzungen für beliebige Finite-Elemente-Methoden und drei Variationsungleichungen, nämlich dem Hindernisproblem, dem Signorini Problem und dem Bingham Problem in zwei Raumdimensionen. Die Fehlerabschätzungen hängen vom zum Problem gehörenden Lagrange Multiplikator ab, der eine Verbindung zwischen der Variationsungleichung und dem zugehörigen linearen Problem darstellt. Effizienz und Zuverlässigkeit werden bezüglich eines totalen Fehlers gezeigt. Die Fehleranschätzungen fordern minimale Regularität. Die Approximation der exakten Lösung erfüllt die Dirichlet Randbedingungen und die Approximation des Lagrange Multiplikators ist nicht-positiv im Falle des Hindernis- und Signoriniproblems, und hat Betrag kleiner gleich 1 für das Bingham Problem. Dieses allgemeine Vorgehen ermöglicht das Einbinden nicht-exakter diskreter Lösungen, welche im Kontext dieser Ungleichungen auftreten. Aus dem Blickwinkel der Anwendungen ist Effizienz und Zuverlässigkeit im Bezug auf den Fehler der primalen Variablen in der Energienorm von großem Interesse. Solche Abschätzungen hängen von der Wahl eines effizienten diskreten Lagrange Multiplikators ab. Im Falle des Hindernis- und Signorini Problems werden postive Beispiele für drei Finite-Elemente Methoden, der konformen Courant Methode, der nicht-konformen Crouzeix-Raviart Methode und der gemischten Raviart-Thomas Methode niedrigster Ordnung hergeleitet. Partielle Resultate liegen im Fall des Bingham Problems vor. Numerischer Experimente heben die theoretischen Ergebnisse hervor und zeigen Effizienz und Zuverlässigkeit. Die numerischen Tests legen nahe, dass der aus den Abschätzungen resultierende adaptive Algorithmus mit optimaler Konvergenzrate konvergiert.
Efficient and reliable a posteriori error estimates are a key ingredient for the efficient numerical computation of solutions for variational inequalities by the finite element method. This thesis studies such reliable and efficient error estimates for arbitrary finite element methods and three representative variational inequalities, namely the obstacle problem, the Signorini problem, and the Bingham problem in two space dimensions. The error estimates rely on a problem connected Lagrange multiplier, which presents a connection between the variational inequality and the corresponding linear problem. Reliability and efficiency are shown with respect to some total error. Reliability and efficiency are shown under minimal regularity assumptions. The approximation to the exact solution satisfies the Dirichlet boundary conditions, and an approximation of the Lagrange multiplier is non-positive in the case of the obstacle and Signorini problem and has an absolute value smaller than 1 for the Bingham flow problem. These general assumptions allow for reliable and efficient a posteriori error analysis even in the presence of inexact solve, which naturally occurs in the context of variational inequalities. From the point of view of the applications, reliability and efficiency with respect to the error of the primal variable in the energy norm is of great interest. Such estimates depend on the efficient design of a discrete Lagrange multiplier. Affirmative examples of discrete Lagrange multipliers are presented for the obstacle and Signorini problem and three different first-order finite element methods, namely the conforming Courant, the non-conforming Crouzeix-Raviart, and the mixed Raviart-Thomas FEM. Partial results exist for the Bingham flow problem. Numerical experiments highlight the theoretical results, and show efficiency and reliability. The numerical tests suggest that the resulting adaptive algorithms converge with optimal convergence rates.
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Chow, Chak-On 1968. "On a posteriori finite element bound procedures for nonsymmetric Eigenvalue problems." Thesis, Massachusetts Institute of Technology, 1999. http://hdl.handle.net/1721.1/85266.

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Pled, Florent. "Vers une stratégie robuste et efficace pour le contrôle des calculs par éléments finis en ingénierie mécanique." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2012. http://tel.archives-ouvertes.fr/tel-00776633.

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Ce travail de recherche vise à contribuer au développement de nouveaux outils d'estimation d'erreur globale et locale en ingénierie mécanique. Les estimateurs d'erreur globale étudiés reposent sur le concept d'erreur en relation de comportement à travers des techniques spécifiques de construction de champs admissibles, assurant l'aspect conservatif ou garanti de l'estimation. Une nouvelle méthode de construction de champs admissibles est mise en place et comparée à deux autres méthodes concurrentes, en matière de précision, coût de calcul et facilité d'implémentation dans les codes éléments finis. Une amélioration de cette nouvelle méthode hybride fondée sur une minimisation locale de l'énergie complémentaire est également proposée. Celle-ci conduit à l'introduction et à l'élaboration de critères géométriques et énergétiques judicieux, permettant un choix approprié des régions à sélectionner pour améliorer localement la qualité des champs admissibles. Dans le cadre des estimateurs d'erreur locale basés sur l'utilisation conjointe des outils d'extraction et des estimateurs d'erreur globale, deux nouvelles techniques d'encadrement de l'erreur en quantité d'intérêt sont proposées. Celles-ci sont basées sur le principe de Saint-Venant à travers l'emploi de propriétés spécifiques d'homothétie, afin d'améliorer la précision des bornes d'erreur locale obtenues à partir de la technique d'encadrement classique fondée sur l'inégalité de Cauchy-Schwarz. Les diverses études comparatives sont menées dans le cadre des problèmes d'élasticité linéaire en quasi-statique. Le comportement des différents estimateurs d'erreur est illustré et discuté sur des exemples numériques tirés d'applications industrielles. Les travaux réalisés constituent des éléments de réponse à la problématique de la vérification dans un contexte industriel.
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Apel, Thomas, and Cornelia Pester. "Clément-type interpolation on spherical domains - interpolation error estimates and application to a posteriori error estimation." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601335.

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In this paper, a mixed boundary value problem for the Laplace-Beltrami operator is considered for spherical domains in $R^3$, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (\varphi, \theta), such that functions on the unit sphere are considered as functions in these coordinates. Careful investigation leads to the introduction of a proper finite element space corresponding to an isotropic triangulation of the underlying domain on the unit sphere. Error estimates are proven for a Clément-type interpolation operator, where appropriate, weighted norms are used. The estimates are applied to the deduction of a reliable and efficient residual error estimator for the Laplace-Beltrami operator.
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Kunert, Gerd. "A posteriori error estimation for convection dominated problems on anisotropic meshes." Universitätsbibliothek Chemnitz, 2002. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200200255.

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A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.
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Books on the topic "A posteriori error bound"

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Ainsworth, Mark, and J. Tinsley Oden. A Posteriori Error Estimation in Finite Element Analysis. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2000. http://dx.doi.org/10.1002/9781118032824.

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A posteriori estimates for partial differential equations. Berlin: Walter de Gruyter, 2008.

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Phillips, M. R. Some a posteriori error estimates for elliptic partial differential equations. Manchester: UMIST, 1997.

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Kunert, Gerd. Advances in a posteriori error estimation on anisotropic finite element discretizations. Berlin: Logos, 2003.

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I, Repin Sergey, ed. Reliable methods for computer simulation: Error control and a posteriori estimates. Amsterdam: Elsevier, 2004.

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Verfürth, Rüdiger. A review of a posteriori error estimation and adaptive mesh-refinement techniques. Chichester: Wiley-Teubner, 1996.

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Han, Weimin. Posteriori error analysis via duality theory: With applications in modeling and numerical ... [S.l.]: Springer, 2004.

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Pester, Cornelia. A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3D vertex singularities. Berlin: Logos-Verl., 2006.

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United States. National Aeronautics and Space Administration. Scientific and Technical Information Division., ed. Model reduction by trimming for a class of semi-Markov reliability models and the corresponding error bound. [Washington, DC]: National Aeronautics and Space Administration, Office of Management, Scientific and Technical Information Division, 1991.

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A Posteriori Error Analysis via Duality Theory. Boston: Kluwer Academic Publishers, 2005. http://dx.doi.org/10.1007/b101775.

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Book chapters on the topic "A posteriori error bound"

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Xuan, Z. C., K. H. Lee, and J. Peraire. "A Posteriori Output Bound for Partial Differential Equations Based on Elemental Error Bound Computing." In Computational Science and Its Applications — ICCSA 2003, 1035–44. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/3-540-44839-x_109.

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Patera, Anthony T., and Jaume Peraire. "A General Lagrangian Formulation for the Computation of A Posteriori Finite Element Bounds." In Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, 159–206. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/978-3-662-05189-4_4.

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Georgoulis, Emmanuil H., and Omar Lakkis. "A Posteriori Error Bounds for Discontinuous Galerkin Methods for Quasilinear Parabolic Problems." In Numerical Mathematics and Advanced Applications 2009, 351–58. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-11795-4_37.

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Korneev, Vadim G. "On a Renewed Approach to A Posteriori Error Bounds for Approximate Solutions of Reaction-Diffusion Equations." In Lecture Notes in Computational Science and Engineering, 221–45. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-14244-5_12.

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Sauter, Stefan A., and Christoph Schwab. "A Posteriori Error Estimation." In Boundary Element Methods, 517–44. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-540-68093-2_9.

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Li, Jichun, and Yunqing Huang. "A Posteriori Error Estimation." In Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials, 173–93. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-33789-5_6.

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Ern, Alexandre, and Jean-Luc Guermond. "A posteriori error analysis." In Texts in Applied Mathematics, 141–56. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-56923-5_34.

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Rüter, Marcus Olavi. "Energy Norm A Posteriori Error Estimates." In Error Estimates for Advanced Galerkin Methods, 171–278. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-06173-9_6.

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Oneto, Luca. "Compression Bound." In Model Selection and Error Estimation in a Nutshell, 59–63. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-24359-3_6.

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Verfürth, R. "The Equivalence of A Posteriori Error Estimators." In Notes on Numerical Fluid Mechanics (NNFM), 273–83. Wiesbaden: Vieweg+Teubner Verlag, 1995. http://dx.doi.org/10.1007/978-3-663-14125-9_23.

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Conference papers on the topic "A posteriori error bound"

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Choi, Hae-Won, and Marius Paraschivoiu. "A-Posteriori Finite Element Bound and Adaptive Discretization Methods for the Electro-Osmotic Flows in Heterogeneous Microchannels." In ASME 3rd International Conference on Microchannels and Minichannels. ASMEDC, 2005. http://dx.doi.org/10.1115/icmm2005-75113.

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Numerical simulation based a design tool devised by an aposteriori error estimation technique termed the ‘bound method’ is applied in this paper to examine and analyze fluidic flow and species transport phenomena of the electro-osmotic flow in a heterogeneously charged T-shaped micro-mixer. This novel technique provides fast, inexpensive and reliable bounds to the ‘output’ for the slip velocity model of the electro-osmotic flow. The bound method presented here-in is extended to use of an adaptive refinement to generate the ‘ideal’ mesh for computations, the direct equilibration to evaluate the ‘hybrid-flux’ very efficiently, and a parallel calculation to accelerate the error estimates of the output of interest.
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II Hong, Bum, Intae Ryoo, and Gon Khang. "On a Posteriori Error Bounds of Trapezoidal Rule." In 2019 International Conference on Information Networking (ICOIN). IEEE, 2019. http://dx.doi.org/10.1109/icoin.2019.8718134.

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Barth, Tim. "An Overview of Combined Uncertainty and A-Posteriori Error Bound Estimates for CFD Calculations." In 54th AIAA Aerospace Sciences Meeting. Reston, Virginia: American Institute of Aeronautics and Astronautics, 2016. http://dx.doi.org/10.2514/6.2016-1062.

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Rozza, G., C. N. Nguyen, A. T. Patera, and S. Deparis. "Reduced Basis Methods and a Posteriori Error Estimators for Heat Transfer Problems." In ASME 2009 Heat Transfer Summer Conference collocated with the InterPACK09 and 3rd Energy Sustainability Conferences. ASMEDC, 2009. http://dx.doi.org/10.1115/ht2009-88211.

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This paper focuses on the parametric study of steady and unsteady forced and natural convection problems by the certified reduced basis method. These problems are characterized by an input-output relationship in which given an input parameter vector — material properties, boundary conditions and sources, and geometry — we would like to compute certain outputs of engineering interest — heat fluxes and average temperatures. The certified reduced basis method provides both (i) a very inexpensive yet accurate output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element discretization. The feasibility and efficiency of the method is demonstrated for three natural convection model problems: a scalar steady forced convection problem in a rectangular channel is characterized by two parameters — Pe´clet number and the aspect ratio of the channel — and an output — the average temperature over the domain; a steady natural convection problem in a laterally heated cavity is characterized by three parameters — Grashof and Prandtl numbers, and the aspect ratio of the cavity — and an output — the inverse of the Nusselt number; and an unsteady natural convection problem in a laterally heated cavity is characterized by two parameters — Grashof and Prandtl numbers — and a time-dependent output — the average of the horizontal velocity over a specified area of the cavity.
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Sabawi, Younis A. "A Posteriori $L_{\infty}(H^{1})$ Error Bound in Finite Element Approximation of Semdiscrete Semilinear Parabolic Problems." In 2019 First International Conference of Computer and Applied Sciences (CAS). IEEE, 2019. http://dx.doi.org/10.1109/cas47993.2019.9075699.

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Gerner, Anna-Lena, Karen Veroy, Theodore E. Simos, George Psihoyios, and Ch Tsitouras. "Reduced Basis A Posteriori Error Bounds for the Stokes Equations in Parametrized Domains: A Penalty Approach." In ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010. AIP, 2010. http://dx.doi.org/10.1063/1.3498348.

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Veroy, Karen, Christophe Prud'homme, Dimitrios Rovas, and Anthony Patera. "A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations." In 16th AIAA Computational Fluid Dynamics Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2003. http://dx.doi.org/10.2514/6.2003-3847.

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Zhang, Jun Jason, Wenfan Zhou, Narayan Kovvali, Antonia Papandreou-Suppappola, and Aditi Chattopadhyay. "On the Use of the Posterior Crame´r-Rao Lower Bound for Damage Estimation in Structural Health Management." In ASME 2009 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. ASMEDC, 2009. http://dx.doi.org/10.1115/smasis2009-1454.

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The use of the posterior Crame´r-Rao lower bound (PCRLB) as a lower bound for the mean-squared estimation error (MSEE) of progressive damage is investigated. The estimation problem is formulated in terms of a stochastic dynamic system model that describes the random evolution of damage and provides measurement uncertainty. Based on whether the system is linear or nonlinear, sequential Monte Carlo techniques are used to approximate the posterior probability density function and thus obtain the damage state estimate. The resulting MSEE is compared to the lower bound offered by the PCRLB that is obtained from the implied state transition probability density function and the measurement likelihood function. The progressive estimation results and the PCRLB are demonstrated for fatigue crack estimation in an aluminum compact-tension (CT) sample subjected to variable-amplitude loading.
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Veeser, A., and C. Kreuzer. "Oscillation in a Posteriori Error Estimation." In 10th International Conference on Adaptative Modeling and Simulation. CIMNE, 2021. http://dx.doi.org/10.23967/admos.2021.067.

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Zhang, X., J. Y. Trepanier, R. Camarero, X. Zhang, J. Y. Trepanier, and R. Camarero. "An a posteriori error estimation method based on error equations." In 13th Computational Fluid Dynamics Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 1997. http://dx.doi.org/10.2514/6.1997-1889.

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Reports on the topic "A posteriori error bound"

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Rabier, Patrick J. A Posteriori Error Estimation New" Approach". Fort Belvoir, VA: Defense Technical Information Center, September 1994. http://dx.doi.org/10.21236/ada284960.

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Babuska, I., l. Plank, and R. Rodriguez. Basic Problems of A-Posteriori Error Estimation. Fort Belvoir, VA: Defense Technical Information Center, February 1992. http://dx.doi.org/10.21236/ada248986.

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Babuska, I., T. Strouboulis, C. S. Upadhyay, S. K. Gangaraj, and K. Copps. Validation of A-Posteriori Error Estimators by Numerical Approach. Fort Belvoir, VA: Defense Technical Information Center, June 1993. http://dx.doi.org/10.21236/ada269493.

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Wildey, Timothy Michael, Eric C. Cyr, Roger Patrick Pawlowski, John Nicolas Shadid, and Thomas Michael Smith. Adjoint based a posteriori error estimation in Drekar::CFD. Office of Scientific and Technical Information (OSTI), October 2012. http://dx.doi.org/10.2172/1055892.

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Paulino, G. H., L. J. Gray, and V. Zarikian. A posteriori pointwise error estimates for the boundary element method. Office of Scientific and Technical Information (OSTI), January 1995. http://dx.doi.org/10.2172/42836.

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Romkes, A., S. Prudhomme, and J. T. Oden. A Posteriori Error Estimation for a New Stabilized Discontinuous Galerkin Method. Fort Belvoir, VA: Defense Technical Information Center, August 2002. http://dx.doi.org/10.21236/ada438102.

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Babuska, Ivo, Lothar Plank, and Rodolfo Rodriguez. Quality Assessment of the A-posteriori Error Estimation in Finite Elements. Fort Belvoir, VA: Defense Technical Information Center, August 1991. http://dx.doi.org/10.21236/ada254767.

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Manzini, Gianmarco, and Lourenco Beirao da Veiga. Residual a posteriori error estimation of a mimetic/virtual element method. Office of Scientific and Technical Information (OSTI), January 2013. http://dx.doi.org/10.2172/1054671.

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El sakori, Ahmed. A Posteriori Error Estimates for Maxwell's Equations Using Auxiliary Subspace Techniques. Portland State University Library, January 2000. http://dx.doi.org/10.15760/etd.7471.

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Bai, Z. D. Exponential Bound for Error Probability in NN-Discrimination. Fort Belvoir, VA: Defense Technical Information Center, April 1985. http://dx.doi.org/10.21236/ada160305.

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