Relevant bibliographies by topics / Raviart-Thomas element

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Academic literature on the topic 'Raviart-Thomas element'

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Published: 6 September 2023
Last updated: 7 September 2023

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Journal articles on the topic "Raviart-Thomas element"

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Bartels, Sören, and Zhangxian Wang. "Orthogonality relations of Crouzeix–Raviart and Raviart–Thomas finite element spaces." Numerische Mathematik 148, no. 1 (May 2021): 127–39. http://dx.doi.org/10.1007/s00211-021-01199-3.

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AbstractIdentities that relate projections of Raviart–Thomas finite element vector fields to discrete gradients of Crouzeix–Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix–Ravaiart space onto elementwise constant functions are deduced.
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Kobayashi, Kenta, and Takuya Tsuchiya. "Error analysis of Crouzeix–Raviart and Raviart–Thomas finite element methods." Japan Journal of Industrial and Applied Mathematics 35, no. 3 (September 6, 2018): 1191–211. http://dx.doi.org/10.1007/s13160-018-0325-9.

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Lu, Zuliang, Yanping Chen, and Weishan Zheng. "A Posteriori Error Estimates of Lowest Order Raviart-Thomas Mixed Finite Element Methods for Bilinear Optimal Control Problems." East Asian Journal on Applied Mathematics 2, no. 2 (May 2012): 108–25. http://dx.doi.org/10.4208/eajam.130212.300312a.

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AbstractA Raviart-Thomas mixed finite element discretization for general bilinear optimal control problems is discussed. The state and co-state are approximated by lowest order Raviart-Thomas mixed finite element spaces, and the control is discretized by piecewise constant functions. A posteriori error estimates are derived for both the coupled state and the control solutions, and the error estimators can be used to construct more efficient adaptive finite element approximations for bilinear optimal control problems. An adaptive algorithm to guide the mesh refinement is also provided. Finally, we present a numerical example to demonstrate our theoretical results.
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Carstensen, Carsten. "Explicit Error Estimates for Courant, Crouzeix-Raviart and Raviart-Thomas Finite Element Methods." Journal of Computational Mathematics 30, no. 4 (June 2012): 337–53. http://dx.doi.org/10.4208/jcm.1108-m3677.

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VOHRALÍK, MARTIN, and BARBARA I. WOHLMUTH. "MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS." Mathematical Models and Methods in Applied Sciences 23, no. 05 (February 21, 2013): 803–38. http://dx.doi.org/10.1142/s0218202512500613.

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In this paper, we study the mixed finite element method for linear diffusion problems. We focus on the lowest-order Raviart–Thomas case. For simplicial meshes, we propose several new approaches to reduce the original indefinite saddle point systems for the flux and potential unknowns to (positive definite) systems for one potential unknown per element. Our construction principle is closely related to that of the so-called multi-point flux-approximation method and leads to local flux expressions. We present a set of numerical examples illustrating the influence of the elimination process on the structure and on the condition number of the reduced matrix. We also discuss different versions of the discrete maximum principle in the lowest-order Raviart–Thomas method. Finally, we recall mixed finite element methods on general polygonal meshes and show that they are a special type of the mimetic finite difference, mixed finite volume, and hybrid finite volume family.
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Dubois, Francois, Isabelle Greff, and Charles Pierre. "Raviart–Thomas finite elements of Petrov–Galerkin type." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 5 (August 6, 2019): 1553–76. http://dx.doi.org/10.1051/m2an/2019020.

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Finite volume methods are widely used, in particular because they allow an explicit and local computation of a discrete gradient. This computation is only based on the values of a given scalar field. In this contribution, we wish to achieve the same goal in a mixed finite element context of Petrov–Galerkin type so as to ensure a local computation of the gradient at the interfaces of the elements. The shape functions are the Raviart–Thomas finite elements. Our purpose is to define test functions that are in duality with these shape functions: precisely, the shape and test functions will be asked to satisfy some orthogonality property. This paradigm is addressed for the discrete solution of the Poisson problem. The general theory of Babuška brings necessary and sufficient stability conditions for a Petrov–Galerkin mixed problem to be convergent. In order to ensure stability, we propose specific constraints for the dual test functions. With this choice, we prove that the mixed Petrov–Galerkin scheme is identical to the four point finite volume scheme of Herbin, and to the mass lumping approach developed by Baranger, Maitre and Oudin. Convergence is proven with the usual techniques of mixed finite elements.
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Braess, D., and R. Verfürth. "A Posteriori Error Estimators for the Raviart–Thomas Element." SIAM Journal on Numerical Analysis 33, no. 6 (December 1996): 2431–44. http://dx.doi.org/10.1137/s0036142994264079.

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Huang, Peiqi, Jinru Chen, and Mingchao Cai. "A Mortar Method Using Nonconforming and Mixed Finite Elements for the Coupled Stokes-Darcy Model." Advances in Applied Mathematics and Mechanics 9, no. 3 (January 17, 2017): 596–620. http://dx.doi.org/10.4208/aamm.2016.m1397.

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AbstractIn this work, we study numerical methods for a coupled fluid-porous media flow model. The model consists of Stokes equations and Darcy's equations in two neighboring subdomains, coupling together through certain interface conditions. The weak form for the coupled model is of saddle point type. A mortar finite element method is proposed to approximate the weak form of the coupled problem. In our method, nonconforming Crouzeix-Raviart elements are applied in the fluid subdomain and the lowest order Raviart-Thomas elements are applied in the porous media subdomain; Meshes in different subdomains are allowed to be nonmatching on the common interface; Interface conditions are weakly imposed via adding constraint in the definition of the finite element space. The well-posedness of the discrete problem and the optimal error estimate for the proposed method are established. Numerical experiments are also given to confirm the theoretical results.
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Lashuk, I. V., and P. S. Vassilevski. "Element agglomeration coarse Raviart-Thomas spaces with improved approximation properties." Numerical Linear Algebra with Applications 19, no. 2 (January 13, 2012): 414–26. http://dx.doi.org/10.1002/nla.1819.

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Lu, Zuliang. "Adaptive Mixed Finite Element Methods for Parabolic Optimal Control Problems." Mathematical Problems in Engineering 2011 (2011): 1–21. http://dx.doi.org/10.1155/2011/217493.

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We will investigate the adaptive mixed finite element methods for parabolic optimal control problems. The state and the costate are approximated by the lowest-order Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise constant elements. We derive a posteriori error estimates of the mixed finite element solutions for optimal control problems. Such a posteriori error estimates can be used to construct more efficient and reliable adaptive mixed finite element method for the optimal control problems. Next we introduce an adaptive algorithm to guide the mesh refinement. A numerical example is given to demonstrate our theoretical results.
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Dissertations / Theses on the topic "Raviart-Thomas element"

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Dib, Serena. "Méthodes d'éléments finis pour le problème de Darcy couplé avec l'équation de la chaleur." Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066294/document.

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Dans cette thèse, nous étudions l'équation de la chaleur couplée avec la loi de Darcy à travers de la viscosité non-linéaire qui dépend de la température pour les dimensions d=2,3 (Hooman et Gurgenci ou Rashad). Nous analysons ce problème en introduisant la formulation variationnelle équivalente et en la réduisant à une simple équation de diffusion-convection pour la température où la vitesse dépend implicitement de la température.Nous démontrons l'existence de la solution sans la restriction sur les données par la méthode de Galerkin et du point fixe de Brouwer. L'unicité globale est établie une fois la solution est légèrement régulière et les données se restreignent convenablement. Nous introduisons aussi une formulation variationnelle alternative équivalente. Toutes les deux formulations variationnelles sont discrétisées par quatre schémas d'éléments finis pour un domaine polygonal ou polyédrique. Nous dérivons l'existence, l'unicité conditionnée, la convergence et l'estimation d'erreur a priori optimale pour les solutions des trois schémas. Par la suite, ces schémas sont linéarisés par des algorithmes d'approximation successifs et convergentes. Nous présentons quelques expériences numériques pour un problème modèle qui confirme les résultats théoriques de convergence développées dans ce travail. L'estimation d'erreur a posteriori est établie avec deux types d'indicateurs d'erreur de linéarisation et de discrétisation. Enfin, nous montrons des résultats numériques de validation
In this thesis, we study the heat equation coupled with Darcy's law by a nonlinear viscosity depending on the temperature in dimension d=2,3 (Hooman and Gurgenci or Rashad). We analyse this problem by setting it in an equivalent variational formulation and reducing it to an diffusion-convection equation for the temperature where the velocity depends implicitly on the temperature.Existence of a solution is derived without restriction on the data by Galerkin's method and Brouwer's Fixed Point. Global uniqueness is established when the solution is slightly smoother and the dataare suitably restricted. We also introduce an alternative equivalent variational formulation. Both variational formulations are discretized by four finite element schemes in a polygonal or polyhedral domain. We derive existence, conditional uniqueness, convergence, and optimal a priori error estimates for the solutions of the three schemes. Next, these schemes are linearized by suitable convergent successive approximation algorithms. We present some numerical experiments for a model problem that confirm the theoretical rates of convergence developed in this work. A posteriori error estimates are established with two types of errors indicators related to the linearisation and discretization. Finally, we show numerical results of validation
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Nguyen, Cong Uy. "Hybrid stress visco-plasticity : formulation, discrete approximation, and stochastic identification." Thesis, Compiègne, 2022. http://www.theses.fr/2022COMP2695.

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Dans cette thèse, une nouvelle approche est développée pour les problèmes de viscoplasticité et de dynamique non linéaire. En particulier, les équations variationnelles sont élaborées selon le principe de Helligner-Reissner, de sorte que les champs de contrainte et de déplacement apparaissent comme des champs inconnus sous la forme faible. Trois nouveaux éléments finis sont développés. Le premier élément fini est formulé pour le problème axisymétrique, dans lequel le champ de contraintes est approximé par des polynômes d’ordre inférieur tels que des fonctions linéaires. Cette approche donne des solutions précises spécifiquement dans les problèmes incompressibles et rigides. De plus, un élément fini de flexion de membrane et de plaque est nouvellement conçu en discrétisant le champ de contraintes en utilisant l’espace vectoriel de Raviart-Thomas d’ordre le plus bas RT0. Cette approche garantit la continuité du champ de contraintes sur tout un domaine discret, ce qui est un avantage significatif dans la méthode numérique, notamment pour les problèmes de propagation des ondes. Les développements sont effectués pour le comportement constitutif visco-plastique des matériaux, où les équations d’évolution correspondantes sont obtenues en faisant appel au principe de dissipation maximale. Pour résoudre les équations d’équilibre dynamique, des schémas de conservation et de décroissance de l’énergie sont formulés en conséquence. Le schéma de conservation de l’énergie est inconditionnellement stable, car il peut préserver l’énergie totale d’un système donné sous une vibration libre, tandis que le schéma décroissant peut dissiper des modes de vibration à plus haute fréquence. La dernière partie de cette thèse présente les procédures d’upscaling du comportement des matériaux visco-plastiques. Plus précisément, la mise à l’échelle est effectuée par une méthode d’identification stochastique via une mise à jour baysienne en utilisant le filtre de Gauss-Markov-Kalman pour l’assimilation des propriétés importantes des matériaux dans les régimes élastique et inélastique
In this thesis, a novel approach is developed for visco-plasticity and nonlinear dynamics problems. In particular, variational equations are elaborated following the Helligner-Reissner principle, so that both stress and displacement fields appear as unknown fields in the weak form. Three novel finite elements are developed. The first finite element is formulated for the axisymmetric problem, in which the stress field is approximated by low-order polynomials such as linear functions. This approach yields accurate solutions specifically in incompressible and stiff problems. In addition, a membrane and plate bending finite element are newly designed by discretizing the stress field using the lowest order Raviart-Thomas vector space RT0. This approach guarantees the continuity of the stress field over an entire discrete domain, which is a significant advantage in the numerical method, especially for the wave propagation problems. The developments are carried out for the viscoplastic constitutive behavior of materials, where the corresponding evolution equations are obtained by appealing to the principle of maximum dissipation. To solve the dynamic equilibrium equations, energy conserving and decaying schemes are formulated correspondingly. The energy conserving scheme is unconditional stable, since it can preserve the total energy of a given system under a free vibration, while the decaying scheme can dissipate higher frequency vibration modes. The last part of this thesis presents procedures for upscaling of the visco-plastic material behavior. Specifically, the upscaling is performed by stochastic identification method via Baysian updating using the Gauss-Markov-Kalman filter for assimilation of important material properties in the elastic and inelastic regimes
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Bertrand, Fleurianne [Verfasser]. "Approximated flux boundary conditions for Raviart-Thomas finite elements on domains with curved boundaries and applications to first-order system least squares / Fleurianne Bertrand." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2014. http://d-nb.info/1063982103/34.

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Biswas, Rahul. "Local Projection Stabilization Methods for the Oseen Problem." Thesis, 2022. https://etd.iisc.ac.in/handle/2005/6067.

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Finite element approximation of fluid flow problems with dominant convection exhibit spurious oscillations. To eliminate these nonphysical oscillations one needs to incorporate stabilizations that can curb the effect of convection. The main aim of this thesis is to design and analyse local projection stabilization based finite element schemes for the Oseen problem. In chapter \ref{intro}, we have established a background for the Oseen problem citing its main difficulties and a literature survey. In the thesis, we have predominantly discussed the use of three different finite elements methods, namely, the non-conforming Crouzeix-Raviart (${\rm CR}$) method, the $H(\Hdiv;\Omega)$ conforming Raviart-Thomas (${\rm RT}$) element method and the hybrid high order method. The thesis is divided into four chapters. Chapter \ref{chap1} analyses the edge patchwise local projection (EPLP) stabilized nonconforming finite element methods for the Oseen problem. For approximating the velocity, the lowest-order Crouzeix-Raviart (CR) nonconforming finite element space is considered, whereas for approximating the pressure, two separate discrete spaces are considered, namely, the piecewise constant polynomial space and the lowest-order CR finite element space. The proposed discrete weak formulations are a combination of the standard Galerkin method, EPLP stabilization and weakly imposed boundary condition (Nitsche's technique). We present stability results for both schemes and provide convergence analysis. {\it A~posteriori} error analysis of the edge patch-wise local projection (EPLP) stabilized Crouzeix-Raviart finite element method is developed in chapter \ref{chap2}. The {\it a~posteriori} analysis is based on the approach of Verf\"urth \cite{verfurth_dual_main}. We prove a stability result for the Oseen equation under a dual norm. The stability result gives an equivalence of error and residual which is independent of the discrete formulation. This gives the freedom of using other stabilizations and finite element spaces in the setting of our analysis. Equivalence of error and residual is exploited to formulate an error estimator which is proven to be reliable. Efficiency estimates show a dependence on the diffusion coefficient. In chapter \ref{chap3}, we define a Local projection stabilization (LPS) scheme with the Raviart-Thomas( ${\rm RT}_k$) elements for the oseen problem. We show that a divergence free, pressure robust LPS scheme can be designed with ${\rm RT}_k$ elements of order $k \geq 1$. We also show that stability under the streamline upwind Petrov-Galerkin (SUPG) norm can be achieved if the ${\rm RT}_k$ space is enriched with tangential bubbles. The enriched scheme also gives divergence free velocity. We present {\it a~priori} error estimates for both the schemes. Chapter \ref{chap4} deals with the use of a local projection stabilized Hybrid High-Order scheme for the Oseen problem. We prove an existence-uniqueness result under a SUPG like norm. We derive an optimal order {\it a~priori} error estimate under this norm for equal order polynomial discretization of velocity and pressure spaces. In the last chapter we provide some concluding remarks on the results proved in the thesis and discuss some future problems to work on.
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Book chapters on the topic "Raviart-Thomas element"

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Dubois, François, Isabelle Greff, and Charles Pierre. "Raviart Thomas Petrov–Galerkin Finite Elements." In Springer Proceedings in Mathematics & Statistics, 341–49. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-57397-7_27.

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Benkhaldoun, Fayssal, and Abdallah Bradji. "A New Error Estimate for a Primal-Dual Crank-Nicolson Mixed Finite Element Using Lowest Degree Raviart-Thomas Spaces for Parabolic Equations." In Large-Scale Scientific Computing, 489–97. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-97549-4_56.

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Hoppe, R. H. W., and B. Wohlmuth. "Hierarchical basis error estimators for Raviart–Thomas discretizations of arbitrary order." In finite element methods, 155–67. Routledge, 2017. http://dx.doi.org/10.1201/9780203756034-12.

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"Some Observations on Raviart–Thomas Mixed Finite Elements in p Extension for Parabolic Problems." In finite element methods, 235–44. CRC Press, 2016. http://dx.doi.org/10.1201/b16924-22.

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El Boukili, A., A. Madrane, and R. Vaillancourt. "Adaptive techniques for semiconductor equations with a Raviart-Thomas element." In Computational Fluid and Solid Mechanics, 1151–54. Elsevier, 2001. http://dx.doi.org/10.1016/b978-008043944-0/50864-7.

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Conference papers on the topic "Raviart-Thomas element"

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Ruas, V., Theodore E. Simos, George Psihoyios, Ch Tsitouras, and Zacharias Anastassi. "A Modified Lowest Order Raviart-Thomas Mixed Element with Enhanced Convergence." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP, 2011. http://dx.doi.org/10.1063/1.3636703.

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Barbi, G., A. Chierici, A. Cervone, V. Giovacchini, S. Manservisi, L. Sirotti, and R. Scardovelli. "A new projection method for Navier-stokes equations by using Raviart-thomas finite element." In 8th European Congress on Computational Methods in Applied Sciences and Engineering. CIMNE, 2022. http://dx.doi.org/10.23967/eccomas.2022.021.

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Bertrand, F. "A Decomposition of the Raviart-Thomas Finite Element into a Scalar and an Orientation-Preserving Part." In 14th WCCM-ECCOMAS Congress. CIMNE, 2021. http://dx.doi.org/10.23967/wccm-eccomas.2020.034.

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