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175 results on '"Valentin, Frédéric"'

1. A Three-Field Multiscale Method

Author

de Barros, Franklin, Madureira, Alexandre L., and Valentin, Frédéric

Subjects
Mathematics - Numerical Analysis, 65N30 (Primary)
Abstract

"A Three-Field Domain Decomposition Method" is the title of a seminal paper by F. Brezzi and L. D. Marini which introduces a three-field formulation for elliptic partial differential equations. Based on that, we propose the Multiscale-Hybrid-Hybrid Method (MH$^2$M) for the Darcy model, a multiscale finite element method that yields, after a series of formal manipulations, a symmetric positive definite formulation that depends only on the trace of the solution. We show stability and convergence results for a family of finite element spaces and establish relationships with other multiscale finite element methods.

Published
2024

2. A low-order locking-free multiscale finite element method for isotropic elasticity

Author

Gomes, Antônio Tadeu Azevedo, Pereira, Weslley da Silva, and Valentin, Frédéric

Subjects
Mathematics - Numerical Analysis, 65N30, 65N12, 65N22
Abstract

The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a family of low-order finite elements for the linear elasticity problem which are free from Poisson locking. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient refinement levels on the fine-scale mesh such that the MHM method is well-posed, optimally convergent under local regularity conditions, and locking-free. Two-dimensional numerical tests assess theoretical results., Comment: 22 pages, 3 figures, 6 tables

Published
2024

3. Physics-Aware Neural Networks for Boundary Layer Linear Problems

Author

Gomes, Antonio Tadeu Azevedo, da Silva, Larissa Miguez, and Valentin, Frederic

Subjects
Mathematics - Numerical Analysis, Computer Science - Machine Learning
Abstract

Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most pieces of work in the area of PINNs tackle non-linear PDEs. Nevertheless, many interesting problems involving linear PDEs may benefit from PINNs; these include parametric studies, multi-query problems, and parabolic (transient) PDEs. The purpose of this paper is to explore PINNs for linear PDEs whose solutions may present one or more boundary layers. More specifically, we analyze the steady-state reaction-advection-diffusion equation in regimes in which the diffusive coefficient is small in comparison with the reactive or advective coefficients. We show that adding information about these coefficients as predictor variables in a PINN results in better prediction models than in a PINN that only uses spatial information as predictor variables. This finding may be instrumental in multiscale problems where the coefficients of the PDEs present high variability in small spatiotemporal regions of the domain, and therefore PINNs may be employed together with domain decomposition techniques to efficiently approximate the PDEs locally at each partition of the spatiotemporal domain, without resorting to different learned PINN models at each of these partitions., Comment: 10 pages, 10 figures

Published
2022

7. On the Implementation of a Scalable Simulator for Multiscale Hybrid-Mixed Methods

Author

Gomes, Antonio Tadeu A., Pereira, Weslley S., Valentin, Frederic, and Paredes, Diego

Subjects
Computer Science - Distributed, Parallel, and Cluster Computing, Mathematics - Numerical Analysis, 65N30, D.1.3, F.2.1
Abstract

The family of Multiscale Hybrid-Mixed (MHM) finite element methods has received considerable attention from the mathematics and engineering community in the last few years. The MHM methods allow solving highly heterogeneous problems on coarse meshes while providing solutions with high-order precision. It embeds independent local problems which are responsible for upscaling unresolved scales into the numerical solution. These local contributions are brought together through a global problem defined on the skeleton of the coarse partition. Since the local problems are completely independent, they can be easily computed in parallel. In this paper, we present two simulator prototypes specifically crafted for the MHM methods, which adopt two different implementation strategies: (i) a multi-programming language approach, each language tackling different simulation issues; and (ii) a classical, single-programming language approach. Specifically, we use C++ for numerical computation of the global and local problems in a modular way; for process distribution in the simulator, we adopt the Erlang concurrent language in the first approach, and the MPI standard in the second approach. The aim of exploring these different approaches is twofold: (i) allow for the deployment of the simulator both in high-performance computing (with MPI) and in cloud computing environments (with Erlang); and (ii) pave the way for further exploration of quality attributes related to software productivity and fault-tolerance, which are key to Exascale systems. We present a performance evaluation of the two simulator prototypes taking into account their efficiency.

Published
2017

12. Foundations of the MHM Method

Author

Harder, Christopher, Valentin, Frédéric, Barth, Timothy J, Series editor, Griebel, Michael, Series editor, Keyes, David E., Series editor, Nieminen, Risto M., Series editor, Roose, Dirk, Series editor, Schlick, Tamar, Series editor, Barrenechea, Gabriel R., editor, Brezzi, Franco, editor, Cangiani, Andrea, editor, and Georgoulis, Emmanuil H., editor

Published
2016
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14. Mutations in Three Genes Encoding Proteins Involved in Hair Shaft Formation Cause Uncombable Hair Syndrome

Author

Ü. Basmanav, F. Buket, Cau, Laura, Tafazzoli, Aylar, Méchin, Marie-Claire, Wolf, Sabrina, Romano, Maria Teresa, Valentin, Frederic, Wiegmann, Henning, Huchenq, Anne, Kandil, Rima, Garcia Bartels, Natalie, Kilic, Arzu, George, Susannah, Ralser, Damian J., Bergner, Stefan, Ferguson, David J.P., Oprisoreanu, Ana-Maria, Wehner, Maria, Thiele, Holger, Altmüller, Janine, Nürnberg, Peter, Swan, Daniel, Houniet, Darren, Büchner, Aline, Weibel, Lisa, Wagner, Nicola, Grimalt, Ramon, Bygum, Anette, Serre, Guy, Blume-Peytavi, Ulrike, Sprecher, Eli, Schoch, Susanne, Oji, Vinzenz, Hamm, Henning, Farrant, Paul, Simon, Michel, and Betz, Regina C.

Published
2016
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15. Flux approximation on unfitted meshes and application to multiscale hybrid-mixed methods

Author

Chaumont-Frelet, Théophile, Paredes, Diego, Valentin, Frédéric, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Universidad de Concepción - University of Concepcion [Chile], and Laboratorio Nacional de Computação Cientifica [Rio de Janeiro] (LNCC / MCT)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The flux variable determines the approximation quality of hybridizationbased numerical methods. This work proves that approximating flux variables in discontinuous polynomial spaces from the L2 orthogonal projection is super-convergent on meshes that are not aligned with jumping coefficient interfaces. The results assume only the local regularity of exact solutions in physical partitions. Based on the proposed flux approximation, we demonstrate that the mixed hybrid multiscale (MHM) finite element method is superconvergent in unfitted meshes, supporting the numerics presented in MHM seminal works.

Published
2022

16. The MHM Method for Linear Elasticity on Polytopal Meshes.

Author

Gomes, Antônio Tadeu A, Pereira, Weslley S, and Valentin, Frédéric

Subjects
BOUNDARY value problems, ELASTICITY, NEUMANN problem, DEGREES of freedom, LINEAR equations
Abstract

The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed and optimally convergent under local regularity conditions. Also, a multi-level error analysis demonstrates that the MHM method achieves convergence without refining the coarse partition. The upshot is that the Poincaré and Korn's inequalities do not degenerate, and then the convergence arises on general meshes. Two- and three-dimensional numerical tests assess theoretical results and verify the robustness of the method on a multi-layer media case. [ABSTRACT FROM AUTHOR]

Published
2023
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24. Bridging the Multiscale Hybrid-Mixed and Multiscale Hybrid High-Order methods

Author

Chaumont-Frelet, Théophile, Ern, Alexandre, Lemaire, Simon, Valentin, Frédéric, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Laboratorio Nacional de Computação Cientifica [Rio de Janeiro] (LNCC / MCT), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Laboratoire Jean Alexandre Dieudonné (LJAD), and Université de Lille-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis (math.NA), 010103 numerical & computational mathematics, 01 natural sciences, Highly heterogeneous diffusion, 010101 applied mathematics, High-order methods, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], General polytopal meshes, Mathematics - Numerical Analysis, 0101 mathematics, Multiscale methods, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, we prove that the equivalence holds for general polytopal (coarse) meshes and arbitrary approximation orders. We also leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods.

Published
2022
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25. PINNs for the time-domain Maxwell equations - Preliminary results

Author

Bennini, Amine, Lanteri, Séphane, Valentin, Frédéric, Gomes, Tadeu, Miguez da Silva, Larissa, Lanteri, Stéphane, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and Laboratorio Nacional de Computação Cientifica [Rio de Janeiro] (LNCC / MCT)

Subjects
[SPI]Engineering Sciences [physics], [SPI] Engineering Sciences [physics], [MATH] Mathematics [math], [MATH]Mathematics [math]
Abstract

International audience; The Physics-Informed Neural Network (PINN) corresponds to a machinelearning strategy to approximate the solution of partialdifferential equations by in- cluding the residual PDE in the lossfunction. In a previous work, we found that adding physicalcoefficients as predictor variables in a PINN for boundary layerlinear problems improves the accuracy of the approximation when com-paring it with the approximate solution obtained from PINNs that useonly spatio-temporal inputs. This work explores this same strategyfor the time-dependent Maxwell linear equations in case electric andmagnetic fields present highly oscillatory behavior. Extensivenumerical experiments assess this strategy by simulating electromagneticfields in heterogeneous media with high permittivity variabilityin small spatiotemporal regions of the domain

Published
2022

37. New H(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

Author

Devloo, Philippe, Farias, Agnaldo, Gomes, Sonia, Pereira, Weslley, Dos Santos, Antonio, Valentin, Frédéric, Universidade Estadual de Campinas (UNICAMP), Instituto Federal do Norte de Minas Gerais (IFNMG), Instituto de Matemática, Estatística e Computação Científica [Brésil] (IMECC), UFJF - Universidade Federal de Juiz de Fora [Minas Gerais, Brazil], Universidade Federal da Paraiba (UFPB), and Laboratorio Nacional de Computação Cientifica [Rio de Janeiro] (LNCC / MCT)

Subjects
Multiscale, Mixed finite elements, Linear elasticity, Hybridization, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation variables using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing rigid-body modes element wisely. In the second scale level, the methods are made effective by solving completely independent local Newmann elasticity problems written in a mixed form with weak symmetry enforced via a rotation multiplier. Since the finite-dimensional space for the traction variable constrains the discrete space for the local stress tensor, the discrete stress field lies in the H(div) space globally and stays in local equilibrium with external forces. We propose different choices to approximate local problems based on locally stable pairs of finite elements defined on affine second-level meshes. Those choices generate the family of multiscale finite element methods for which stability and convergence are proved in a unified framework. Notably, we prove that the methods are optimal and high-order convergent in the natural norms. Also, it emerges that the approximate displacement and stress divergence are super-convergent in the L2-norm. Numerical verification assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.; On propose une famille de méthodes mixtes hybrides multi-échelles pour le problème d'élasticité linéaire bidimensionnelle sur des maillages polygonaux généraux. Les nouvelles méthodes approchent le déplacement, la contrainte et la rotationà l'aide de discrétisationsà deuxéchelles. Le premier niveau consisteà approcher la traction (multiplicateur de Lagrange) dans des espaces polynomiaux discontinus, età calculer judicieusement les mouvements de corps rigides. Dans le deuxième niveau d'échelle, les méthodes sont rendues efficaces en résolvant des problèmes d'élasticité locaux du type Neumann complètement indépendantsécrits sous une forme mixte avec une symétrie faible, imposée via un deuxième multiplicateur de Lagrange. Étant donné que l'espace de dimension finie pour la traction contraint l'espace discret pour le tenseur de contrainte local, le champ de contraintes discrètes se situe dans l'espace H(div) global et reste enéquilibre local avec des forces externes. Nous proposons différents choix pour approcher les problèmes locaux basés sur des paires d'éléments finis localement stables définies sur des mailles affines de second niveau. Ces choix génèrent la famille de méthodes d'éléments finis multi-échelles pour lesquelles la stabilité et la convergence sont prouvées dans un cadre unifié. Notamment, nous prouvons que les méthodes sont optimales et convergentes dans les normes naturelles. De plus, on démontre que le déplacement et la divergence des contraintes discrètes sont super-convergents dans la norme L2. Des essais numériques valident les résultats théoriques et mettent enévidence la bonne précision des nouvelles méthodes sur des mailles grossières pour des problèmes de matériaux hétérogènes multicouches.

Published
2020

45. The Multiscale Hybrid Mixed Method in General Polygonal Meshes

Author

Barrenechea, Gabriel, Jaillet, Fabrice, Paredes, Diego, Valentin, Frédéric, University of Strathclyde [Glasgow], Simulation, Analyse et Animation pour la Réalité Augmentée (SAARA), Laboratoire d'InfoRmatique en Image et Systèmes d'information (LIRIS), Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École Centrale de Lyon (ECL), Université de Lyon-Université Lumière - Lyon 2 (UL2)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Université Lumière - Lyon 2 (UL2), Departamento de Ingenieria Matematica, Pontificia Universidad Católica de Valparaíso (PUCV), Department of Mathematical and Computational Methods, Laboratorio Nacional de Computação Cientifica [Rio de Janeiro] (LNCC / MCT), and LNCC - Laboratorio Nacional de Computacao Cientifica

Subjects
[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

This work extends the general form of the Multiscale Hybrid-Mixed (MHM) method for the second-order Laplace (Darcy) equation to general non-conforming polygonal meshes. The main properties of the MHM method, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometry of the elements used for the first level mesh. More precisely, it is proven that piecewise polynomials of degree k and k + 1, k ≥ 0, for the Lagrange multipliers (flux), along with continuous piecewise polynomial interpolations of degree k + 1 posed on second-level sub-meshes are stable if the latter is fine enough with respect to the mesh for the Lagrange multiplier. We provide an explicit sufficient condition for this restriction. Also, we prove that the error converges with order k + 1 and k + 2 in the broken H 1 and L 2 norms, respectively, under usual regularity assumptions, and that such estimates also hold for non-convex; or even non-simply connected elements. Numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides.

Published
2019

46. The multiscale hybrid-mixed finite element method in polygonal meshes

Author

Barrenechea, Gabriel Raúl, Jaillet, Fabrice, Paredes, Diego, Valentin, Frédéric, Jaillet, Fabrice, Department of Mathematics and Statistics [Univ Strathclyde], University of Strathclyde [Glasgow], Simulation, Analyse et Animation pour la Réalité Augmentée (SAARA), Laboratoire d'InfoRmatique en Image et Systèmes d'information (LIRIS), Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École Centrale de Lyon (ECL), Université de Lyon-Université Lumière - Lyon 2 (UL2)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Université Lumière - Lyon 2 (UL2), Instituto de Matematica [Valparaiso] (IMA - PUCV ), Pontificia Universidad Católica de Valparaíso (PUCV), Department of Computational and Applied Mathematics, and Laboratorio Nacional de Computação Cientifica [Rio de Janeiro] (LNCC / MCT)

Subjects
Multiscale finite element method, MSC (2010) 65N15, optimal convergence, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [INFO.INFO-MO] Computer Science [cs]/Modeling and Simulation, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, hybrid formulation, polygonal mesh, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In this talk the recent extension of the Multiscale Hybrid-Mixed (MHM) method, originally proposed in [1], to the case of general polygonal meshes (that can be non-convex and non-conforming as well) will be presented. We present new stable multiscale finite elements such that they preserve the well-posedness, super-convergence and local conservation properties of the original MHM method under mild regularity conditions on the polygons. More precisely, we show that piecewise polynomial of degree k−1 and k, k≥1, for the Lagrange multipliers (flux) along with continuous piecewise polynomial interpolations of degree k posed on second-level sub-meshes are stable if the latter is refined enough. Such one- and two-level discretization impact the error in a way that the discrete primal (pressure) and dual (velocity) variables achieve super-convergence in the natural norms under extra local regularity only. Numerical tests illustrate theoretical results and the flexibility of the approach.

Published
2019

47. New H(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

Author

Devloo, Philippe R. B., Farias, Agnaldo M., Gomes, Sônia M., Pereira, Weslley, dos Santos, Antonio J. B., Valentin, Frédéric, Devloo, Philippe R. B., Farias, Agnaldo M., Gomes, Sônia M., Pereira, Weslley, dos Santos, Antonio J. B., and Valentin, Frédéric

Abstract

This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing elementwise rigid body modes. In the second level, the methods are made effective by solving completely independent local boundary Neumann elasticity problems written in a mixed form with weak symmetry enforced viathe rotation multiplier. Since the finite-dimensional space for the traction variable constraints the local stress approximations, the discrete stress field lies in the H(div) space globally and stays in local equilibrium with external forces. We propose different choices to approximate local problems based on pairs of finite element spaces defined on affine second-level meshes. Those choices generate the family of multiscale finite element methods for which stability and convergence are proved in a unified framework. Notably, we prove that the methods are optimal and high-order convergent in the natural norms. Also, it emerges that the approximate displacement and stress divergence are super-convergent in the L2-norm. Numerical verifications assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.

Published
2021
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