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 via the 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.
Keywords: Multiscale, mixed finite elements, linear elasticity, hybridization
@article{M2AN_2021__55_3_1005_0,
author = {Devloo, Philippe R. B. and Farias, Agnaldo M. and Gomes, S\^onia M. and Pereira, Weslley and dos Santos, Antonio J. B. and Valentin, Fr\'ed\'eric},
title = {New $H ( \mathrm{div} )$-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1005--1037},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {3},
doi = {10.1051/m2an/2021013},
mrnumber = {4253171},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021013/}
}
TY - JOUR
AU - Devloo, Philippe R. B.
AU - Farias, Agnaldo M.
AU - Gomes, Sônia M.
AU - Pereira, Weslley
AU - dos Santos, Antonio J. B.
AU - Valentin, Frédéric
TI - New $H ( \mathrm{div} )$-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes
JO - ESAIM: Mathematical Modelling and Numerical Analysis
PY - 2021
SP - 1005
EP - 1037
VL - 55
IS - 3
PB - EDP-Sciences
UR - https://www.numdam.org/articles/10.1051/m2an/2021013/
DO - 10.1051/m2an/2021013
LA - en
ID - M2AN_2021__55_3_1005_0
ER -
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%A Devloo, Philippe R. B.
%A Farias, Agnaldo M.
%A Gomes, Sônia M.
%A Pereira, Weslley
%A dos Santos, Antonio J. B.
%A Valentin, Frédéric
%T New $H ( \mathrm{div} )$-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes
%J ESAIM: Mathematical Modelling and Numerical Analysis
%D 2021
%P 1005-1037
%V 55
%N 3
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2021013/
%R 10.1051/m2an/2021013
%G en
%F M2AN_2021__55_3_1005_0
Devloo, Philippe R. B.; Farias, Agnaldo M.; Gomes, Sônia M.; Pereira, Weslley; dos Santos, Antonio J. B.; Valentin, Frédéric. New $H ( \mathrm{div} )$-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 3, pp. 1005-1037. doi: 10.1051/m2an/2021013
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