Multipoint stress mixed finite element methods for the linear Cosserat equations
Wietse M. Boon, Alessio Fumagalli, Jan M. Nordbotten, Ivan Yotov
TL;DR
The paper addresses efficient discretization of linear Cosserat materials by introducing four MS-MFE schemes that localize the Cauchy and couple stresses through low-order quadrature, producing a reduced Schur complement system in displacement $u$ and rotation $r$. Each scheme uses elasticity-stable finite element triplets to guarantee $\ell$-robust stability and convergence, with both reduced (Schur-complement) and full formulations analyzed. The authors provide rigorous a priori error estimates and demonstrate linear or quadratic convergence depending on regularity and the Cosserat parameter $\ell$, supported by numerical experiments in 2D and 3D that confirm the theory and show significant reductions in degrees of freedom compared to full MFE. The methods extend MSMFE ideas from elasticity to Cosserat systems with weak stress symmetry, offering computationally efficient, robust tools for micropolar media and related applications.
Abstract
We propose mixed finite element methods for Cosserat materials that use suitable quadrature rules to eliminate the Cauchy and coupled stress variables locally. The reduced system consists of only the displacement and rotation variables. Four variants are proposed for which we show stability and convergence using a priori estimates. Numerical experiments verify the theoretical findings and higher order convergence is observed in some variables.