New<i>H</i>(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new approximate displacement, stress, and rotation using two-scale discretizations. first scale level setting consists approximating traction variable (Lagrange multiplier) in discontinuous polynomial spaces, computing elementwise rigid body modes. In second level, are made effective by solving completely independent local boundary Neumann problems written mixed form with weak symmetry enforced via multiplier. Since finite-dimensional space constraints stress approximations, discrete field lies H ( div ) globally stays equilibrium external forces. We propose different choices to based pairs finite element spaces defined affine second-level Those generate which stability convergence proved unified framework. Notably, we prove that optimal high-order convergent natural norms. Also, it emerges displacement divergence super-convergent L 2 -norm. Numerical verifications assess theoretical results highlight high precision coarse meshes multilayered heterogeneous material problems.