Multiscale finite element coarse spaces for the application to linear elasticity

• DOI: 10.2478/s11533-012-0166-8
• Publication Date: 2013-01-28T04:49:51Z
• Authors: Marco Buck; Oleg Iliev; Heiko Andrä
• Keywords: Linear elasticity; Elasticity; Homogenization

Abstract We extend the multiscale finite element method (MsFEM) as formulated by Hou and Wu in [Hou T.Y., X.-H., A for elliptic problems composite materials porous media, J. Comput. Phys., 1997, 134(1), 169–189] to PDE system of linear elasticity. The application, motivated analysis highly heterogeneous materials, is twofold. Resolving heterogeneities on finest scale, we utilize MsFEM basis construction robust coarse spaces context two-level overlapping domain decomposition preconditioners. motivate explain show that constructed space contains all rigid body modes. Under assumption material jumps are isolated, they occur only interior grid elements, our numerical experiments uniform convergence rates independent contrast Young’s modulus within material. Elsewise, if no restrictions position high coefficient inclusions imposed, robustness cannot be guaranteed any more. These results justify expectations obtain coefficient-explicit condition number bounds elasticity similar existing ones scalar PDEs given work Graham, Lechner Scheichl [Graham I.G., P.O., R., Domain PDEs, Numer. Math., 2007, 106(4), 589–626]. Furthermore, numerically observe properties an upscaling framework. Therefore, present experimental showing approximation errors w.r.t. fine-scale solution.