Multiscale finite element coarse spaces for the application to linear elasticity

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