On Hierarchical Convergence of the Heterogeneous Multiscale Finite Element Method Using Polyhedral Supports

Numerical simulation of physical processes in such complex formations as permafrost is impossible without modern multiscale methods. For example, the heterogeneous multiscale finite element method (FE-HMM) can be used to simulate elastic deformation of solids. However, it is necessary to have some control tools of computational errors as well as a priori information on limitations of the method in solving practical problems. In this article, we investigate the influence of different mesh hierarchy levels on the accuracy and speed of solving the elastic deformation problem using FE-HMM with polyhedral supports at the macrolevel and with tetrahedral supports at the microlevel. The obtained estimates allow us to adjust the planning strategies of computational experiments, which are largely related to the construction of a set of meshes with different accuracy. We apply the h-refinement technology at the edges of macro-polyhedra in the microlevel mesh. Increasing in accuracy of computational solution up to two orders of magnitude with maintaining the total size of discretizations is obtained. Also, the applicability of the method for numerical simulation of physical processes in media with elongated inhomogeneities intersecting several macroelements is shown.