The goal of this research is to merge state of the art developments in homogenization theory with multilevel discretization and solution techniques for elliptic equations. Complex multiscale problems are integral part of modeling and numerical simulations in a number of engineering, environmental and biomedical sciences. These problems have physical phenomena in hierarchical structures with multiple, poorly separated length scales. This can result in very large discretizations, which often require advanced supercomputing equipment. Today’s supercomputers, however, allow a limited number of high fidelity simulations. We propose to develop fast semi-analytic methods for multiscale simulations of elliptic systems, when the media has poor scale separation. The key idea is to incorporate analytical approximation of fine-scale local solutions into multilevel methods. These solutions, currently developed for scalar elliptic equation, have been used to approximate both the cell solution of classical homogenization as well as to compute upscaled tensor coefficients. By incorporating those in multilevel iterations on can achieve considerable computational savings compared to state of the art numerical multiscale techniques. The approximations to the fine-scale cell solution will allow to implement both efficient and accurate prolongation operators from coarse to fine levels, as well as coarsening strategies involving the analytical effective tensor coefficient. The procedure will then be extended to the elasticity operator, targeting biomedical applications, such as simulations of bone tissue, relevant to osteoporosis disease. Similar procedure can be also applied for field scale environmental problems, such as carbon sequestration. The expected transfer of knowledge in the area of multiscale methods will add to the host’s current efforts in biomedical modeling and simulations. The proposed research also coincides with long-term goals of the host institution, IPP-BAS.
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