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Heterogeneous problems with high contrast, multiscale and possibly also random coefficients arise frequently in practice, e.g., reservoir simulation and material sciences. However, due to the disparity of scales, their efficient and accurate simulation is notorious challenging. First, I will describe some impor- tant applications, and review several state-of-the-art multiscale model reduction algorithms, especially the Generalized Multiscale Finite Element Method (GMsFEM). Then I will describe recent efforts on developing a mathematical theory for GMsFEM, and ongoing works on algorithmic developments and novel applications.

References

[1] Guanglian Li, On the Convergence Rates of GMsFEMs for Heterogeneous Elliptic Problems without Oversampling Techniques, submitted to Multiscale Modeling & Simulation, 2018.

[2] Shubin Fu, Eric Chung and Guanglian Li, Edge Multiscale Methods for elliptic problems with hetero- geneous coefficients, submitted to J. Comput. Phys, 2018.