Many problems of fundamental and practical importance contain multiple scale solutions. Composite materials, flow and transport in porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the wide range of length scales in the underlying physical problems. In this talk, I describe some of our recent efforts in developing multiscale computational methods to upscale two-phase flows in strongly heterogeneous porous media. Further, we introduce a new multiscale analysis for convection dominated incompressible flow with multiscale solutions. The main idea is to construct semi-analytic multiscale solutions locally in space and time, and use them to construct the coarse grid approximation to the global multiscale solution. For both problems, geometric information plays an important role in our upscaling analysis. Our multiscale analysis provides an important guideline in designing a multiscale method for computing flow in strongly heterogeneous media.
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