Many natural and man-made materials (e.g. porous media and biological tissues on the one hand, and composites used in technology on the other) exhibit vast spatial variability in most of their properties, such as hydraulic and electrical conductivities and dielectric permeability. These are called high contrast media. Because these materials are encountered in a wide range of application areas, such as subsurface flows, bioengineering, suspensions, and medical and geophysical imaging, they have received increased attention within the mathematical community in recent years. The investigator studies high contrast composite materials that exhibit an additional feature, which is of geometric nature, namely, the heterogeneous domain either has highly concentrated inhomogeneities or contains multiple scales (that is, its inhomogeneities are very small compared to some length characteristic of the material). Graduate and undergraduate students are included in the work of the project.

Mathematically, high contrast media are described by partial differential equations with coefficients that take on an extremely large range of values in the domain. The investigator studies problems of this type that also have rapidly varying coefficients, meaning that they fluctuate on a length scale that is much smaller than the size of the domain occupied by the composite. These problems require special efforts in establishing strategies for designing multiscale models. Recent advances in the field have led to the development of many approaches that were designed mostly for problems with bounded oscillations in problem parameters. However, problems whose coefficients have high aspect ratio pose additional challenges for developing both analytical and numerical solution methods. One therefore seeks techniques that are robust with respect to problem parameters. The investigator develops efficient analytical and numerical tools for describing strongly heterogeneous flows in media with large variations in material properties and complex geometry.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Victor Roytburd
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University of Houston
United States
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