Mathematical Modeling / Computational Mathematics

My research is focused on mathematical modeling in the interdisciplinary areas of materials science and crystal growth and on development of advanced computational methods for tracking material/fluid surfaces and interfaces. Primarily, I am interested in modeling formation of a micro(nano)structure arrays as a result of the morphological (shape) instabilities of a solid film surfaces, material deposition, long-range molecular interactions with a substrate, self-assembly and external control. Such ordered surface structures, as well as a single crystals grown on a substrate have numerous applications in micro(nano)electronics, optoelectronics and photonics. Another side of this activity is modeling damage due to some of the mentioned factors under favorable conditions, for instance the loss of a structural integrity of a thin film by “dewetting” from a substrate or, in the polycrystalline film, by grain-boundary grooving (void formation). A fundamental problem in these fields is to predict the growth rate, the morphology of the film surface, and the stability of the morphology as functions of process parameters such as, for instance, the temperature, the material deposition rate and the strength of intermolecular interactions. The main challenges are formulation of the mathematical model from available experimental data and general laws of physics, development of mathematical/computational methods, and generation of results that advance understanding of a natural or man-made phenomena under consideration. My work puts an equal emphasis on the three components above, and thus it has to be viewed in the spirit of classical modeling in the natural sciences. Interaction with experimentalists, the ability to accommodate ideas from such diverse areas as the physical experiment and theory, fluid and solid mechanics, heat and mass transfer, crystal growth, non-equilibrium phenomena, pattern formation, nonlinear partial and ordinary differential equations (PDEs and ODEs), numerical analysis and methods, and familiarity with literature in these areas is crucial. Mathematically, the models are typically formulated in terms of highly nonlinear PDEs which do not admit exact solution. Once the solutions have been characterized