The Frontier of Numerical Analysis for Dynamics of Interfaces and Developments in Sciences and Engineering

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We were concerned with the following numerical methods to the phenomena appearing in the repre-sentative dynamical interfaces :i) Pattern dynamics in the reaction-diffusion system,ii) Viscous fingering phenomena in Hele-Shaw Cell,iii) Dynamical behavior of the region occupied by the water in the process of evaporation.We obtained several results :1) The TCD (Threshold Competition Dynamics) method is developed for the numerical computation in reaction-diffusion system, and enables us to realize the dynamical behavior of free boundary in R^n (n=1, 2, 3.). The idea of this method is based on the theory of "Singular limit method".2) The mathematical model for the crystal growth is considered in the, form of the reaction-diffusion equation with the effect of a convection, and gives us interesting mathematical results.3) In viscous fingering phenomena, the buoyancy-driven path instabilities of bubble rising in Hele-Shaw Cell are examined. As an interesting phenomenon there appears a wake which is similar to a comet. However, such a wake is not realized in numerical method yet.4) Multi-scale FEM based on crystallographic homogenization method is developed to predict the dynamics of interfaces in the formability of sheet metal.5) The repeated support splitting and connecting property in the process of evaporation is investigated, where the the support means the region occupied by the water. The numerical methods for this process are established and the shape of the initial distribution for which such a property appear is explicitly obtained.