Development of Numerical Methods for Dynamics of Interfaces and its Applications to Experiments in Science and Engineering

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We obtained several results, which are concerned with the following representative phenomena of inter-faces : "Belousov-Zhabotinsky reaction", "Viscous fingering" and "Behavior of support in an absorbingmedium".1)"Singular limit method", "Bifurcation theory", "Proper viscosity solutions" and "Level set method"are developed and enable us to analyze the mechanism of the appearance of such phenomena. For example, the appearance of a travelling wave and a helical wave, and the global existence of the solution of the model equation describing some bunching phenomena observed in the epitaxial growth of crystals.2)A difference scheme based on the singular limit method is constructed and enables us to compute the 3-dimensional Stefan problem numerically.3)The growth of finger patterns can be interpreted by taking into account the non-linearity in the rheological properties ; that is, it becomes easy to construct the mathematical model.4)A numerical method which realizes the behavior of the support of the flow through an absorbing medium is developed, and the support splitting and non-splitting phenomena are justified by the mathematical analysis.5)A numerical method with the multiple precision arithmetic for tracking the level set is constructed, and can capture the travelling wave solutions.6)A finite element method with error estimates and a domain decomposition method are constructed, and enable us to realize the dynamical behavior of the cerebrospinal fluid and the Earth's mantle convection, which are unable to be observed directly. Furthermore, a high-performance domain decomposition method is proposed for the parallel finite element analysis using meshless virtual nodes along the domain interface.