The research on homogenization, elasitic problems and the algorithms for numerical simulations of flows of liquids and development

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(1) Algorithm for numeric scheme on liquids flows : First, a theorem on errors of between the exact boundaries and numeric construction of boundaries are proposed by colleague and examined by numerically. Second, convergent finite element scheme for multi-flows miscible among different liquids, are proposed and proved. The governed equations are the Navier-Stokes equation for the first topic and the Stokes equation for the second one.(2) The Laplace equation is considered as the corresponding scalar equation to the Navier-Stokes equation, in order to treat in a unified way under the notion of the generalized boundary value probleml for elliptic partial differential equations by using the variational calculus. The approach is verified by numerical examples.(3) Among elastic waves, the Rayleigh surface waave is researched. In particular, the propagation of the Rayleigh surface wave is interested by us, and some estimates on decay of the local energy of the Rayleigh waves are obtained in many types of regions.(4) Generally in the total space, it is well known that waves are represented as superposition of plane waves. In this research, it is showd that, in the half-space region, a concrete form of superposition to elastic waves are obtained. This expression is expected to be applied to the construction of the Fourier transformation in the half-space.