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

(1)液体流动数值格式的算法：首先，作者提出了精确边界与边界数值构造之间误差的一个定理，并用数值方法进行了检验。其次，提出并证明了求解不同液体间多流混溶问题的收敛有限元格式。(2)将Laplace方程视为对应于Navier-Stokes方程的标量方程，以便用变分的方法统一处理椭圆型偏微分方程广义边值问题。通过数值算例验证了该方法的有效性。(3)研究了弹性波中的瑞利面波。特别是，瑞利面波的传播引起了我们的兴趣，并在许多类型的区域得到了一些关于瑞利波局部能量衰减的估计。(4)在整个空间中，众所周知，波可以表示为平面波的叠加。研究表明，在半空间区域，得到了弹性波的一种具体叠加形式。该表达式有望应用于半空间中傅里叶变换的构造。

(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.