Study on the numerical analysis and its applications to engineering processes for free surface or interface flows

为了发展和分析自由表面或自由界面流动的有限元格式，并考虑其在工业过程中的应用，我们在2003-2004年间开展了这项研究。2003年，我们提出了自由界面两流体流动的无通量有限元方法，该方法利用拉格朗日乘子技术建立了包含无通量约束的变分格式。然后，我们考虑了问题新的变分形式的数学有效性，定常问题的唯一性和存在性，以及Navier-Stokes方程的分数投影格式的稳定性。此外，我们还对溃坝问题和水池晃荡问题进行了大量的数值实验，验证了该方法的有效性。因此，我们的方法显示了高质量的质量守恒。2004年，我们还利用Stokes界面问题的有限元逼近的论点，讨论了由间断粘性和密度的Navier-Stokes方程建立的双流体流动的无通量有限元方法的收敛问题。事实上，我们已经提出了Strang型的误差估计，并证明了在界面光滑性和Stokes界面问题解的正则性的假设下，无通量有限元方法是收敛的。然而，对于高密度粘性比两种流体流动的新的有限元格式，我们的方法在实际应用于充型模拟中仍有许多需要考虑的地方。

This study has been carried out during 2003-2004 in order to develop and analyze the finite element scheme for free-surace or free interface flows and to consider its application to industrial processes.In 2003, we have proposed the flux-free finite element method for two-fluid flows with free interface, which is based a new variational formulation including the flux-free constraint by using Lagrange multiplier technique. Then we have considered the mathematical validity of the new variational formulation of our problems, the uniqueness and existence of the stationary problem and the stability of the fractional projection scheme for Navier-Stokes equations. Furthermore we have performed many numerical experiments for dam-break problems and sloshing tank problems to demonstrate the efficiency of our method. As a result, our method shows the high quality for the mass conservation.In 2004, we have also considered the convergence of the flux-free finite element method for two-fluid flows which is modelled by Navier-Stokes equations with discontinuous viscosity and density by using the arguments on the finite element aproximation for Stokes interface problems. In fact, we have proposed an error estimate of Strang type and have shown the convergence of the flux-free finite element method under the assumption on the smoothness of the interface and the regularity of the solution of Stokes interface problem.However, so much remains to be considered about new finite element schemes for two-fluid flows with high ratios of the density and viscosity in the real application of our method to the mold filling simulations.