Study on the finite element method for fluid flows with moving boundary

本研究已在2000-2002年期间进行了研究，以便订购并分析与移动边界一起流动的有限元素方法，这些方法经常出现在自然和某些工业过程中。在2000年，我们考虑到在有限元素近似流动中接近界面的聚合，以实现不可能的immiscttwo-fluid流动。我们已经估计了L^p （\）norm of the difference between the positive value of the pseudo-density function及其approximation by using the Heaviside operator H（·）。根据我们的结果，在P_k有限元素的情况下，收敛率为O（h^<2k/3 p>）。这一结果是由数字实验测试的。在2001年，我们改进了先前的结果，该结果是由常规重型操作器的含义来衡量的。然后，在P_1或P_1isoP_2元素的情况下，通过L^2（Ω）-规范的近似接口的收敛速率是O（h^<1/2>）。本结果也是由一些数字实验测试的。在2003年，我们认为质量保守的有限元素方案是一个不复杂的不相容的两流流。我们已经提出了一种混合的变体公式，用于Navier-Stokes equations。我们已经为连续和接近的问题、如何、对非静态问题和数字计算的应用提供了存在和独特性，是我们未来的主题。

This study has been carried out during 2000-2002 in order to develop and analyze the finite element method for fluid flows with moving boundary, which are often appeared in nature and some industrial processes.In 2000, we have considered the convergence of the approximate interface in the finite element approximation for the incompressible immiscible two-fluid flows. We have estimated the L^p (Ω)norm of the difference between the measure of the positive value of the pseudo-density function and its approximation by using the Heaviside operator H(・). Due to our result, the convergence rate is O(h^<2k/3p>) in the case of P_k-finite element. This result has been tested by numerical experiments.In 2001, we have improved the previous result by means of the regularized Heaviside operator. Then the convergence rate of the approximate interface measured by the L^2(Ω)-norm is O(h^<1/2>) in the case of P_1 or P_1isoP_2-element. This result is also tested by some numerical experiments.In 2003, we have considered the mass conservative finite element scheme for incompressible immiscible two-fluid flows. We have proposed the mixed variational formulation with the flux functional for Navier-Stokes equations. We have proved the existence and the uniqueness for continuous and approximate problems, however, the application to the non-stationary problem and the numerical computation are our theme in the near future.