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

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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.