Co-operative Research on Numerical Solutions in Seience and Technology

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1. When the problem is axisymmetric, we can transform original three-dimensional problems to two-dimensional ones by introducing cylidrincal coordinates. The amount of required computation is reduced drastically but a singularity appears on the axis of symmetry. We have introduced suitable finite element subdivisions and weighted function spaces adapted to the singularity and obtained convergence results and optimal error estimates for both mixed and stabilized finite element approximations to flow problems.2. We have developed a new method for getting much more precise force exerted from fluid to bodies in flows. Using a weak form we transform the surface integrals to body integrals for the computation of drag and lift forces. We have establish error analyzes and found more precise values of drag and lift coefficients of bodies of any shape.3. The vortex layr is one of main fundamental processes of the turbulence and it is important ot find out the mechanism. We have studied numerically the behavior of two-dimensional vortex leyers in shear flows. We have observed that the phenomena differ much depending on the signs of vortices of vortex layrs and shear layrs.4. Leak boundary conditions often appear in phenomena of engineering and environmental problems. We have formulated and analyzed the boundary condition. We have also developed a unmerical method.5. We have combined higher-order finite elements and a residual iteration method in order to estimate errors of numerical solutions of nonlinear elliptic problems in the maximum norm. We have improved drastically the verification accuracy of numerical solutions.6. We have proposed collocation type two-step Runge-Kutta methods. They preserve the orders and stability and are easily implemented on parallel computers. We have constructed higher-order two-step Runge-Kutta methods by a suitable setting of collocation points, implemented it in the prediction and correction type, and realized a high-performance computation.