Synthetic approach for new developments of self-validating numerics

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In this research, we newly developed the self-validating numerical methods which can be applied to wide mathematical and analytical problems as well as extended or improved the existing techniques.And we actually applied these methods to particular problems such as equations in the mathematical fluid mechanics and oscillation problems. The important research results obtained by investigators and co-investigators are as follows :1. Nakao, N.Yamamoto, Watanabe established several refinements and extensions for the numerical verification methods of solutions for elliptic problems. Namely, they succeeded the numerical computation with guaranteed error bounds for the inverse eigenvalue problems of second order elliptic operator. They also obtained some results for enclosing the solutions for elliptic variational inequlities. Moreover, they computed an optimal constant with guaranteed accuracy appearing in the a priori error estimates for the finite element projection of the Poisson problem, which is an important contribution for the numerical verification for nonlinear elliptic problems.2. Nagatou and Minamoto obtained interesting computer assisted proofs for the Kolmogorov problem and for the perturbed Gelfand equation, respectively.3. Oishi established some fast algorithms for the fundamental validated computations for the solutions of linear equations.4. Nishida et al. computed with guaranteed error bounds for the non-trivial solution of heat convection problems, which is an important result for a computer assisted proof in the fluid mechanics.5. T. Yamamoto obtained some convergence results of the finite difference scheme for the singular solutions of two point boundary value problems.