Director,Mathematical Research Center for Industrial Technology

092-642-2779

Ph.D

Computational Mathematics

My current research interest is the numerical verification methods for
the existence of solutions for nonlinear partial differential equations and
related problems by using the finite element or the spectral methods with
computable error estimates.

In the last decade, several kinds of numerics with result verifications have
been proposed for differential equations. Such a methodology is known as the
computer assisted proofs in analysis. Some of them worked upon the ordinary
differential equations with sufficient practical level. However, there are
not so many such works for partial differential equations up to now. In
1988, the author showed, first in the world, that the exact solution for the
elliptic boundary value problem can be numerically verified by the
effectively use of the finite element approximation and the error estimates
combining with the infinite dimentional fixed point theorem. Our method can
also be applied to obtain the finite element solution with guaranteed error
bounds, even if we have no information about the existence of exact solution
for the original problems.

In the meantime, several refinements have been established. Namely, we
extended the method to the nonconvex nonsmooth domain, nondifferentiable
problems, maximum-norm error estimates and so on. Also, we improved on the
accuracy of the error estimates by using higher order finite element.
Next, we found an a posteriori and constructive a priori error estimates for the
finite element solution of the Stokes problem, which plays an important
role in the verification of solutions for the Navier-Stokes equation.
Furthermore, the numerical enclosure method of eigenvalue problems was
formulated and some applications, including the inverse eigen value problems, have been presented, as well as for solutions of the free boundary problem described as a variatioal inequality.
Recently, we succeeded to apply our verification principle in proving numerically existence of solutions for the heat convection problems in fluid dynamics and to establish other computer assisted proofs in nonlinear analysis.　

Research Interests

• Numerical verification method of solutions for partial differential equations
  keyword : Validated numerical computation, constructive error estimates of approximate solutions for partial differential equations, computer assisted analysis
  1986.01～2012.03.

Current and Past Project

• In the fiscal year 2003 among 86 proposals 24 grants were awarded in the mathematics, physics and earth science division of the 21st Century COE (Center Of Excellence) Program by the Ministry of Education. Among them 7 were awarded to mathematics-related fields, and we are the only mathematical institution in the western half of Japan that had submitted a successful proposal. Our proposal "Development of Dynamic Mathematics with High Functionality(DMHF for short)" aims at development of new theories in mathematics based on other natural sciences, especially with the help of the computer. Such theories are expected to play crucial roles in various disciplines in science and technology. In educational side we try to reform our graduate program in such a way that our graduates can make contributions not only in the academic world but also in various sectors in the industry.

Academic Activities