## Research field

numerical analysis, verified numerical computation

## Key words

finite element method, spectral method, semilinear elliptic/parabolic PDE, reaction-diffusion equation, advection equation, blow-up solution of ODEs/PDEs

## Outline

*Verified numerical computation* is a numerical method that takes all errors of numerical computations into account, which yields mathematically rigorous result. My research interest is *verified computations* of solutions to differential equations (ODEs and PDEs). Verified computations reveal all errors that occur in an approximation of solutions to differential equations. It follows that the sufficient condition of a fixed-point theorem can be validated by verified numearical computations. Then, by using verified computations, we can prove existence and local uniqueness of a solution to differential equations in the neighborhood of an approximate solution.

A number of verified numerical methods for various elliptic equations have been developed and improved in the recent two decades by many researchers. The finite element method (FEM) is a powerful approach of obtaining approximate solutions to PDEs. We have developed a verified numerical method based on the FEM for semilinear elliptic equations defined on arbitrary polygonal domains. Our method can easily deal with a variety of problems.

We have also developed a verified numerical method for the initial-boudary value problems of semilinear parabolic equations. This method is based on the semigroup theory (analytic semigroups and evolution operators). My current interest is to validate the blow-up solutions of ODEs by combining verified numerical coputations with theories of resolution of singularity in the field of algebraic geometry. I am also interested in developing a method of verified computing for hyperbolic equations on the basis of sparctral methods.

## Recent research topics

- Verified computations of solutions for elliptic boundary value problems on arbitrary polygonal domains
- Computable a priori error estimates for the FEM
- Computer-assisted analysis for bifurcation structure of elliptic PDEs
- Applications to equilibrium of Reaction-diffusion systems
- Verified numerical methods for parabolic equations
- Numerical verification of blow-up solutions for ODEs
- Verified numerical mehtods for hyperbolic equations