numerical analysis, verified numerical computation
spectral method, finite element method, semigroup theory, time-dependent PDEs, DDEs, ODEs, blow-up solution of ODEs/PDEs, dynamical systems
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, DDEs, 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, in particular, various elliptic equations have been developed and improved in the recent two decades by many researchers. In this field, the finite element method (FEM) is a powerful approach of obtaining appropriate 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, which depends on the shape of domain.
We have also developed a verified numerical method for the initial-boundary value problems of semilinear parabolic equations. This method is based on the semigroup theory (analytic semigroups and evolution operators). Furthermore, we have constructed an approach for validating the blow-up solutions of ODEs by combining dynamical systems with theories of resolution of singularity in the field of algebraic geometry. My current interest is to develop a method of verified computing for DDEs, parabolic/hyperbolic/dispersive PDEs on the basis of spectral methods.