Kaori Nagatou

An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness

We propose a numerical method to verify the existence and local uniqueness of solutions to nonlinear elliptic equations. We numerically construct a set containing solutions which satisfies the hypothesis of Banachs fixed point theorem in a certain Sobolev space. By using the finite element approximation and constructive error estimates, we calculate the eigenvalue bound with smallest absolute value to evaluate the norm of the inverse of the linearized operator. Utilizing this bound we derive a verification condition of the Newton-Kaiitorovich type. Numerical examples are presented.

Numerical Verifications for Eigenvalues of Second-Order Elliptic Operators

In this paper, we consider a numerical technique to verify the exact eigenvalues and eigenfunctions of second-order elliptic operators in some neighborhood of their approximations. This technique is based on Nakao’s method [9] using the Newton-like operator and the error estimates for the C∘ finite element solution. We construct, in computer, a set containing solutions which satisfies the hypothesis of Schauder’s fixed point theorem for compact map on a certain Sobolev space. Moreover, we propose a method to verify the eigenvalue which has the smallest absolute value. A numerical example is presented.

A numerical method to verify the elliptic eigenvalue problems including a uniqueness property

Abstract.We propose a numerical method to enclose the eigenvalues and eigenfunctions of second-order elliptic operators with local uniqueness. We numerically construct a set containing eigenpairs which satisfies the hypothesis of Banachs fixed point theorem in a certain Sobolev space by using a finite element approximation and constructive error estimates. We then prove the local uniqueness separately of eigenvalues and eigenfunctions. This local uniqueness assures the simplicity of the eigenvalue. Numerical examples are presented.

Numerical Verification Method for Infinite Dimensional Eigenvalue Problems

We consider an eigenvalue problem for differential operators, and show how guaranteed bounds for eigenvalues (together with eigenvectors) are obtained and how non-existence of eigenvalues in a concrete region can be assured. Some examples for several types of operators will be presented.

A Numerical Verification Method for a System of FitzHugh-Nagumo Type

We propose a numerical method to enclose a solution of the FitzHugh-Nagumo equation with Neumann boundary conditions. We construct, on a computer, a set which satisfies the hypothesis of Schauders fixed point theorem for a compact map in a certain Sobolev space, which, therefore contains a solution. Several verified results are presented.

Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators

Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at . It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinsteins bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account.

Validated Computations for Fundamental Solutions of Linear Ordinary Differential Equations

We present a method to enclose fundamental solutions of linear ordinary differential equations, especially for a one dimensional Schrodinger equation which has a periodic potential. Our method is based on Floquet theory and Nakao’s verification method for nonlinear equations. We show how to enclose fundamental solutions together with characteristic exponents and give a numerical example.

Validated computation for infinite dimensional eigenvalue problems

In this paper we will show how guaranteed bounds for eigenvalues (together with eigenvectors) are obtained and how non-existence of eigenvalues in a concrete region could be assured. Some examples for several types of operators in bounded and unbounded domains will be presented. We will furthermore discuss possible future applications to eigenvalue enclosing/excluding of Schrodinger operator, hopefully in its spectral gaps.

Spectral Problem on 3‐D Photonic Crystals

In this research we will study some fundamental mathematical properties of photonic crystals which have many applications in optics and nanotechnology. A good model for these crystals is described by the Maxwell’s equations with periodic electric permittivity. The spectral problem for Maxwell’s equations possesses bands of essential spectrum which may be separated by gaps. Waves of a frequency in such a gap cannot propagate in the crystal, while other frequencies can. This makes photonic crystals very interesting for applications in nanotechnology and other areas. It is important not just to have quantitative numerical evidence for the location of these gaps, but to be able to prove the existence of spectral gaps with mathematical certainty. We propose to use computer assisted methods to achieve this.