Nobito Yamamoto

A Numerical Verification Method for Solutions of Boundary Value Problems with Local Uniqueness by Banach's Fixed-Point Theorem

In this paper, we propose a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computers. Choosing a set which possibly includes a solution, we transform it by an approximate linearization of the operator appearing in the equation. Then we calculate the radii of the transformed set in order to check sufficient conditions for Banachs fixed-point theorem. This method is applied to elliptic problems and numerical examples are given.

Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element

The verifications of solutions to weakly nonlinear elliptic equations by the method described e.g. by Nakao (1988, 1989), etc. are sometimes hardly accomplished when the right-hand sides of the equations are very large. To overcome such difficulties, a residual iteration technique with approximate solution was introduced by Nakao (1993). In the present paper, we propose an a posteriori method for the residual iteration, and show that a remarkable improvement in efficiency and in accuracy of the verification can be obtained when we use a higher order finite element.

Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains

SummaryIn this paper, methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied. In order to verify solutions using computer, it is necessary to determine some constants which appear in a priori error estimations. We propose some methods for determination of these constants. In numerical examples, calculating these constants for anL-shaped domain, we verify the solution of a nonlinear elliptic equation.

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 Guaranteed Bound of the Optimal Constant in the Error Estimates for Linear Triangular Element

We consider a numerical method to get a guaranteed bound of the optimal constant in the error estimates of a finite element method with linear triangular elements in the plane. The problem is reduced to a kind of smallest eigenvalue problem for an elliptic operator in a certain function space on the reference triangle. In order to solve the problem, we formulate a numerical verification procedure based on finite element approximations and constuctive error estimates. Consequently, we obtain a sufficiently sharp bound of the desired constant by a computer assisted proof. In this paper, we provide the basic idea and outline the concept of verification procedures as well as show the final numerical result. The detailed description of procedures for actual computations will be presented in the forthcoming paper [11].

A Numerical Verification Method of Solutions for the Navier-Stokes Equations

A numerical verification method of the solution for the stationary Navier-Stokes equations is described. This method is based on the infinite dimensional fixed point theorem using the Newton-like operator. We present a verification algorithm which generates automatically on a computer a set including the exact solution. Some numerical examples are also discussed.

A posteriori and constructive a priori error bounds for finite element solutions of the Stokes equations

We describe a method to estimate the guaranteed error bounds of the finite element solutions for the Stokes problem in mathematically rigorous sense. We show that an a posteriori error can be computed by using the numerical estimates of a constant related to the so-called inf-sup condition for the continuous problem. Also a method to derive the constructive a priori error bounds are considered. Some numerical examples which confirm us the expected rate of convergence are presented.

A simple method for error bounds of eigenvalues of symmetric matrices

Abstract We propose a simple method for validated computation of eigenvalues of symmetric matrices. The method is based on LDL T decomposition and its error estimation. The indices of eigenvalues with respect to magnitude can also be obtained by this method.

Numerical verifications of solutions for elliptic equations with strong nonlinearity

Numerical methods for automatic proof of the existence and the local uniqueness of weak solutions of elliptic boundary value problems with strongly nonlinear terms are proposed. They are based on the infinite dimensional fixed point theorems and the explicit error estimates for finite element approximations. We present detailed verification procedures and numerical examples for the typical model problem: -▵u = e u.